QUANTUM WORLDS

Quantum theory underpins much of modern physics, and its implications draw theattention of industry, academia, and public funding agencies. However there aremany unsettled conceptual and philosophical problems in the interpretation ofquantum mechanics, which are a matter of extensive debate. These hotly debatedtopics include the meaning of the wave function, the nature of the quantum objects,the role of the observer, the nonlocality of the quantum world, and the emergenceof classicality from the quantum domain. Containing chapters written by eminentresearchers from the fields of physics and philosophy, this book provides interdis-ciplinary, comprehensive, and up-to-date perspectives of the problems related tothe interpretation of quantum theory. It is ideal for academic researchers in physicsand philosophy who are working on the ontology of quantum mechanics.

olimpia lombardi is a principal researcher at the National Scientific andTechnical Research Council (In Spanish: Consejo Nacional de InvestigacionesCientíficas y Técnicas [CONICET]). She is the director of the research group inthe philosophy of physics and philosophy of chemistry at the University of BuenosAires, and has been awarded grants from the Foundational Questions Institute andJohn Templeton Foundation.

sebastian fortin and federico holik are research fellows at the NationalScientific and Technical Research Council (CONICET). cristian lόpez is aPhD student at the University of Buenos Aires and the University of Lausanne. Allare members of the group headed by Olimpia Lombardi and collaborated inorganizing the international workshop Identity, indistinguishability and non-localityin quantum physics (Buenos Aires, 2017) on which this volume is based.

QUANTUM WORLDS

Perspectives on the Ontology of Quantum Mechanics

Edited by

OLIMPIA LOMBARDIUniversity of Buenos Aires

SEBASTIAN FORTINUniversity of Buenos Aires

CRISTIAN LÓPEZUniversity of Buenos Aires

FEDERICO HOLIKNational University of La Plata

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Holik, Federico, editor.Title: Quantum worlds : perspectives on the ontology of quantum mechanics / edited by

Olimpia Lombardi (Universidad de Buenos Aires, Argentina), Sebastian Fortin(Universidad de Buenos Aires, Argentina),

Cristian López (Universidad de Buenos Aires, Argentina), Federico Holik(Universidad Nacional de La Plata, Argentina).

Description: Cambridge ; New York, NY : Cambridge University Press, 2019. |Includes bibliographical references and index.

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Contents

List of Contributors page viiPreface xi

Introduction 1

Part I Ontology from Different Interpretations ofQuantum Mechanics 7

1 Ontology for Relativistic Collapse Theories 9wayne c. myrvold

2 The Modal-Hamiltonian Interpretation: Measurement,Invariance, and Ontology 32olimpia lombardi

3 Quantum Mechanics and Perspectivalism 51dennis dieks

4 Quantum Physics Grounded on Bohmian Mechanics 71nino zanghı

5 Ontology of the Wave Function and the Many-Worlds Interpretation 93lev vaidman

6 Generalized Contexts for Quantum Histories 107marcelo losada, leonardo vanni, and roberto laura

Part II Realism, Wave Function, and Primitive Ontology 119

7 What Is the Quantum Face of Realism? 121james ladyman

8 To Be a Realist about Quantum Theory 133hans halvorson

v

9 Locality and Wave Function Realism 164alyssa ney

Part III Individuality, Distinguishability, and Locality 183

10 Making Sense of Nonindividuals in Quantum Mechanics 185jonas r. b. arenhart, otavio bueno, and decio krause

11 From Quantum to Classical Physics: The Role of Distinguishability 205ruth kastner

12 Individuality and the Account of Nonlocality: The Case for theParticle Ontology in Quantum Physics 222michael esfeld

13 Beyond Loophole-Free Experiments: A Search for Nonergodicity 245alejandro a. hnilo

Part IV Symmetries and Structure in Quantum Mechanics 267

14 Spacetime Symmetries in Quantum Mechanics 269cristian lopez and olimpia lombardi

15 Symmetry, Structure, and Emergent Subsystems 294nathan harshman

16 Majorization, across the (Quantum) Universe 323guido bellomo and gustavo m. bosyk

Part V The Relationship between the Quantum Ontology andthe Classical World 343

17 A Closed-System Approach to Decoherence 345sebastian fortin and olimpia lombardi

18 A Logical Approach to the Quantum-to-Classical Transition 360sebastian fortin, manuel gadella, federico holik, and

marcelo losada

19 Quantum Mechanics and Molecular Structure:The Case of Optical Isomers 379juan camilo martınez gonzalez, jesus jaimes arriaga, and

sebastian fortin

Index 393

vi Contents

Contributors

Jonas R. B. ArenhartFederal University of Santa Catarina

Guido BellomoUniversity of Buenos Aires – CONICET

Gustavo M. BosykNational University of La Plata – CONICET

Otávio BuenoUniversity of Miami

Dennis DieksUtrecht University

Michael EsfeldUniversity of Lausanne

Sebastian FortinUniversity of Buenos Aires – CONICET

Manuel GadellaUniversity of Valladolid

Hans HalvorsonPrinceton University

vii

Nathan HarshmanAmerican University

Alejandro HniloCITEDEF – CONICET

Federico HolikNational University of La Plata – CONICET

Jesús Jaimes ArriagaUniversity of Buenos Aires – CONICET

Ruth KastnerUniversity of Maryland

Décio KrauseFederal University of Santa Catarina

James LadymanUniversity of Bristol

Roberto LauraNational University of Rosario

Olimpia LombardiUniversity of Buenos Aires – CONICET

Cristian LópezUniversity of Buenos Aires – University of Lausanne – CONICET

Marcelo LosadaNational University of Rosario – CONICET

Juan Camilo Martínez GonzálezUniversity of Buenos Aires – CONICET

Wayne C. MyrvoldUniversity of Western Ontario

viii List of Contributors

Alyssa NeyUniversity of California, Davis

Lev VaidmanTel Aviv University

Leonardo VanniUniversity of Buenos Aires

Nino ZanghìUniversity of Genova

List of Contributors ix

Preface

Developing a research group in philosophy of physics is not an easy task inArgentina, the southernmost country in South America. In addition to languagebarriers and the lack of a tradition on the matter, distance is the main obstacle toattend the best academic meetings and to visit renowned research centers. For thisreason, the Grant 57919 that we were awarded by the John Templeton Foundation(JTF) represented an invaluable support to our work: It allowed us to develop anintense activity manifested in publications and participation in specialized confer-ences, which gave a qualitative boost to our research development.

In the context of this JTF grant, we organized the International WorkshopIdentity, indistinguishability and non-locality in quantum physics, held in BuenosAires from June 26 to June 29, 2017. We were proud to welcome some of the mostsalient international specialists on the interpretation of quantum mechanics, whokindly accepted our invitation to participate in the workshop and to contribute tothe present volume: Dennis Dieks, Michael Esfeld, Hans Halvorson, NathanHarshman, Alejandro Hnilo, Ruth Kastner, Décio Krause, James Ladyman, WayneMyrvold, Alyssa Ney, Lev Vaidman, and Nino Zanghì. Our first acknowledgmentis for them. Those who want to know about the workshop can access to the linkwww.filoexactas.exactas.uba.ar/project-ontology/workshop.html, which containsthe videos of the full talks and of the lively final discussion.

Of course, our most special acknowledgment goes to the John TempletonFoundation, which made possible the successful development of our project. Butsince institutions do not exist without the people who embody them, we want toparticularly thank Alexander Arnold for his continued support during the threeyears of work.

We also want to acknowledge the Academia Nacional de Ciencias Exactas yNaturales, which supplied the venue for the meeting. However, the workshopwould not have been successful without the essential assistance of the membersof the Group of Philosophy of Science led by Olimpia Lombardi and based both in

xi

the Faculty of Exact and Natural Sciences and in the Faculty of Philosophy andLetters of the University of Buenos Aires: Hernán Accorinti, Guido Bellomo,Martín Bosyk, Mariana Córdoba, María José Ferreira Ruiz, Manuel Herrera, JesúsJaimes Arriaga, Martín Labarca, Marcelo Losada, Juan Camilo Martínez González,Erick Rubio, Leonardo Vanni, and Alfio Zambon. Their strong commitment andunlimited enthusiasm made the organization of the meeting an enjoyable task.

Although the presentations from the workshop were the basis for this book, weare also grateful to Jonas Arenhart, Otávio Bueno, and Manuel Gadella, whograciously accepted the invitation of the editors to contribute to the present volumein different chapters. Last, but not least, we want to express our gratitude toCambridge University Press, in the person of Simon Capelin, Editorial Director(physical sciences) and Sarah Lambert, Editorial Assistant, for their support andassistance during all the stages of this project.

xii Preface

Introduction

In its original meaning, ‘ontology’ is the study of what there is – not only of whatentities exist but also of the very structure of reality. For the most part of the historyof philosophy, ontology was the core of metaphysics, perhaps the major branch ofphilosophy. Nowadays, however, the word has different meanings and nuances. Inthe analytic tradition, for instance, ontology is the study not only of what there is,but also of the most general features of and the relations among what there is. Thisstudy commonly starts out from our intuitions about reality or from an a priorireasoning. Yet, another, increasingly growing sense of ontology has to do withreality itself in relation to our best scientific theories: When one asks for “theontology of” a certain scientific theory, the question is about what reality would belike if the theory were true. Although this second meaning does not exactly matchthe etymology of the word (from Ancient Greek: on, what is; logos, discourse,account), the meaning drift is completely natural in the light of the fact that, at leastafter the Renaissance, scientific knowledge was crucial with respect to how thestructure of reality and the nature of its entities were conceived.

Quantum mechanics is probably the most successful and the least understoodphysical theory that we have ever had. Even though this claim has become almost acliché, its frequent repetition does not make it less true. Indeed, after almost acentury of its first formulations, quantum mechanics is still posing unsolvedpuzzles with respect to our understanding of the microscopic world. Of course,numerous important results have been obtained during years of research, and manyof them are relevant to the foundations and the interpretation of the theory.However, it is not completely clear yet how the ontology of the theory is, inparticular, how reality would be if quantum mechanics were true. Not only doesthe question remain in force just as in the first decades of the twentieth century, buta century of philosophical and scientific discussions has brought to light thatquantum reality is far more complex than originally supposed. The numerousand varied perspectives developed up to the present time just manifest this

1

complexity. If each different perspective tells us a different story about thequantum realm, then the current variety of perspectives point out to the many“quantum worlds” we have come to conceive to date.

The aim of this volume is, precisely, to present this variety of “quantum worlds”in the most unbiased way. The different perspectives on the ontology of quantummechanics that this volume compiles rely on different metaphysical commitments,diverse formal tools, diverging aims, or even disparate readings of the theory’sformalism. All this not only makes manifest how rich and puzzling quantummechanics is for our understanding of the physical world, but also how bridgesbetween philosophy and physics can be built in order to make progress in suchunderstanding. To unfold the wide variety of perspectives in an organized way, thevolume is structured in five parts.

Part I, “Ontology from Different Interpretations of Quantum Mechanics,”groups the chapters focusing on particular interpretations already proposed in theliterature on the matter. This part opens with the chapter, “Ontology for CollapseTheories,” where Wayne Myrvold claims that the natural ontology for a collapsetheory is a distributional ontology, according to which dynamical quantities, suchas charge or mass within a specified region, do not take on precise values, butrather have associated with them a distribution of values; this chapter discusses theextension of such a picture to the context of a relativistic spacetime. The secondchapter, “The Modal-Hamiltonian Interpretation: Measurement, Invariance, andOntology” by Olimpia Lombardi, draws the attention to this interpretation byexposing it in a conceptually clear and concise way, stressing its advantages bothfor dealing with the traditional interpretive problems of quantum mechanics andfor supplying a physically meaningful account of relevant aspects of the theory.The main aim of the third chapter, “Quantum Mechanics and Perspectivalism” byDennis Dieks, is to argue for a perspectival noncollapse interpretation that, byassigning relational or perspectival states, makes it possible to reconcile universalunitary evolution, and the resulting omnipresence of entangled states, with theoccurrence of definite values of physical quantities. In the fourth chapter, “Quan-tum Physics Grounded on Bohmian Mechanics,” Nino Zanghì rejects the inter-pretations based on the concepts of measurement and observer; according to him,the theory must be based on a clear primitive ontology that provides the spatio-temporal histories of its basic entities. In the fifth chapter, “Ontology of the WaveFunction and the Many-Worlds Interpretation,” Lev Vaidman undertakes a vigor-ous defense of this interpretation by claiming that what we see is only a tiny part ofwhat there is: There are multiple parallel worlds similar to ours, and our experi-ences supervene on the wave function of the universe. In the sixth and closingchapter of the first part, “Generalized Contexts for Quantum Histories,” MarceloLosada, Leonardo Vanni, and Roberto Laura propose a formalism of generalized

2 Quantum Worlds

contexts for quantum histories, in which the contexts of properties at each timemust satisfy compatibility conditions given by commutation relations in the Hei-senberg representation; any family of histories satisfying these conditions isorganized in a distributive lattice with well-defined probabilities obtained by anatural generalization of the Born rule.

Part II, “Realism, Wave function, and Primitive Ontology,” is devoted to thequestion of realism in the quantum domain in general and, in particular, of realismregarding the wave function. In the first chapter, “What Is the Quantum Face ofRealism?,” James Ladyman explores the interaction between different forms ofrealism and different forms of quantum physics, showing the tension between usualarguments for scientific realism in the philosophy of science literature and theinvocation of realism in certain interpretations of quantum mechanics. In the secondchapter of this part, “To Be a Realist about Quantum Theory,” Hans Halvorson takesa closer look at the distinction between realist and antirealist views of the quantumstate, and argues that this binary classification should be reconceived as a continuumof different views about which properties of the quantum state are representationallysignificant. The final chapter of the second part, “Locality and Wave FunctionRealism” by Alyssa Ney, advocates for wave function realism, according to whichthe fundamental quantum entity is the wave function, understood as a scalar field on ahigh-dimensional space with the structure of a configuration space; according to her,this kind of realism is an attempt to explain nonlocal influences, instead of takingthem as brute facts of the world.

In Part III, “Individuality, Distinguishability, and Locality,” the ontologicalproblems related to the identity and nature of quantum particles are addressed.This part begins with the chapter by Jonas Arenhart, Otávio Bueno, and DécioKrause, “Making Sense of Nonindividuals in Quantum Mechanics,” which focuseson a very specific question: Assuming that quantum theories deal with “particles”of some kind, what kind of entities can such particles be? The authors respond thatquantum entities are nonindividuals and that a metaphysics of nonindividualsrequires a system of logic where the basic items have no identity. In the secondchapter of this part, entitled “From Quantum to Classical Physics: The Role ofDistinguishability,” Ruth Kastner reviews the derivations of the classical and thequantum statistics in order to argue that a form of separability is a key feature ofthe quantum-to-classical transition; on this basis, she considers the question ofwhat allows separability to serve as a form of distinguishability in the classicallimit. The third chapter, “Individuality and the Account of Nonlocality: The Casefor the Particle Ontology in Quantum Physics” by Michael Esfeld, examinesdifferent solutions to the measurement problem to conclude that the particleontology of Bohmian mechanics provides the least deviation from the ontologyof classical mechanics that is necessary so as to accommodate quantum physics,

Introduction 3

both in the case of quantum mechanics and in the case of quantum field theory. Thisthird part closes with the chapter by Alejandro Hnilo, “Beyond Loophole-FreeExperiments: A Search for Nonergodicity,” where he analyzes the experimentsdesigned to measure violations of Bell’s inequalities and argues that, besides localityand realism, the measurement of the inequalities implicitly assumes the ergodichypothesis; therefore, in order to save the validity of local realism in nature, it isnecessary to search for evidence of nonergodic dynamics in Bell’s experiments.

The chapters composing Part IV, “Symmetries and Structure in QuantumMechanics,” deal with structural features of the quantum theory. The first chapterof this part, “Spacetime Symmetries in Quantum Mechanics” by Cristian Lópezand Olimpia Lombardi, stresses the relevance of symmetries to interpretation; onthis basis, the authors consider the behavior of nonrelativistic quantum mechanicsunder the Galilean group and critically analyze the widely-accepted view about theinvariance of the Schrödinger equation under time reversal. In the second chapter,“Symmetry, Structure, and Emergent Subsystems,” Nathan Harshman focuses onthe particular structures called irreducible representations of symmetry groups, inorder to explore the connections between the mathematical units of symmetryembodied by those irreducible representations and the conceptual units of realitythat form the basis for the interpretation of quantum theories. Finally, the thirdchapter of this fourth part, “Majorization, across the (Quantum) Universe” byGuido Bellomo and Gustavo Bosyk, reviews the wide applicability of majorizationin the quantum realm and stresses that such applicability emerges as a consequenceof deep connections among majorization, partially ordered probability vectors,unitary matrices, and the probabilistic structure of quantum mechanics.

The chapters in the fifth and final part of this volume, “The Relationshipbetween the Quantum Ontology and the Classical World,” address the classicallimit of quantum mechanics from different perspectives. In the first chapter,“A Closed-System Approach to Decoherence,” Sebastian Fortin and OlimpiaLombardi argue that the conceptual difficulties of the orthodox approach todecoherence are the result of its open-system perspective; so, they propose aclosed-system approach that not only solves or dissolves the problems of theorthodox approach, but is also compatible with a top-down view of quantummechanics. In the second chapter of this part, “A Logical Approach to theQuantum-to-Classical Transition,” Sebastian Fortin, Manuel Gadella, FedericoHolik, and Marcelo Losada present a logical approach to the emergence ofclassicality based on the Heisenberg picture, which describes how the logicalstructure of the elementary properties of a quantum system becomes classicalwhen the classical limit is reached. In the chapter, “Quantum Mechanics andMolecular Structure: The Case of Optical Isomers,” which closes the last partand the volume, Juan Camilo Martínez González, Jesús Jaimes Arriaga, and

4 Quantum Worlds

Sebastian Fortin address the difficulty of giving a quantum explanation of chirality,that is, of the difference between the members of a pair of optical isomers orenantiomers; according to them, the solution of the problem requires an interpret-ation of quantum mechanics that conceives measurement as a breaking-symmetryprocess.

As this brief review shows, the plurality of perspectives and “quantum worlds”collected by this volume is an excellent opportunity not only to show how alivethe debate on the longstanding puzzles of quantum mechanics remains, butalso to present an updated state of affairs regarding our understanding of thequantum reality.

Introduction 5

Part I

Ontology from Different Interpretations ofQuantum Mechanics

1

Ontology for Relativistic Collapse Theories

wayne c. myrvold

1.1 Introduction

Dynamical collapse theories, such as the Ghirardi–Rimini–Weber (GRW) theory(Ghirardi, Rimini, and Weber 1986), the Continuous Spontaneous Localizationtheory, or CSL (Pearle 1989, Ghirardi, Pearle, and Rimini 1990), QuantumMechanics with Universal Position Localization, or QMUPL (Diósi 1989), andtheir respective relativistic extensions (Dove 1996, Dove and Squires 1996,Tumulka 2006, Bedingham 2011a, b, Pearle 2015), modify the usual deterministic,unitary quantum dynamics such as to produce something like the textbook collapseprocess. See Bassi and Ghirardi (2003), Bassi et al. (2013), and Ghirardi (2016) foroverviews.

If some sort of dynamical collapse theory is correct, what might the world belike? Can a theory of that sort be a quantum state monist theory, or must suchtheories supplement the quantum state ontology with additional beables? In aprevious work (Myrvold 2018), I defended quantum state monism. The viewdefended involves a natural extension of the usual eigenstate-eigenvalue link,which provides a sufficient condition for a quantum state to be one in which asystem has a definite value of some dynamical variable, namely, that the quantumstate be an eigenstate of that variable. The usual eigenstate-eigenvalue link leavesopen the question of what to say about states that are not eigenstates. A state that isnot an eigenstate of some dynamical variable, but is very close to an eigenstate,exhibits behaviour that closely approximates that of the eigenstate. In accordancewith a proposal of Ghirardi, Grassi, and Pearle (1990), in such a case the quantitymay be treated as if it were definite. However specification of the quantities that aredefinite or near-definite does not exhaustively specify the condition of the physicalworld, as there are matters of fact about such things as the spread of values of adynamical variable in a given state. The natural ontology for a collapse theory is adistributional ontology along the lines advocated by Philip Pearle (2009). On such

9

an account, dynamical quantities such as charge or mass within a specified regiondo not take on precise values, but, rather, have associated with them a distributionof values.

This chapter discusses the extension of such a picture to the context of arelativistic spacetime. This will not be presumed to be Minkowski spacetime, aswe will want to consider curved spacetimes; furthermore, we do not wish toexclude the possibility of discrete spacetimes. What the spacetimes we willconsider have in common is a relativistic causal structure. In particular, we willfocus attention on spacetimes in which, for any spacetime point p, there aretemporally extended processes that go on at spacelike separation from p, prohibitedby the spacetime structure from either causally influencing p or being influencedby it.

In Section 1.2 I give a brief recapitulation of the argument, presented in moredetail in Myrvold (2018), for distributional ontology. This is based on a principlethat, I claim, ought to be respected by any project of seeking to draw ontologicalconclusions from nonfundamental physical theories, a principle that I call thePrinciple of Metaphysical Continuity, outlined in Section 1.2.1. This principlepermits us to draw conclusions about ontology for nonideal collapse theories – thatis, collapse theories that yield, not exact eigenstates of the dynamical quantities onewould like to be definite, but close approximations to them – from ontologicalconsiderations regarding ideal collapse theories. Section 1.3 presents a fairlygeneral schema for collapse theories in a relativistic spacetime. Finally, in Section1.4, we turn to the identification of local beables for theories of that sort.

1.2 The Case for Distributional Ontology

1.2.1 Ontology for Nonfundamental Theories

If we want to know something about the make-up of the world, we can do no betterthan to look to our best scientific theories. Doing this poses a prima facie problem,however, as there is not and never has been a convincing candidate on the table fora complete and fundamental physical theory.

One reaction to this fact might be take on the task of giving an account of whatthe world would be like if such-and-such physical theory were a complete andfundamental theory. On this view, metaphysics is a subgenre of fiction, thoughstripped of plot and character and, indeed, of everything that motivates us to readfiction. This strikes me as an uninteresting enterprise, except insofar as consider-ations of unrealistic theories yields insights regarding the ontology of the actualworld. For instance: Though we currently lack a theory that incorporates bothquantum and gravitational phenomena, one could, perhaps, investigate the

10 Wayne C. Myrvold

structure of a world in which there is no gravitation and in which the standardmodel of quantum field theory is exactly correct. However such a world would be alonely and boring place, as it would contain no stars or planets, and, since virtuallyall nuclei heavier than helium are formed in stars, would contain little in the way ofchemical reactions. We could not ask what it would be like to be a denizen of aworld like that, because a world like that would contain no life.

Another reaction might be to abandon all ontological inquiry as hopeless, on thegrounds that we can expect future theories to generate radical ontological shifts.This strikes me as overly pessimistic. The objects dealt with in classical physics doafter all exist, even if their behaviour is not exactly what classical physics wouldlead one to expect. Any theory that can lay claim to the title of a viable successortheory to our current theories is obliged to recover the empirical successes of ourcurrent theories, and as long as we resist the temptation to draw stronger conclu-sions from our current physical theories than we have warrant for, there are reasonsfor optimism that those conclusions will weather the storms of future theorychange.

This sort of attitude recommends due caution in our metaphysical musings. Theevidence we have concerning our physical theories warrants only the conclusions thatthey hold to a good approximation within their domains of applicability, and that anyviable successor theory will have to entail something like the current theories withinthose domains. If our current theories have metaphysical consequences that aresensitive to the precise details of the theory, consequences that would not hold ifthe theory were slightly different, then we have no warrant for taking those conse-quences to hold of our world. Our metaphysical conclusions should satisfy a Principleof Metaphysical Continuity: they should be robust under small perturbations oftheory. This is a principle that we will put to work, in Section 1.2.3.

1.2.2 The Requirement of Local Beables

Consider a region of spacetime that is bounded in both time and space, say, thespatial region inside your office, during some specified hour of time. Of the thingsthat are true of that bounded spacetime region, some are local to that region: Theyrefer only to intrinsic properties of that region. These are to be contrasted withthings that involve relations to states of affairs outside the region, or either implicitor explicit reference to things outside.

For example, on the usual way of thinking about things, if your office, duringthe hour we are considering, contains a cabinet-shaped piece of steel, this is a localfact about that spacetime region. If the proposition that the spacetime region underconsideration contains an object of that sort is true, its truth is compatible withcompletely arbitrary states of affairs outside the region, and its truth value cannot

Ontology for Relativistic Collapse Theories 11

be changed by goings-on outside the region unless those goings-on have an effecton local matters within the region. By contrast, if the contents of your office areapproximately 150 million kilometers from the nearest star, this fact is clearly afact about relations between the things in your office and the world outside of it.A symptom of this fact is that it can be changed by making changes outside youroffice that do not affect anything within it.

By a local beable, I will mean something that is, in this sense, local to abounded spacetime region. The ontology of a physical theory might contain bothlocal and nonlocal beables. If it is the case that, for an arbitrarily fine covering ofspacetime with open sets, the full ontology of the theory supervenes on beablesthat are local to elements of that covering, we will say that the ontology satisfiesthe condition of separability (see Myrvold 2011 for further discussion).

Quantum state realism entails rejection of separability. It does not followthat there are no local beables. For one thing, there could be local beablespostulated in addition to the quantum state. Additionally, some aspects of thequantum state – in particular, the reduced state that is the restriction of the state toobservables pertaining to a bounded spacetime region – might be counted as localbeables.

Need there be any local beables at all? If we are willing to countenance arejection of separability, might we not go all the way and accept a radically holisticview in which there are no beables intrinsic to any region short of the whole ofspacetime?

The difficulty with this is that, if the theory is meant to be one that is in principlecomprehensive, it must have room for such things as experimental apparatus that issubject to local manipulations and whose experimental readouts are, presumably,matters of fact local to the laboratory. In the absence of things like these, the theoryruns the risk of undermining its own evidential base (see Maudlin 2007 for a luciddiscussion of this point).

A brief comment, before we continue. What it means to say that a structurefound within a physical theory plays the role of spacetime for that theory is that ithas the appropriate connections with dynamics. In speaking of spacetime, I willalways mean that structure that plays the role in the theory of affording spatio-temporal relations, such as distances, temporal intervals, causal connectability andthe like, distances and temporal intervals and causal relations that are relevant tothe dynamics. It is necessary to say this because it has been claimed that quantumtheory motivates the introduction of a so-called fundamental space, or fundamentalarena, a high-dimensional space that would be such that quantum states involvenothing more than assignments of local beables to points in that fundamental space(see Albert 1996 and the various contributions to Ney and Albert 2013). In aquantum theory, even if such a space can be found, that space is not the structure

12 Wayne C. Myrvold

on which the distances, temporal intervals, and causal relations relevant to thedynamics are defined. For that reason, such a space, even if it were to exist, is notspacetime in the sense of the word being used in this chapter. Hence, even if afundamental space of the sort sought by Albert and others did exist, a quantumstate realist ontology violates separability as we are using the term.

1.2.3 Ontology for Ideal Collapse Theories

According to the textbook collapse postulate, after an experiment the quantumstate of the system subjected to the experiment is an eigenstate of the observablewhose value has been obtained. Naively, one might expect a dynamical collapsetheory to be like that. There are good reasons for thinking that this is an unattain-able goal. If, however, we could have a theory like that – a theory that yieldedeigenstates of an appropriate dynamical variable – then, I claim, there would be noproblem of ontology for the theory once we have settled on a choice of dynamicalvariable to collapse to eigenstates of (a suitable choice seems to be that of asmeared mass density, as advocated by Ghirardi, Grassi, and Benatti 1995). Thatthere is any question about the ontology for a collapse theory is an aspect of whathas been called the tails problem (first flagged as an issue by Shimony 1991 and byAlbert and Loewer 1991), which stems from the fact that collapse theories do notlead to eigenstates of familiar dynamical quantities.

Consider a quantum theory on a discrete spacetime, one on which spaceconsists of elementary cells of size vastly smaller than the scales on which wedeal with things. Suppose we had a collapse theory that tended to suppresssuperpositions of distinct mass densities smeared over regions (which couldconsist of a great many of these elementary cells) of order 10�5 cm, small on ahuman scale, but large compared to atomic dimensions. Suppose that our collapsetheory induced collapse, within a finite time, to eigenstates of the operatorscorresponding to total mass within regions of this size, and that states that arenot eigenstates of these mass operators could persist only for a minuscule fractionof a second.

On such a theory, for every region of space of sufficient size, the quantum statewould, most of the time, be an eigenstate corresponding to a definite mass withinthat region. Hence, by the eigenstate-eigenvalue link, there would be a matter offact about the amount of mass within that region. Thus, a possible state of the roomin which I am sitting would be one in which there was a desk-shaped region ofhigher mass density than its surroundings. Provided that these regions of high massdensity exhibited the right sort of dynamical behaviour, there would be no problemin identifying them with desks, chairs, and laboratory equipment, and there wouldbe no problem of ontology for collapse theories.

Ontology for Relativistic Collapse Theories 13

1.2.4 Distributional Ontology

Prospects are dim for a viable collapse theory that yields precise eigenstates of totalmass in any bounded region, or indeed, precise eigenstates of any local beables.A collapse theory can, however, yield close approximations to eigenstates ofappropriate local beables, such as mass smeared over sufficiently large regions.

Whether the dynamics is linear, unitary, and deterministic, as in the Schrödingerequation, or nonunitary and stochastic, initial states that are close to each other, inHilbert space norm, evolve in approximately the same way. Thus, a state that isclose to being an eigenstate of a given dynamical quantity will evolve in approxi-mately the same way as the eigenstate that it is close to.

If we accept (as we should; see Myrvold 2018 for a fuller discussion, also Albert2015: 127ff.) that to be a physical body is nothing more and nothing less than tohave a certain place in a network of dynamical and causal relations of an appropri-ate sort, and if we accept (as we should) that there would be no problem ofinterpretation of an ideal collapse theory that yielded eigenstates of the right sortof dynamical quantities, then, by the Principle of Metaphysical Continuity, weshould accept that regions of space whose states are very near to eigenstates of totalmass can serve as physical objects just as well as would regions of space in exacteigenstates of total mass.

Considerations such as this have led to a proposed modification of theeigenstate-eigenvalue link, as follows:

if one wishes to attribute objective properties to individual systems one has to accept thatsuch an attribution is legitimate even when the mean value of the projection operator on theeigenmanifold associated to the eigenvalue corresponding to the attributed property is notexactly equal to 1, but is extremely close to it.

(Ghirardi, Grassi, and Pearle 1990: 1298)

This modification has been dubbed, by Clifton and Monton (1999), the fuzzylink.

To say that we can ascribe a property to a system when the quantum state is suchthat its variance is negligibly small requires that there be a matter of fact aboutwhat the variance is. Considerations of this sort suggest a revision of the way wethink about dynamical quantities, along the lines advanced by Pearle (2009). Onthis view, dynamical variables typically do not take on sharp values as they wouldclassically. What they have, instead, is a distribution associated with them. Thesedistributions, though having the formal characteristics of probability distributions,are to be thought of not as a probability distribution over precise but unknownpossessed values, but as reflecting a physical, ontological lack of determinacyabout what the value is. A limiting case would be the classical case, in which thedistribution is a delta function.

14 Wayne C. Myrvold

On this view, the value of every dynamical variable is distributional. A collapsetheory will tend to narrow the spread of the distributions of some of thesequantities. When the distribution is sufficiently narrow, things will be almostexactly as if the quantity has a precise value, and under such circumstances, wecan treat the variable as if it does possess a precise value. In seeking objects thatbehave like our familiar macroscopic objects, it is to those variables that we shoulddirect our attention. But the spread-out distributions of the other variables are noless part of physical reality.

1.2.5 Primitive Ontology as an Alternative?

Given a family of operators M xð Þ, corresponding to a smeared mass densitycentered at the point x, for any quantum state ψ, one can define a function m xð Þ,whose value at the point x is equal to the expectation value of M xð Þ in state ψ.When Ghirardi et al. (1995) introduced the smeared mass density as a basis for theontology of collapse theories, their proposal was an application of the fuzzyeigenstate-eigenvalue link. They argued that the quantity m xð Þ behaves like a massdensity when – and only when – the variance of M xð Þ is sufficiently small as to benegligible, in which case the mass density is said to be objective. When thiscondition is not satisfied, the quantity m xð Þ, though well defined, cannot beinterpreted as a mass density, as other systems do not behave as if a quantity ofmass corresponding to m xð Þ is present. In some later works (Ghirardi and Grassi1996, Ghirardi 1997a, b) the mass density is said to be ‘accessible’ if its variance issufficiently small (this shift is attributed by Ghirardi and Grassi 1996: fn. 5 to aconversation with S. Goldstein).

There is, at least apparently, a rival interpretation of m xð Þ. On this view,introduced by Goldstein (1998) and discussed extensively by Allori et al. (2008),a mass density equal at every point to the expectation value of M xð Þ is posited asadditional, primitive ontology over and above the quantum state.

The quantities m xð Þ are well defined for any quantum state. However, since insituations in which the objectivity, or accessibility, condition is not satisfied – thatis, situations in which the variance of M xð Þ is not small enough to be neglected –

other objects do not respond as if a mass density equal to m xð Þ is present, m xð Þ actslike a mass density only when the accessibility condition is satisfied. Somethingthat does not act like a mass density is not a mass density. Thus, on the supposedrival interpretation, despite what is said, a mass density is present only when thereis a mass density present on the original, quantum state monist proposal, that is,when the accessibility condition is satisfied. The proposal to take m xð Þ as add-itional, primitive ontology does not present a genuine alternative to the originalproposal of Ghirardi et al. (1995).

Ontology for Relativistic Collapse Theories 15

1.3 A Schema for Relativistic Collapse Theories

1.3.1 Relativistic Spacetimes

We assume a spacetime equipped with a causal order, that is, a relation� of causalprecedence, assumed to be transitive and antisymmetric (that is, if p causallyprecedes q, then q does not causally precede p). Two spacetime points are saidto be causally unconnected if they stand in no causal order, that is, if neither p � qnor q � p obtains. Because no point is in the causal past of itself, the relation ofbeing causally unconnected is reflexive. That it is symmetric follows straightfor-wardly from its definition. Two distinct points that are causally unconnected aresaid to be spacelike separated.

In Galilean spacetime, the relation of being causally unconnected is transi-tive, and therefore, is an equivalence relation, and the spacetime can beportioned into equivalence classes of simultaneity. In Minkowski spacetime,on the other hand, for any two points p, q that are spacelike separated fromeach other, there are other points r that are spacelike separated from p but notfrom q. Define a relativistic spacetime as one in which, for any spacelikeseparated p, q, there exists a point r that is spacelike separated from p, suchthat r � q.

A causal curve is a curve such that, for any pair of distinct points p, q, eitherp � q or q � p. A Cauchy surface is a set of spacetime points that is intersectedexactly once by every inextendible causal curve. A spacetime that contains Cauchysurfaces is said to be globally hyperbolic.

We will, in what follows, assume a globally hyperbolic relativistic spacetime.We can define the relation � between Cauchy surfaces: If σ, σ0 are two Cauchysurfaces, then σ � σ0 when no part of one if no part of σ0 is in the causal past of σ.This relation is reflexive and transitive, and hence is a partial order on Cauchysurfaces.

1.3.2 Collapse Theories in Relativistic Spacetime

A collapse theory modifies the deterministic, unitary evolution so as to producesomething like the textbook collapse. Gisin (1989) has demonstrated, on theassumption that the evolution is Markovian (meaning that future states dependonly on the present state and not on any details about the past that are notreflected in the present), that any deterministic, nonlinear dynamics for quantumstates that does not respect a certain linearity condition permits signalling – if twospatially separated systems are in an entangled state, a choice of experiment onone can influence probabilities of outcomes of experiments performed onthe other.

16 Wayne C. Myrvold

The relevant condition is the following:

Linearity. Let T be a dynamical map on the set of pure states of a system. Letψi; i ¼ 1; . . . ; nf g and φj; j ¼ 1; . . . ;m

� �be sets of pure states such that, for some

non-negative weights {xi}, {yj},

Xni¼1

xiψi ¼Xmj¼1

yj φj:

Then

Xni¼1

xi T ψið Þ ¼Xmj¼1

yj T φj� �

:

As Kent (2005) has argued, although violations of linearity permit signalling,this need not be superluminal signalling. Nonetheless, ordinary quantum mech-anics does not allow operations on one system to be used for signalling to anothersystem, unless there is an interaction term between the two systems in theHamiltonian, and we will assume that the no-signalling condition holds, andhence that the evolution is linear. This means that (unsurprisingly) a theory thatproduces collapse must be a theory with indeterministic dynamics.

It is convenient to work within what may be called the stochastic Tomonaga–Schwinger picture. The usual Tomonaga–Schwinger picture (see Schweber1961: 419–422 for an introduction) is an extension of the interaction pictureto a relativistic spacetime. One divides the Lagrangian density into two parts(typically regarded as the free Lagrangian density and the interaction Lagrangian),

L xð Þ ¼ L0 xð Þ þ L1 xð Þ: (1.1)

The operators representing observables are Heisenberg-picture operators for thefree theory. We utilize, however, evolving state vectors; with each Cauchy surfaceσ is associated a state vector ψ σð Þj i. Evolution from a surface σ to another, σ0,differing by a small deformation δσ about a point x, satisfies the Tomonaga-Schwinger equation:

iℏcδ ψ σð Þj i

δσ¼ H1 xð Þ ψ σð Þj i: (1.2)

Integration of this equation yields, for any Cauchy surfaces σ and σ0, a unitarymapping from ψ σð Þj i to ψ σ0ð Þj i.

We wish to modify this equation so as to produce collapse. On the stochasticTomonaga–Schwinger picture, we work with Heisenberg-picture operators that aresolutions to the standard field-theoretic equations, for free or interacting fields. The

Ontology for Relativistic Collapse Theories 17

difference between states on different Cauchy surfaces is due to the stochasticmodifications to the usual evolution. We will assume that the new dynamics isMarkovian: that is, that if σ � σ0, the set of possible states on σ0 and theirrespective probabilities are determined by ψ σð Þj i, and not by other facts aboutthe history leading up to that state.

Given two Cauchy surfaces, σ, σ0, with σ � σ0, and a state vector ψ σð Þj i, therewill be some state vector ψ σ0ð Þj i, but what this vector will be is not determined byψ σð Þj i and the dynamics. Instead, there will be some set of alternatives��ψγ σ0ð Þ�n o

, which we take to be indexed by a parameter γ that takes on values

in a set Γ. We expect our theories to specify, given two surfaces σ, σ0, with σ � σ0,

and the state vector ψ σð Þj i, the set of alternatives��ψγðσ0Þ

�; γ 2 Γ

n o, and a

probability distribution over the possible values of γ. Suppose that, with respectto some background measure μ, this probability distribution is represented by adensity function p γð Þ.

With this apparatus in place, we can define a mixed state ρ σ0; σð Þ as theweighted average over the possibilities for the state on σ0, given ψ σð Þj i.

ρ σ0; σð Þ ¼ðγp γð Þ ��ψγ σ0ð Þ��ψγ σ0ð Þ��dμ: (1.3)

This would be the state used by someone who knows the state on σ andthe possible state transitions from ψ σð Þj i to ψ σ0ð Þj i and their respectiveprobabilities, but does not know the outcome of the process that occurs betweenσ and σ0.

Gisin’s proof, mentioned earlier, generalizes to stochastic theories. If we takeT to be the mapping that takes a pure state on σ to a mixed state ρ σ0; σð Þ, no-signalling entails that this map must satisfy the linearity condition (Simon,Bužek, and Gisin 2001, Bassi and Hejazi 2015), and from this, together withthe condition that, applied to subsystems in entangled states, the mappingextends to a positive map on the state space of the wider system, it entails thatthe map from the state on σ to the mixed state ρ σ0; σð Þ be a completelypositive map.

We will therefore take the mapping from a pure state on σ to the mixed stateρ σ0; σð Þ to be a nonselective completely positive map, which is a mixture ofselective completely positive maps that takes us from ψ σð Þj i to

��ψγ σ0ð Þ�. Thisentails that there is a set of operators Kγ; γ 2 Γ

� �, which we will call evolution

operators, such that, for some γ,

ψ σ0ð Þj i ¼ ��ψγ σ0ð Þ� ¼ Kγ ψ σð Þj i= Kγ ψ σð Þj i , (1.4)

18 Wayne C. Myrvold

with probabilities for which state is realized given by

Pr γ 2 Δð Þ ¼ðΔp γð Þdμ: (1.5)

The linearity condition entails that

p γð Þ ¼ Kγ ψ σð Þj i 2: (1.6)

Any probabilities that depart from these would lead to signalling. The conditionthat p always be normalized is the condition thatð

ΓK†

γ Kγ dμ ¼ 1: (1.7)

The evolutions should also satisfy the semi-group property, which requires that, forCauchy surfaces σ � σ0 � σ00, the possible evolutions from σ to σ00 be the compos-itions of evolutions from σ to σ0 with evolutions from σ0 to σ00.

The theory of quantum dynamical semi-groups is well studied (see Bassi andGhirardi 2003 or Alicki and Lendi 2007 for an introduction). Provided that theevolution satisfies an appropriate continuity condition, the mixed-state density oper-ators on Cauchy surfaces σ to the future of some surface σ0 will satisfy a Lindbladequation. We consider the change in ρ σ; σ0ð Þ as we pass from one surface σ toanother σ0 differing by a small deformation about a point x on σ with spacetimevolume δσ. Let H xð Þ be the Hamiltonian density, that is, the component of theenergy-momentum density along the normal to σ at x. For a Lindblad-type evolution,there is also a countable set Lα xð Þf g of operators, such that the change δρ satisfies

δρδσ

¼ 1iℏc

H xð Þ;ρ½ � þXα

Lα xð ÞρL†α xð Þ � 12

Xα

L†α xð ÞLα xð Þρþ ρXα

L†α xð ÞLα xð Þ !

:

(1.8)

Consider two Cauchy surfaces σ, σ0, with σ � σ0, that coincide everywhereexcept on the boundaries of two bounded regions δ and δ0 (see Figure 1.1). Theevolution from σ to σ0 through δ [ δ0 must equal the composition of the evolutionthrough δ and the evolution through δ0, in either order. The necessary and sufficient

s¢d¢

sd

Figure 1.1 Cauchy surfaces σ, σ0, with σ � σ0, that coincide everywhere except onthe boundaries of two bounded regions δ and δ0.

Ontology for Relativistic Collapse Theories 19

condition for this is that evolution operators corresponding to spacelike separatedregions commute.

Moreover, for computing probabilities for the results of experiments on theoverlap of σ and σ0, it should not matter whether ψ σð Þj i or ρ σ0; σð Þ is used.Someone located in the overlap, who knows the state on σ and believes collapsewill occur between σ and σ0, but does not know (because it occurs at spacelikeseparation) what the outcome of that collapse is, should be able to use ψ σð Þj i orρ σ0; σð Þ for computing probabilities of results of experiments that he or she is aboutto undertake, and these should yield the same probabilities for outcomes of thoseexperiments. The necessary and sufficient condition for this is that evolutionoperators that implement evolution through a given spacetime region δ shouldcommute with operators representing observables at spacelike separation from δ.

These conditions give us a rather general schema for a quantum theory withstochastic dynamics on a relativistic spacetime. It includes, as a special case,deterministic, unitary evolution, in which case the set of evolution operatorspertaining to any region of spacetime is a singleton set. Concrete theories will fillin the details, specifying in particular what the sets of evolution operators are.

1.4 Beables for Relativistic Collapse Theories

1.4.1 Intrinsic and Extrinsic States of a Spacetime Region

Consider a bounded spacelike region α, that is common to Cauchy surfacesσ; σ0; σ00; . . .f g. In the stochastic Tomonaga–Schwinger picture, there will bequantum states ρ σð Þ, ρ σ0ð Þ, ρ σ00ð Þ, . . . Each of these states yields probabilities ofoutcomes of experiments to the future of its Cauchy surface, conditional on events,including any collapses, to the past of the Cauchy surface. For each of these states,we can consider the reduced state that consists of the restriction of the state toobservables in the forward domain of dependence of α. Call these reduced statesρα σð Þ, ρα σ0ð Þ, etc. If the evolution between two surfaces σ and σ0 is purely unitary,then ρα σð Þ will coincide with ρα σ0ð Þ. If, however, collapse occurs between σ andσ0, then they need not coincide (see Figure 1.2).

Since the reduced state ρα σð Þ is conditioned on any collapses to the past of σ,including any that are spacelike separated from α, it should be clear that, though itis associated with the region α, ρα σð Þ cannot in general be regarded as a beablelocal to α. Thus, if the reduced states ρα σð Þ and ρα σ0ð Þ differ, they do not offercompeting accounts of intrinsic properties of the region α.

The intrinsic state of a bounded spacelike region α must be conditioned only oncollapses to the past of α. We can define this state by a limiting procedure.Consider a sequence of Cauchy surfaces σ1; σ2; . . . ; σn; . . .f g, that is such that α

20 Wayne C. Myrvold

is contained as a common part of all σn and for all n, σnþ1 � σn, and the sequenceconverges on the past light cone of α (that is, the set of points that are to the past ofall σn is precisely the causal past of α). Define the past light-cone state of α as thelimit, if it exists, of ρα σnð Þ, as n increases indefinitely. Though a state derived froma Cauchy surface with events to its past that are spacelike separated from α cannotbe regarded as the intrinsic state of α, its past light-cone state can.

1.4.2 Compatibility of Extrinsic States

Maudlin (1996: 301–302) raised the question of consistency of state assign-ments derived from different hypersurfaces passing through a given region. Iftwo hypersurfaces σ, σ0, having a region α in common, yielded reduced statesthat were orthogonal to each other, yielding conflicting definite (probabilityequal to unity) predictions for the outcome of some experiment, this would beproblematic.

The question arises: Do the conditions on collapse dynamics outlined previouslyguarantee that the differing extrinsic state assignments obtained from differentCauchy surfaces are not in outright conflict with each other? It can be shown (seeMyrvold 2003: 489, 2016: 255–257) that these conditions suffice to guarantee thatthe states ρα σð Þ and ρα σ0ð Þ are not orthogonal.

In fact, a stronger sense of compatibility obtains. The question of the compati-bility of reduced states derived from states on different Cauchy surfaces is essen-tially the same as that addressed by Brun, Finkelstein, and Mermin (2002). Theydemonstrate that state assignments that can represent information about a systemavailable to different observers are compatible, in the sense that they have overlap-ping support.

The context in which Brun et al. work is that of finite-dimensional Hilbertspaces. However, essentially the same conclusion holds in a setting appropriate toquantum field theory. In this context we cannot assume a finite-dimensional Hilbertspace, nor can we assume that the mixed state of a bounded region α obtained froma pure state on a Cauchy surface containing α admits of a decomposition into pure

s²

s²s¢

sa

Figure 1.2 Bounded spacelike region α, that is common to Cauchy surfacesσ, σ0, σ00.

Ontology for Relativistic Collapse Theories 21

states. We must take care to formulate the condition in a manner that is independ-ent of assumptions such as these.

We assume a von Neumann algebra R αð Þ, whose self-adjoint elements representthe bounded observables pertaining to the forward domain of dependence of α. Letρ be a normal state of R αð Þ (that is, a completely additive state). We define thesupport projection for ρ as the orthogonal complement of the union of all projec-tions in R αð Þ to which ρ assigns expectation value zero.

With these definitions in hand, it can be shown that, given a set σ; σ0; σ00; . . .f gof Cauchy surfaces containing α, then on the assumption that there is a Cauchysurface containing α that is nowhere to the past of any of them (which, inparticular, will always be the case for any finite set of Cauchy surfaces), thecorresponding set of states ρα σð Þ; ρα σ0ð Þ; ρα σ00ð Þ; . . .f g have nonzero overlappingsupport. See Appendix for details.

1.4.3 Local Beables for Collapse Theories

Suppose that we have a collapse theory that yields near-eigenstates of an appropri-ate dynamical quantity. For example: The natural extension to the relativisticcontext of a mass density would be the components of the stress-energy tensor.Assume that we have an appropriate relativistically invariant smearing function(see Bedingham 2011a, b) and formulate smeared operators T μν xð Þ, representing asmeared stress-energy density centered at the point x. For any state ρ, we can define

T μν xð Þ ¼ T μν xð Þ� �ρ: (1.9)

The 00-component of this is the relativistic analogue of the mass density that hasbeen proposed as an appropriate ontology for nonrelativistic collapse theories.

For a bounded region α contained in distinct Cauchy surfaces σ and σ0, thereduced states ρα σð Þ and ρα σ0ð Þ may yield differing values for T μν xð Þ, with xwithin α. But, obviously, these do not yield rival accounts of local beables withinα, as they are defined via the extrinsic states ρα σð Þ and ρα σ0ð Þ, which are notthemselves local beables.

Where then, may we find local beables for a relativistic collapse theory? Thereare, in the literature, two proposals for extending the fuzzy link to a relativisticcontext. One is what might be called the agreement criterion, formulated byGhirardi, Grassi, and Pearle (1991):

We think that the appropriate attitude is the following: when considering a local observableA with its associated support we say that an individual system has the objective property a(a being an eigenvalue of A), only when the mean value of Pa is extremely close to one,when evaluated on all spacelike hypersurfaces containing the support of A.

(Ghirardi et al. 1991: 1310).

22 Wayne C. Myrvold

This means that, for x within α, T μν xð Þ will be regarded as representing anobjective property if and only if the accessibility criterion is satisfied in ρα σð Þ forevery Cauchy surface containing α.

The other criterion is the past light cone criterion, formulated by Ghirardi andGrassi (1994: 419, see also Ghirardi 1996: 336, 1999: 139, 2000: 1364). On thiscriterion, a system is said to possess the property A = a when the expectation valueof Pa is extremely close to one, evaluated on the past light-cone state.

If the criterion for property attribution were an exact eigenstate-eigenvaluelink – that is, if we were ascribing a property A = a only when the expectationvalue of Pa is exactly equal to one – then the two would be equivalent. The pastlight-cone state of a region α is an eigenstate of an observable A pertaining to α,with eigenvalue a, if and only if the state on every Cauchy surface containing α is.On the fuzzy link, the agreement criterion entails the past light-cone criterion, butthe past light-cone criterion does not guarantee satisfaction of the agreementcriterion; it only entails that the agreement criterion will hold with high probability.

If the property attribution criterion is meant to supply local beables, then it isclear that what is wanted is the past light-cone criterion and not the agreementcriterion. The agreement criterion makes reference to events at spacelike separationfrom the region in question. Moreover, as Ghirardi, Grassi, Butterfield, andFleming (1993: 358) have shown, a choice regarding experiments performed atspacelike separation from α can affect the probability that the agreement criterion issatisfied. Consider a case of two spin-½ particles, located in world-tubes A and B.We take initial conditions on a Cauchy surface σ0, and suppose that the particle inA is undisturbed in the interval between σ0 and some later Cauchy surface σ1. Let αbe a spacelike slice of A between σ0 and σ1. Let β0, β1 be the intersections of B withσ0 and σ1, respectively. We will take σ0 and σ1 such that β0 and β1 are to the pastand future, respectively, of α (see Figure 1.3).

Suppose that the state of the pair of particles on σ0 is

ψ σ0ð Þj i ¼ffiffiffiffiffiffiffiffiffiffiffi1� ε

pþj iA þj iB þ

ffiffiε

p �j iA �j iB, (1.10)

where þj i and �j i are spin eigenstates in some designated direction (say, the z-direction), and ε is an extremely small number, small enough that the state ψ σ0ð Þj iis sufficiently close to an eigenstate of spin for the particle in A that the accessibil-ity criterion is satisfied. Thus, on the past light-cone criterion, we ascribe + spin inthe z-direction to the particle in A as a possessed property.

Suppose that Bob, located near B, has a choice of whether to perform a spinexperiment on B. Suppose that, if he does not, the pair of spin-½ systems iseffectively isolated from outside interference and that in that case our collapsetheory assigns, for some δ smaller than ε, probability 1� δ that the state willremain undisturbed in the interval between σ0 and σ1, in which case the agreement

Ontology for Relativistic Collapse Theories 23

criterion for property attribution is satisfied. If Bob chooses to do a spin experimenton the particle in B, there is probability ε that he will obtain the result �j i. If hedoes, then the state of the combined system on a Cauchy surface σ2 that includes αand runs to the future of Bob’s experiment will not be a state in which the particlein A is close to a + eigenstate for spin-z; on the contrary, it will close to a –

eigenstate for spin-z. In such an eventuality, the agreement criterion for ascribing“spin-z = +” to the particle in A is not satisfied.

Now, if the threshold for satisfaction of the accessibility criterion is stringentenough – say, 10–40, as suggested by Pearle (1997) – then the probability ofdisagreement between the past light-cone criterion and the agreement criterion issufficiently low as to be negligible, whether or not Bob chooses to do an experi-ment. However, it is still true that the value of this negligibly low probabilitydepends on Bob’s choice regarding his experiment, and hence, if we were to applythe agreement criterion for outcome attribution, this would require acceptance thatthe theory exhibits parameter dependence, albeit a very weak parameter depend-ence (see Ghirardi et al. 1993 for discussion). If, however, we adopt the past light-cone criterion, then (as noted already by Ghirardi and Grassi 1994), there is noparameter dependence at all, not even very weak dependence. The conclusion to bedrawn is that local beables for a relativistic collapse theory are to be identifiedaccording to the past light-cone criterion.

Figure 1.3 Two spin-½ particles, located in world-tubes A and B. The initialconditions are taken on a Cauchy surface σ0, and the particle in A is undisturbedin the interval between σ0 and some later Cauchy surface σ1. α is a spacelike sliceof A between σ0 and σ1. β0, β1 are the intersections of B with σ0 and σ1,respectively, such that they are to the past and future, respectively, of α.

24 Wayne C. Myrvold

As mentioned previously, Ghirardi et al. (1995) have argued that a theory onwhich, at the macroscopic scales, a smeared mass density is almost always near-definite yields an adequate picture of the world. Combined with the past light-conecriterion, this gives a past light-cone matter density ontology, which is discussedby Tumulka (2007) and in more detail, by Bedingham et al. (2014).

1.5 Conclusion

There is a sensible ontology for collapse theories in a relativistic context. More-over, considerations of what it takes for a theory to represent a world that contains,among other things, objects like our experimental apparatus, to be thought of alocal beables, determine the form that this ontology takes. It is one on which alldynamical quantities are distributional in character. In spite of this distributionalcharacter, dynamical quantities may have effectively precise values (in the sensethat they behave, to a high degree of approximation, as if they have precise values);it is the goal of a collapse theory to ensure that the properties of macroscopicobjects almost always have this character. Beables local to a bounded spacetimeregion are to be evaluated via the past light-cone state of that region.

Ontology for Relativistic Collapse Theories 25

Appendix

We consider a finite set of Cauchy surfaces σ1; σ2; . . . ; σnf g, all containing an opensubset α. Our goal is to show that, given the conditions on a relativistic collapsetheory, the states ρα σ1ð Þ, ρα σ2ð Þ, . . . , ρα σnð Þ have common support.

We assume a Hilbert space that contains vectors ψ σ1ð Þj i, ψ σ2ð Þj i, . . . , ψ σnð Þj i.We assume also that, if σ � σ0, there exists Kγ such that

ψ σ0ð Þj i ¼ Kγ ψ σð Þj i= Kγ ψ σð Þj i : (A1.1)

If σ � σ0, and α is in the overlap of σ and σ0, then the region between σ and σ0 isspacelike separated from α. Therefore, Kγ commutes with all self-adjoint elementsof R αð Þ.

The restrictions of the states on the Cauchy surfaces σ1, σ2, . . . , σn are states(which will typically be mixed states) of R αð Þ, ρα σ1ð Þ, ρα σ2ð Þ, . . . , ρα σnð Þ. We donot assume that these are represented by density operators in R αð Þ or that they aremixtures of pure states of R αð Þ, as this is not needed in what follows.

As mentioned in the chapter text, the projector onto the null space of any stateρ of R αð Þ is the union of all projections P that have zero expectation value in ρ,and the support projection of ρ is the orthogonal complement of the projectoronto the null space. We will call the null space and the support of ρ, Null ρ½ �,and Supp ρ½ �.

Lemma 1. Let σ and σ0 be Cauchy surfaces containing a common open subset α. Ifσ � σ0, then for any positive operator E 2 R αð Þ, if ψ σð Þh jE ψ σð Þj i ¼ 0, thenψ σ0ð Þh jE ψ σ0ð Þj i ¼ 0.

Proof. Any positive operator E has a square root E1=2. Suppose that

ψ σð Þh jE ψ σð Þj i ¼ 0: (A1.2)

Therefore, since

ψ σð Þh jE ψ σð Þj i ¼ E1=2 ψ σð Þj i 2, (A1.3)

26

then

E1=2 ψ σð Þj i ¼ 0: (A1.4)

For some Kγ that commutes with all self-adjoint E 2 R αð Þ,ψ σ0ð Þj i ¼ Kγ ψ σð Þj i= Kγ ψ σð Þj i : (A1.5)

Therefore,

E1=2 ψ σ0ð Þj i ¼ E1=2Kγ ψ σð Þj i= Kγ ψ σð Þj i ¼ KγE1=2 ψ σð Þj i= Kγ ψ σð Þj i ¼ 0 (A1.6)

and so

ψ σ0ð Þh jE ψ σ0ð Þj i ¼ 0 (A1.7)

□.Lemma 1 gives us a relation between the supports of ρα σð Þ and ρα σ0ð Þ when

σ � σ0.

Proposition 1. Let σ and σ0 be Cauchy surfaces containing a commonopen subset α. If σ � σ0, then Null ρα σð Þ½ � � Null ρα σ0ð Þ½ �; equivalently,Supp ρα σ0ð Þ½ � � Supp ρα σð Þ½ �.Proof. This is immediate from Lemma 1.

From this follows the result concerning overlapping support ofρα σ1ð Þ; ρα σ2ð Þ; . . . ; ρα σnð Þf g.

Proposition 2. Let σ1, σ2, . . . , σn be Cauchy surfaces sharing a common opensubset α. Then the supports of ρα σ1ð Þ, ρα σ2ð Þ, . . . , ρα σnð Þ have nonzero intersection.

Proof. We can construct a Cauchy surface σþ that contains α and is such thatσi � σþ for each i, by taking the least upper bound of the set σ1; σ2; . . . ; σnf g underthe ordering �. Consider, now, the state ρα σþð Þ. By Proposition 1,Supp ρα σþð Þ½ � � Supp ρα σið Þ½ � for all i, and hence the support of σþ lies within theintersection of the supports of ρα σ1ð Þ, ρα σ2ð Þ, . . . , ρα σnð Þ.Remark I. The restriction to a finite set is unnecessary; the result holds for an infiniteset of Cauchy surfaces provided that there is a Cauchy surface that is the upperbound of all of them. Furthermore, if there is a future light-cone state that is the limitof an increasing (in the ordering �) set of Cauchy surfaces that converges on thefuture light cone of α, then the support of the future light-cone states is the intersec-tion of the supports of all ρα σð Þ for all σ containing α.

Remark II. A quantum field is an assignment of a field operator φ xð Þ to each point ofspacetime. In standard quantum field theories on Minkowksi spacetime, it is assumedthat there is a unitary representation of the group of spacetime translations, withinfinitesimal generators Pμ that satisfy the spectrum condition:

For any future-directed timelike vector a, the spectrum of Pa is in Rþ

This ensures positivity of the energy, with respect to any reference frame.

Ontology for Relativistic Collapse Theories 27

We assume a unique vacuum state that is invariant under all spacetime symmet-ries. Define the standard Hilbert space of the theory as the closure in norm of theset of all vectors that can be obtained by operating on the vacuum state withoperators constructed from standard fields. It follows from the Reeh–Schliedertheorem that, for any state ρ that is analytic in energy, for any α that is such that theset of points spacelike separated from α contains an open set, the null space of ρ isempty. If ρα σ1ð Þ, ρα σ2ð Þ, . . . , ρα σnð Þ are all states in the standard Hilbert space ofthe theory that are analytic in the energy, each of their null spaces consists solely ofthe zero vector, and hence Proposition 2 holds trivially.

The proposition is less trivial for theories that introduce nonstandard fields andwhose states go beyond the standard Hilbert space, as do the relativistic versions ofCSL due to Bedingham (2011a, b) and Pearle (2015). It can be shown, for anyindeterministic theory formulated within the framework sketched in Section 1.3that is set in Minkowski spacetime, it is necessary to go beyond the standardHilbert space (see Myrvold 2017).

Acknowledgments

Many thanks are due to the organizers of the workshop Identity, indistinguish-ability and non-locality in quantum physics (Buenos Aires, June 2017) and to theparticipants in that workshop, for their helpful comments. I would also like tothank Philip Pearle for comments and advice. I am grateful to Graham and GaleWright, who generously sponsor the Graham and Gale Wright DistinguishedScholar Award at the University of Western Ontario, for financial support ofthis work.

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Ghirardi, G. C., Grassi, R., and Benatti, F. (1995). “Describing the macroscopic world: Closingthe circle within the dynamical reduction program,” Foundations of Physics, 25: 5–38.

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Ghirardi, G. C., Grassi, R., Butterfield, J., and Fleming, G. N. (1993). “Parameter depend-ence and outcome dependence in dynamical models for state vector reduction,”Foundations of Physics, 23: 341–364.

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30 Wayne C. Myrvold

Shimony, A. (1991). “Desiderata for a modified quantum mechanics,” pp. 49–59 in A.Fine, M. Forbes, and L. Wessels (eds.), PSA 1990: Proceedings of the 1990 BiennialMeeting of the Philosophy of Science Association, Volume Two: Symposia and InvitedPapers. East Lansing, MI: Philosophy of Science Association. Reprinted in (1993),Search for a Naturalistic Worldview, Volume II: Natural Science and Metaphysics.Cambridge: Cambridge University Press.

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Ontology for Relativistic Collapse Theories 31

2

The Modal-Hamiltonian Interpretation: Measurement,Invariance, and Ontology

olimpia lombardi

2.1 Introduction

In the seventies, Bas van Fraassen (1972, 1974) proposed an approach to quantummechanics different than those of the best known interpretations. According tohim, although the quantum state always evolves unitarily (with no collapse), it is amodal element of the theory: It describes not what is the case but what may be thecase. This idea led several authors since the eighties to propose the so-called modalinterpretations (Kochen 1985, Dieks 1988, 1989, Vermaas and Dieks 1995, Dieksand Vermaas 1998, Bacciagaluppi and Dickson 1999, Bene and Dieks 2002), thatis, realist, noncollapse interpretations of the standard formalism of quantum mech-anics, according to which the quantum state assigns probabilities to the possiblevalues of all the properties of the system. But since the contextuality of quantummechanics (Kochen and Specker 1967) implies that it is not possible to consist-ently assign definite values to all the properties of a quantum system at a singletime, it is necessary to pick out, from the set of all observables of a quantum systemthe subset of definite-valued properties. The different modal interpretations differfrom each other mainly with respect to their rule of definite-value ascription (seeLombardi and Dieks 2017 and references therein).

Like most interpretations of quantum mechanics, the traditional modalinterpretations were specifically designed to solve the measurement problem. Infact, they successfully reached this goal in the case of ideal measurements.However, a series of articles of the nineties (Albert and Loewer 1990, 1991,1993, Elby 1993, Ruetsche 1995) showed that those traditional approaches basedon the modal views did not pick out the right properties for the apparatus innonideal measurements, that is, in measurements that do not introduce a perfectcorrelation between the possible states of the measured system and the possiblestates of the measuring apparatus. As ideal measurements can never be achieved inpractice, this shortcoming was considered a “silver bullet” for killing modal

32

interpretations (Harvey Brown, cited in Bacciagaluppi and Hemmo 1996). Thisexplains the decline of the interest in modal interpretations since the end of thenineties.

What was not sufficiently noticed in the nineties was the fact that the difficultiesof those original modal interpretations to deal with nonideal measurements wasdue not to their modal nature, but to the fact that their rule of definite-valueascription made the set of definite-valued observables to depend on the instantan-eous state of the system. An author who did notice this was Jeffrey Bub, whosepreference for Bohmian mechanics in those days can be understood in this context.In fact, if Bohmian mechanics is conceived as a member of the modal familywhose definite-valued observables are defined by the position observable (Bub1997), it turns out to be a natural alternative given the difficulties of the originalmodal interpretations.

Bub showed that the shortcomings of the original modal interpretations can beovercome by making the rule of definite-value ascription independent of thesystem’s state and only dependent on an observable of the system. This wascertainly an important step. Nevertheless, it was not sufficient to rehabilitate modalinterpretations in the eyes of most philosophers of physics. What was not realizedat that time is that position is not the only observable that can be appealed to inorder to define the state-independent rule of definite-value ascription of a modalinterpretation. It is in this point that the modal-Hamiltonian interpretation (MHI;Castagnino and Lombardi 2008, Lombardi and Castagnino 2008) entered thescene: The MHI endows the Hamiltonian of the quantum system with the role ofselecting its definite-valued observables. With this strategy, it not only solves theproblems of the original modal interpretations, but can also be successfully appliedto many physical situations. However, perhaps due to the shadow of doubt that stillcovers the entire modal interpretation project, the MHI did not receive a seriousattention by the community of the philosophers of physics. The present chapterintends to contribute toward modifying this situation by introducing the MHI in aconceptually clear and concise way, stressing its advantages both for facing thetraditional interpretive problems of quantum mechanics and for supplying a phys-ically meaningful account of relevant aspects of the theory.

For this purpose, the chapter is organized as follows. In Section 2.2, thetwo main interpretive postulates of the MHI will be introduced, emphasizingthe role played by the Hamiltonian in them. In Section 2.3, the measurementproblem is addressed from the MHI perspective; in particular, it will be arguedthat, beyond the formal von Neumann model, quantum measurement is asymmetry-breaking process that renders empirically accessible an otherwiseinaccessible observable of the system. Section 2.4 will be devoted to assessingthe MHI from the viewpoint of the invariances of the theory, in particular, of the

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 33

Galilei group. Finally, in Section 2.5, the ontological picture suggested by the MHIwill be described, stressing how that picture supplies a conceptually clear solutionto some traditional interpretive problems of quantum mechanics.

2.2 The Modal-Hamiltonian Interpretation

In this section, we shall introduce the MHI without discussing its advantages overother proposals. The arguments in its favor will become clear in the followingsections, where we will argue for its physical relevance and we will apply it tosolve some traditional interpretive challenges.

By adopting an algebraic perspective, the MHI defines a quantum system Sas a pair O;Hð Þ such that (i) O is a space of self-adjoint operators representingthe observables of the system, (ii) H 2 O is the time-independent Hamiltonianof the system S, and (iii) if ρ0 2 O

0(where O

0is the dual space of O) is the

initial state of S, it evolves according to the Schrödinger equation. Here we willassume that the space O is a C*-algebra, which can be represented in terms of aHilbert space H (Gelfand-Naimark-Segal [GNS] theorem). In this particularcase, O ¼ O

0and, therefore, O and O

0are represented by H ⊗H . Nevertheless,

O may be a different *-algebra, under the necessary conditions for itsrepresentation.

In this algebraic framework, the observables that constitute the quantum systemare the basic elements of the theory, and the states are secondary elements, definedin terms of the basic ones. The adoption of an algebraic perspective is not a merelyformal decision. As we will see in Section 2.5, when the logical priority ofobservables over states is transferred to the ontological domain, the space ofobservables turns out to embody the representation of the elemental items of theontology, and this has relevant interpretive consequences.

A quantum system so defined can be decomposed into parts in many ways;however, not any decomposition will lead to parts which are, in turn, quantumsystems. The expression “tensor product structure” (TPS) is used to call anypartition of a closed system S, represented in the Hilbert space H ¼ H A⊗H B,into parts SA and SB represented in H A and H B, respectively. Quantum systemsadmit a variety of TPSs, each one leading to a different entanglement between theirparts. However, there is a particular TPS that is invariant under time evolution: TheTPS is dynamically invariant when there is no interaction between the parts(Harshman and Wickramasekara 2007a, b). In other words, in the dynamicallyinvariant case the components’ behaviors are dynamically independent from eachother; each one evolves unitarily according to the Schrödinger equation. On thisbasis, according to the MHI, a quantum system can be split into subsystems whenthere is no interaction among the subsystems.

34 Olimpia Lombardi

Composite systems postulate: A quantum system S : O;Hð Þ, with initial stateρ0 2 O

0, is composite when it can be partitioned into two quantum systems

S1 : O1;H1� �

and S2 : O2;H2� �

, such that (i) O ¼ O1⊗O2 and (ii)H ¼ H1⊗I2 þ I1⊗H2 (where I1 and I2 are the identity operators in the correspond-ing tensor product spaces). In this case, we say that S1 and S2 are subsystems of thecomposite system S ¼ S1 þ S2. If the system is not composite, it is elemental.

With respect to the definite-valued observables, the basic idea of the MHI is thatthe Hamiltonian of the system, with its own symmetries, defines the subset of theobservables that acquire definite actual values. The group of the transformationsthat leave the Hamiltonian invariant is usually called Schrödinger group (Tinkham1964). In turn, each symmetry of the Hamiltonian leads to an energy degeneracy.The degeneracies with origin in symmetries are called “normal” or “systematic,”and those that have no obvious origin in symmetries are called “accidental”(Cohen-Tannoudji, Diu, and Lalöe 1977). However, a deeper study usually showseither that the accidental degeneracy is not exact or else that a hidden symmetry inthe Hamiltonian can be found that explains the degeneracy. For example, thedegeneracy in the hydrogen atom of states of different angular momentum l butthe same principal quantum number n arises from a four-dimensional rotationalsymmetry of the Hamiltonian in momentum space (Fock 1935). For this reason it isassumed that once all the symmetries of the Hamiltonian have been considered, abasis for the Hilbert space of the system is obtained and the “good quantumnumbers” are well defined.

Once these symmetry considerations are taken into account, the basic ideaof the MHI can be expressed by the classical Latin maxim Ubi lex nondistinguit, nec nos distinguere debemus (where the law does not distinguish,neither ought we to distinguish). The Hamiltonian of the system, with itssymmetries, is what determines which observables acquire definite values.This means that any observable whose eigenvalues would distinguish amongeigenvectors corresponding to a single degenerate eigenvalue of the Hamilto-nian does not acquire definite value, because such an acquisition wouldintroduce in the system an asymmetry not contained in the Hamiltonian. Oncethis idea is understood, the rule of definite-value ascription can be formulatedin a very simple way:

Actualization rule: Given an elemental quantum system S : O;Hð Þ, the actualdefinite-valued observables of S are H, and all the observables commuting with Hand having, at least, the same symmetries as H.

The justification for selecting the Hamiltonian as the preferred observableultimately lies in the physical relevance of the MHI and in its ability to solveinterpretive difficulties. These issues will be the content of the following sections.

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 35

2.3 The Modal-Hamiltonian View of Quantum Measurement

2.3.1 Measurement and Correlations

In general, the quantum measurement problem is presented in terms of the vonNeumann model, without framing it in the context of the measurement practices.But the purpose of a quantum measurement is not to discover the preexisting valueof a system’s observable, but to reconstruct, at least partially, the state of themeasured system. Therefore, the following distinction is in order:

• Single measurement: It is a single process, in which the reading of the pointer isregistered. A single measurement, considered in isolation, does not yet supplyrelevant information about the state of a measured system.

• Frequency measurement: It is a repetition of identical single measurements,whose purpose is to obtain the certain coefficients of the measured system’sstate on the basis of the frequencies of the pointer readings in many singlemeasurements.

A frequency measurement supplies relevant information about the state of thesystem, but is not yet sufficient to completely identify such a state. In order toreconstruct the state of the measured system it is necessary to perform a collectionof frequency measurements with different experimental arrangements.

The von Neumann model addresses the quantum measurement problem in theframework of the single measurement. This is completely reasonable to the extentthat, if we do not have an adequate explanation of the single case, we cannotaccount for the results obtained by the repetition of single cases. Nevertheless, itshould not be forgotten that a single measurement is always an element of ameasurement procedure by means of which, finally, frequencies are to be obtained.

Let us begin, then, by the single measurement. If, as in the original modalinterpretations, the rule of definite-value ascription depends on the instantaneousstate of the system, it is not surprising that it does not supply the expected resultin nonideal measurements: When the state of the composite system measuredsystem+apparatus does not introduce a perfect correlation between the eigenstatesof the measured system’s observable and the eigenstates of the apparatus’ pointer,it is not difficult to see that the pointer will not belong to the context of definite-valued observables. By contrast, if the rule of definite-value ascription does notdepend on the instantaneous state of the system, this problem does not arise. It canbe proved that the MHI explains the definite value of the measurement apparatus’observable both in the ideal and in the nonideal single measurements (for a formaldemonstration and physical examples, see Lombardi and Castagnino 2008: section6; for the account of consecutive measurements, see Ardenghi, Lombardi, andNarvaja 2013).

36 Olimpia Lombardi

While definite records of the apparatus’ pointer are obtained even in nonidealsituations, one can legitimately ask whether all nonideal measurements are equallyunsatisfactory. The MHI supplies a clear criterion to distinguish between reliableand nonreliable frequency measurements (for a detailed explanation and applica-tions, see Lombardi and Castagnino 2008: section 6). In the former case, thecoefficients of the measured system’s state can be computed on the basis of thefrequencies of the pointer’s readings is spite of the imperfect correlation; in thelatter case, the same computation would give inaccurate results (for a presentationof the reliability criterion in informational terms, see Lombardi, Fortin, and López2015). Albert and Loewer (1990, 1993) were right in claiming that the idealmeasurement is a situation that can never be achieved in practice: The interactionbetween measured system and measurement apparatus never introduces a com-pletely perfect correlation; in spite of this, physicists usually perform successfulmeasurements. The MHI account of the quantum measurement shows that perfectcorrelation is not a necessary condition for “good” measurements: The coefficientsof the system’s state at the beginning of the process can be approximately obtainedeven when the correlation is not perfect, if the reliability condition is satisfied.Nevertheless, both in the reliable and in the nonreliable case, a definite reading ofthe apparatus’ pointer is obtained in each single measurement.

2.3.2 Measurement and Symmetries

In the von Neumann model of a single measurement, the observable A of themeasured system S, whose eigenstates will be correlated with those of the pointerP, is considered in formal terms and deprived of any physical content. Then, theinteraction between S and the measuring apparatus M is only endowed with therole of introducing the correlation between A and P. However, the analysis ofphysical situations of measurement shows that there are further aspects to beconsidered beyond correlations.

Let us consider the free hydrogen atom, characterized by the Coulombicinteraction between nucleus and electron. Since the Hamiltonian is degeneratedue to its space-rotation invariance, the hydrogen atom is described in terms ofthe basis n; l;mlj if g defined by the complete set of commuting observables(CSCO) H; L2; Lz

� �. Nevertheless, that space-rotation invariance makes the selec-

tion of Lz a completely arbitrary decision: Given that space is isotropic, we canchoose Lx or Ly to obtain an equally legitimate description of the free atom. Thearbitrariness in the selection of the z-direction is manifested in spectroscopy by thefact that the spectral lines of the free hydrogen atom give no experimental evidenceabout the values of Lz: We have no empirical access to the number ml of the freeatom. The MHI agrees with those experimental results, because it does not assign a

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 37

definite value to Lz; the definite value of Lz would break the symmetry of theHamiltonian of the free hydrogen atom in a completely arbitrary way.

If we want to have empirical access to Lz, we need to apply a magnetic field Balong the z-axis, which breaks the isotropy of space and, as a consequence, thespace-rotation symmetry of the atom’s Hamiltonian. In this case, the symmetrybreaking removes the energy degeneracy in ml: Now Lz is not arbitrarily chosenbut selected by the direction of the magnetic field. However this, in turn, impliesthat the atom is no longer free: The Hamiltonian of the new system includes themagnetic interaction. As a consequence, the original degeneracy of the 2lþ 1ð Þ-fold multiplet of fixed n and l is now removed, and the energy levels turn out to bedisplaced by an amount Δωnlml , which is also function of ml: This is the manifest-ation of the so-called Zeeman effect. This means that the Hamiltonian, witheigenvalues ωnlml , is now nondegenerate: It constitutes by itself the CSCO Hf gthat defines the preferred basis n; l;mlj if g. According to the MHI’s rule of definite-value ascription, in this case H and all the observables commuting with H aredefinite-valued: Since this is the case for L2 and Lz, in the physical conditionsleading to the Zeeman effect, both observables acquire definite values.

Besides the free hydrogen atom and the Zeeman effect, the MHI was applied tomany other physical situations, leading to the results expected from a physicalviewpoint; e.g., the free-particle with spin, the harmonic oscillator, the fine struc-ture of atoms, the Born-Oppenheimer approximation (see Lombardi and Castag-nino 2008: section 5). Recently, the interpretation was applied to solve the problemof optical isomerism (Fortin, Lombardi, and Martínez González 2018), which isconsidered one of the deepest problems for the foundations of molecularchemistry.

All those physical situations show that we have no empirical access to theobservables that are generators of the symmetries of the system’s Hamiltonian;and, in the context of measurement, the observable A of the measured system Smay be one of those observables. This is also the case in the Stern–Gerlachexperiment, where Sz is a generator of the space-rotation symmetry ofHspin ¼ k S2; it is the interaction with the magnetic field B ¼ Bz that breaks theisotropy of space by privileging the z-direction and, as a consequence, breaks thespace-rotation symmetry of Hspin. Therefore, when the observable A to be meas-ured is a generator of a symmetry of the Hamiltonian of S, the interaction with theapparatus M must not only establish a correlation between A and the pointer P, butalso must break that symmetry. Therefore, from a physical viewpoint, measure-ment can be conceived as a process that breaks the symmetries of the Hamiltonianof the system to be measured and, in this way, turns an otherwise nondefinite-valued observable into a definite-valued and empirically accessible observable.This means that the formal von Neumann model of quantum measurement must be

38 Olimpia Lombardi

complemented by a physical model, in terms of which, measurement is a symmetrybreaking process that renders a symmetry generator of the system’s Hamiltonianempirically accessible.

2.4 The Modal-Hamiltonian Interpretation and the Role of Symmetries

2.4.1 The MHI and the Galilei Group

In contrast with the great interest of physicists in the symmetries of physicaltheories, the discussion on this topic has been scarce in the field of quantummechanics (Lévi-Leblond 1974). This situation is reflected in the field of theinterpretation of the theory, where the relevance of the Galilei group � thesymmetry group of nonrelativistic quantum mechanics � is rarely discussed inthe impressive amount of literature on the subject. This is a serious shortcoming inthe foundational context, because the fact that a theory is invariant under a groupdoes not guarantee the same property for its interpretations, to the extent that, ingeneral, they add interpretive assumptions to its formal structure. The MHI, on thecontrary, addresses the issue of the role and meaning of the Galilei transformationsin the interpretation: the study of whether and under what conditions the MHIsatisfies the physical constraints imposed by the Galilei group leads to interestingconsequences.

As it is well known, the invariance of the fundamental law of a theory under itssymmetry group implies that the behavior of the system is not altered by theapplication of the transformation: In terms of the passive interpretation of symmet-ries, the original and the transformed reference frames are equivalent. In theparticular case of nonrelativistic quantum mechanics, the application of a Galileitransformation does not introduce a modification in the physical situation, but onlyexpresses a change of the perspective from which the system is described.

Harvey Brown, Mauricio Suárez, and Guido Bacciagaluppi (1998) correctlypointed out that any interpretation that selects the set of the definite-valuedobservables of a quantum system is committed to explaining how that set istransformed under the Galilei group. This question is particularly pressing forrealist interpretations of quantum mechanics, which conceive a definite-valuedobservable as a physical property that objectively acquires an actual definite valueamong all its possible values: The actualization of one of the possible values has tobe an objective fact. Therefore, to the extent that the theory preserves its invarianceunder the Galilei group, the set of the definite-valued observables of a systemshould be left invariant by the Galilei transformations. From a realist viewpoint, itwould be unacceptable that such a set changed as the mere result of a change in theperspective from which the system is described. The MHI meets the challenge and

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 39

overcomes it successfully. In fact, it can be proved that, in the situations in whichthe Schrödinger equation remains invariant under the group, the set of definite-valued observables picked out by the modal-Hamiltonian rule of definite-valueascription also remains invariant (see Ardenghi, Castagnino, and Lombardi 2009,Lombardi, Castagnino, and Ardenghi 2010).

However the argument can also be developed in the opposite direction. Insteadof starting by the interpretation and considering its behavior under the transform-ations of the relevant group, one can begin by the group of symmetry and ask forthe constraints that it imposes on interpretation. In the case of nonrelativisticquantum mechanics, the objectivity of the definite-valued observables must bepreserved by making them invariant under the Galilei group. The natural way toreach this goal is to appeal to the Casimir operators of the Galilei group, which bydefinition are the operators invariant under all the transformations of the Galileigroup. If the interpretation has to select a Galilei-invariant set of definite-valuedobservables, the members of such a set must be the Casimir operators of the groupor functions of them. The central extension of the Galilei group has three Casimiroperators which, as such, commute with all the generators of the group: They arethe mass operatorM, the squared-spin operator S2, and the internal energy operatorW ¼ H � P2=2mW. The eigenvalues of the Casimir operators label the irreduciblerepresentations of the group; so, in each irreducible representation, the Casimiroperators are multiples of the identity: M ¼ mI, where m is the mass;S2 ¼ s sþ 1ð ÞI, where s is the eigenvalue of the spin S; and W ¼ wI, where w isthe scalar internal energy.

This result, which places the Casimir operators of the group in the center of thestage, may seem to disagree with an interpretation such as the MHI, which endowsthe Hamiltonian with the leading role. The definite-valuedness of the mass operatorM and the squared-spin operator S2 are compatible with the MHI rule of definite-value ascription, because both commute with H and do not break its symmetries(they are multiples of the identity). But the Hamiltonian is not a Casimir operatorof the Galilei group; the Casimir operator is the internal energy. Nevertheless, thedisagreement is only apparent. The Hamiltonian is the sum of the internal energyand the kinetic energy of the system. But the kinetic energy can be disregarded:When the system is described in a reference frame at rest with respect to its centerof mass, then the kinetic energy turns out to be zero and the Hamiltonian isidentified with the internal energy. This means that the internal energy is themagnitude that carries the physically meaningful structure of the energy spectrum,whereas the kinetic energy represents an energy shift that is physically nonrelevantand merely relative to the reference frame used for the description.

Summing up, the modal-Hamiltonian interpretation can be reformulated in anexplicitly invariant form, according to which the definite-valued observables of a

40 Olimpia Lombardi

quantum system are (i) the observables Ci represented by the Casimir operators ofthe Galilei group in the corresponding irreducible representation, and (ii) all theobservables commuting with the Ci and having, at least, the same symmetries asthe Ci (Lombardi, Castagnino, and Ardenghi 2010). Therefore, the interpretationshould be more precisely referred to by the name “modal-Casimir interpretation,”although in the case of nonrelativistic quantum mechanics the original name is alsoadequate.

2.4.2 Interpretation and Symmetry

Now we can come back to the question about the constraints that the Galilei groupimposes on the interpretation of quantum mechanics, but now independently of theMHI. Let us recall that the application of a transformation belonging to thesymmetry group of a theory does not introduce a modification in the physicalsituation, but only expresses a change of the perspective from which the system isdescribed. This leads to the natural idea, expressed by a wide spectrum of authors(e.g., Minkowski 1923, Weyl 1952, Auyang 1995, Nozick 2001), that the invari-ance under the relevant group is a mark of objectivity.

On the other hand, as a consequence of the Kochen-Specker theorem (1967), itis necessary to pick out, from the set of all observables of a quantum system, thesubset of observables that may have definite values. In turn, from a realistviewpoint, the fact that certain observables acquire an actual definite value is anobjective fact in the behavior of the system; therefore, the set of definite-valuedobservables selected by a realist interpretation must be also Galilei-invariant. Butthe Galilei-invariant observables are always functions of the Casimir operators ofthe Galilei group. As a consequence, one is led to the conclusion that any realistinterpretation that intends to preserve the objectivity of the set of the definite-valued observables may not stand very far from the MHI (Lombardi and Fortin2015).

The invariance of the Schrödinger equation holds for the case of isolatedsystems, that is, in the case that there are no external fields applied on the system.Since, in nonrelativistic physics, fields are not quantized, the effect of externalfields on the system has to be accounted for by its Hamiltonian: The potentialshave to modify the form of the Hamiltonian because it is the only observableinvolved in the time-evolution law. As a consequence, in the presence of externalfields, the Hamiltonian is no longer the generator of time-displacements; it onlyretains its role as the generator of the dynamical evolution (see Laue 1996,Ballentine 1998). In turn, since the Hamiltonian includes the action of the fields,the result of the action of the Galilei transformation on it must be computed in eachcase, and the Galilei invariance of the Schrödinger equation can no longer be

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 41

guaranteed. This fact suggests the possibility of generalizing the idea of relying onsymmetry groups in two senses.

It cannot be expected that relativistic quantum mechanics be invariant underthe Galilei group, given the fact that it includes the action of electric andmagnetic fields described by a theory that is not Galilei invariant, but Poincaréinvariant. In turn, in quantum field theory, fields are quantum items, notexternal fields acting on a quantum system; as a consequence, the generatorsof the Poincaré group do not need to be reinterpreted in the presence of externalfactors. These facts lead to generalize the group-based interpretive ideas: Therealist interpretation, expressed in terms of the Casimir operators of the Galileangroup in nonrelativistic quantum mechanics, can be transferred to the relativisticdomain by changing the symmetry group accordingly – the definite-valuedobservables of a system in relativistic quantum mechanics and in quantum fieldtheory would be those represented by the Casimir operators of the Poincarégroup. Since the mass operator M and the squared-spin operator S2 are the onlyCasimir operators of the Poincaré group, they would always be definite-valuedobservables. This conclusion agrees with a usual physical assumption: Elemen-tal particles always have definite values of mass and spin, and those values areprecisely what define their different kinds. Moreover, the classical limit ofrelativistic theories manifests the limit of the corresponding Casimir operators(see Ardenghi, Castagnino, and Lombardi 2011): There is a meaningful limitingrelation between the observables that acquire definite values according torelativistic theories and those that acquire definite values according to nonrela-tivistic quantum mechanics.

These group-based interpretive ideas can be further generalized in a secondsense. If invariance is a mark of objectivity, there is no reason to focus only onspacetime global symmetries. Internal or gauge symmetries should also be con-sidered as relevant in the definition of objectivity and, as a consequence, in theidentification of the definite-valued observables of the system. For instance, inrelativistic quantum mechanics a gauge symmetry is what identifies the charge asan objective quantity. Therefore, a realist interpretation can be extended to thegauge symmetries of the theory: The observables represented by operators invari-ant under those symmetries are also definite-valued observables according to thetheory.

In summary, besides its wide applicability in the nonrelativistic quantumdomain, the MHI opens the way for a general interpretive strategy, valid for anyrealistic view of quantum theories – the definite-valued observables of a system,whose behavior is governed by a certain theory, are the observables invariant underall the transformations corresponding to the symmetries of the theory, both exter-nal and internal.

42 Olimpia Lombardi

2.5 A Modal Ontology of Properties

2.5.1 The Structure of the Ontology

Traditionally, the interpretations of quantum mechanics concentrate their efforts onthe interpretive challenges of the theory. For instance, they focus on searching asolution of the measurement problem without falling beyond the limitationsimposed by the no-go quantum theorems. Due to their difficulty, these tasksusually lead people to disregard ontological issues, in particular, the questionsabout the nature of the items referred to by quantum mechanics. The MHI hastackled the ontological questions from the very beginning.

As explained in Section 2.2, the MHI adopts an algebraic perspective. Thisdecision about the formalism is not confined to the formal domain, but rather hasrelevant consequences about the structure of the ontology referred to by quantummechanics, in particular, about the basic categories of such an ontology. In fact,when the logical priority of observables over states is transferred to the ontologicaldomain, the space of observables turns out to embody the representation of theelemental items of the ontology – observables (mathematically represented by self-adjoint operators) ontologically represent type-properties, and the values of theobservables (mathematically represented by the eigenvalues of the correspondingoperators) ontologically represent the possible case-properties corresponding tothose type-properties. Among the possible case-properties of a type-property, onlyone acquires a definite value (Lombardi and Castagnino 2008: section 8).

In this modal ontology of properties, a quantum system is a bundle of properties:type-properties with their corresponding case-properties. The notion of bundle ofproperties is a well-known idea in contemporary metaphysics: Philosophers of theempiricist tradition have preferred to replace the traditional picture of properties“stuck” to an underlying and unobservable substance by an ontological realmwhere individuals are nothing but bundles of properties. Properties have meta-physical priority over individuals; therefore, they are the fundamental items of theontology. However, the view of bundles of properties that is appropriate forquantum mechanics does not agree with the “bundle theory” of twentieth-centuryanalytic metaphysics concerning two aspects.

In the first place, according to the traditional versions of the bundle theory, anindividual is the confluence of certain case-properties, under the assumption thatthe corresponding type-properties are all determined in terms of actual definitevalues. For instance, a particular ball is the confluence of a definite position, say,on the chair; a definite shape, say, round; a definite color, say, white; etc. The ballis the bundle of those actual case-properties. In general, bundle theories identifyindividuals with bundles of actual properties. By contrast, in the framework of theMHI, a system is identified by its space of observables, which defines all the

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 43

admissible type-properties with their corresponding possible case-properties.Therefore, a quantum system is a bundle of possible case-properties; it inhabitsthe realm of possibility and manifests itself only partially in the realm of actuality.

This ontological interpretation embodies a possibilist conception of possibility,as opposed to an actualist view, which reduces possibility to actuality. Accordingto possibilism, possibility is an ontologically irreducible feature of reality. Possibleitems � possibilia � constitute a basic ontological category (see Menzel 2007). Inother words, possibility is a way in which reality manifests itself, a way independ-ent of and not less real than actuality. The reality of possibilia is manifested by thefact that they may produce definite effects on actual reality even if they neverbecome actual (e.g., “non-interacting experiments” of Elitzur and Vaidman 1993,Vaidman 1994).

The second specific aspect of this quantum-bundle view is related to the way inwhich bundles are conceived. In the traditional versions of the bundle theory, theclaim is that individuals are bundles of properties; therefore, it is necessary to findwhat confers individuality to individuals. In general, the task is fulfilled by somesubset of the bundle’s properties, together with some further principle that ensuresthat no other individual must possess that subset and that preserves the identity ofthe individual through change. By contrast, due to the indistinguishability of“identical particles,” quantum mechanics poses a serious challenge to the notionof individual, either in the substratum-properties picture or in the bundle picture(see French and Krause 2006 and references therein). The identification of thecomplexions resulting from the permutations of identical particles makes thenotion of individual run into trouble.

The MHI tackles the problem by endorsing the idea that quantum systems arenot individuals – they are strictly bundles, and there is no principle that permitsthem to be subsumed under the ontological category of individual. Regrettably,this ontological picture is not properly captured by any formal theory whoseelemental symbols are variables of individual. An ontology populated by bundlesof possible properties cries for a “logics of predicates,” in the spirit of the “calculusof relations” proposed by Alfred Tarski (1941), where individual variables areabsent.

2.5.2 One Ontology, Many Solutions

Quantum mechanics poses different ontological problems – contextuality preventsthe simultaneous assignment of determinate values to all the properties of thesystem, nonseparability seems to undermine the independent existence of nonin-teracting systems, and indistinguishability challenges the traditional category ofindividual. The usual strategies focus only on one of these problems: They design

44 Olimpia Lombardi

an interpretation to solve it, disregarding the remaining difficulties. With itsontology of possible properties, the MHI aspires to provide a “global” approach,which solves most problems in terms of a single ontology.

The Kochen-Specker theorem expresses the impossibility of ascribing actualcase-properties to all the type-properties of the system in a noncontradictorymanner. The classical idea of a bundle of actual properties does not work in thequantum ontology. But this is not a difficulty for the MHI, which conceives thequantum system as the bundle of all the possible type-properties with theircorresponding case-properties, as defined by the space of observables. This onto-logical view is immune to the challenge represented by the Kochen-Speckertheorem, because this theorem imposes no restriction on possibilities (see da Costa,Lombardi, and Lastiri 2013).

Quantum nonseparability is the consequence of the nonfactorization of entan-gled states. When the states are assigned to individual systems that interacted in thepast, the difficulty is to explain the correlations between the values of observablesbelonging to noninteracting systems, which typically are separated in space. Theassumption of collapse leads to understand nonseparability as nonlocality, at riskof falling into the “spooky action at a distance” reported by Albert Einstein.Without collapse, nonseparability seems to imply a kind of holism, in the sensethat quantum systems are not composed by what are commonly conceived as theirsubsystems; but this idea can hardly be compatibilized with the view of systems asindividuals, that is, entities that preserve their identity through change. For theMHI, the interpretation of nonseparability as holism does not represent a difficulty.Since quantum systems are strictly bundles and not individuals, there is noprinciple of individuality that preserves the individuality of the component systemsin the composite system (see da Costa and Lombardi 2014). The composite systemis a single bundle, where the identity of the components is not retained. Therefore,the new bundle-system acts and reacts as a whole – there are not subsystems whosestate nonseparability must be explained or whose correlations seem to implyinstantaneous action at a distance.

The same idea of the “dissolution” of component bundles in the compositebundle is what allows the MHI to face the problem of indistinguishability. In thediscussions about the indistinguishability of “identical particles,” the problem isusually formulated in terms of the possible combinations (complexions) that can beobtained in the distribution of particles over possible states. The problem is, then,to explain why a permutation of particles does not lead to different complexions inquantum statistics. This feature is introduced in the formalism as a restriction onnonsymmetric states, but the strategy has an unavoidable ad hoc flavor in thecontext of the theory. According to the MHI ontology, when a bundle is the resultof the combination of identical bundles, it can be expected that the result does not

The Modal-Hamiltonian Interpretation: Measurement, Invariance, and Ontology 45

depend on the order in which the original identical bundles are considered; thecombination of identical bundles must be commutative. This commutativity ismanifested by the fact that the observables that constitute the resulting bundle-system are represented by operators symmetric with respect to the permutation ofthe indices coming from the original identical bundles. Here symmetry is not an adhoc assumption but a consequence of an ontological feature. When the expectationvalues of these symmetric observables are computed, only the symmetric part ofthe state has an effect. The nonsymmetric part is superfluous, because it plays norole in the physically measurable magnitudes (see details in da Costa et al. 2013).Therefore, symmetrization is not the result of an ad hoc strategy, but is due toontological reasons: The symmetry properties of states are a consequence of thesymmetry of the observables of the whole composite system, which is, in turn, aconsequence of the ontological picture supplied by the interpretation. In otherwords, from the perspective given by the modal-Hamiltonian interpretation,indistinguishability is not a relation between particles manifested in statistics, butrather an internal symmetry of a single bundle of properties.

In summary, according to the MHI, the talk of individual entities and theirinteractions can be retained only in a metaphorical sense. In fact, even the numberof particles is represented by an observable, and superpositions of different particlenumbers are theoretically possible. This fact, puzzling from an ontology populatedby individuals, involves no mystery in an ontology of properties: If quantumsystems are bundles of possible properties, the particle picture, with a definitenumber of particles, is only a contextual picture valid exclusively when the numberof particles satisfies the constraints of the rule of definite-value ascription. In othercases, wave packets may remain narrow and more or less localized during arelatively long time. In this way, particle-like behavior can temporarily emerge –

wave packets can represent approximately definite positions and can follow anapproximately definite trajectory (see Lombardi and Dieks 2016). Moreover, theMHI has proved to be compatible with the theory of decoherence (Lombardi 2010,Lombardi, Ardenghi, Fortin, and Castagnino 2011, Lombardi, Ardenghi, Fortin,and Narvaja 2011). Nevertheless, those particular situations do not undermine thefact that quantum systems are nonindividual bundles of properties.

2.6 Conclusions and Perspectives

The MHI has been developed and successfully articulated in many directions sinceits first presentation in 2008. Of course, this does not mean that any interpretivequestion about quantum mechanics has already been solved. Nevertheless, giventhe results obtained up to this moment, it deserves to be taken into account andfurther explored.

46 Olimpia Lombardi

There are several issues that can still be faced from this interpretive frame-work. A very interesting question is that related to the interpretation of externalfields in a theory that, as quantum mechanics, does not treat fields as quantizedentities. In particular, the Aharonov-Bohm effect is worthy of being analyzedfrom an ontology-of-properties view. Another topic to be examined is how theMHI is in resonance with a closed-system view of decoherence (Castagnino andLombardi 2004, 2005a, b, Castagnino, Laura, and Lombardi 2007, Castagnino,Fortin, and Lombardi 2010, 2014), according to which decoherence is a processrelative to the selected partition of a closed system and how this leads to a top-down view of quantum mechanics based on an algebraic view that turnsentanglement and discord also into relative phenomena (for initial ideas, seeLombardi, Fortin, and Castagnino 2012, Fortin and Lombardi 2014, Lombardiand Fortin 2016). Finally, the natural subsequent interpretive step consists inextending the MHI to quantum field theory, not only regarding the definite-valued observables, but also with respect to the ontology referred to by thetheory. In particular, an ontology-of-properties view seems to favor a field viewin the debate on fields vs. particles, but without representing an obstacle toexplaining the emergence of the nonrelativistic quantum ontology. These differ-ent problems will guide the future research on the further development andextrapolation of the MHI.

Acknowledgments

I am grateful to the participants of the workshop Identity, indistinguishability andnon-locality in quantum physics (Buenos Aires, June 2017) for their useful com-ments. This work was made possible through the support of Grant 57919 from theJohn Templeton Foundation and Grant PICT-2014–2812 from the NationalAgency of Scientific and Technological Promotion of Argentina.

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Weyl, H. (1952). Symmetry. Princeton, NJ: Princeton University Press.

50 Olimpia Lombardi

3

Quantum Mechanics and Perspectivalism

dennis dieks

3.1 Introduction: Early Hints of Noncollapse and Perspectivalism

In introductions to quantum mechanics it is standard to introduce “collapses of thewave function” in order to avoid the occurrence of superpositions of statesassociated with different macroscopic properties. The paradigm case is the quan-tum mechanical treatment of measurement: If the interaction between a quantumsystem and a measuring device is described by means of unitary Schrödingerevolution, the composite system of object plus device will generally end up in anentangled state that is not an eigenstate of the measured observable, but rather asuperposition of such states. However, successful measurements end with therealization of exactly one of the possible outcomes, so it appears plausible that atsome stage during the measurement interaction unitary evolution is suspended anda collapse to one of the terms in the superposition takes place.

However, experimental research of the last few decades has undermined thismotivation for the introduction of collapses. “Schrödinger cat states,” i.e., super-positions of distinguishable quantum states of mesoscopic or even practicallymacroscopic physical systems, are now routinely prepared in the laboratory, andinterference between the different terms in the superpositions have abundantlybeen verified (see e.g., Johnson et al. 2017 for a sample of recent developments).This lends inductive support to the hypothesis that superpositions never reallycollapse, but are merely difficult to detect in everyday situations. In such situationshuge numbers of environmental degrees of freedom come into play, so that themechanism of decoherence may be invoked as an explanation for the practicalunobservability of interference between macroscopically different states understandard conditions. This line of thought leads in the direction of noncollapseinterpretations of quantum mechanics.

The evidence against collapses has not yet affected the textbook tradition, whichhas not questioned the status of collapses as a mechanism of evolution alongside

51

unitary Schrödinger dynamics. However, the relevant views of the pioneers ofquantum mechanics were not at all clear-cut. The locus classicus for the introduc-tion and discussion of collapses is chapter 6 of von Neumann’s (1932) Mathemat-ical Foundations of Quantum Mechanics. In this chapter von Neumann underlinesthe fundamental difference between collapses � occurring in measurements � andunitary evolution, but connects this difference to the distinction between, on theone hand, the experience of an observer and, on the other hand, external descrip-tions (in which the observer is treated in the same way as the other physicalsystems involved in the measurement interaction). In the external description vonNeumann assumes unitary evolution, with superpositions (also involving theobserver) as an inevitable consequence. Nevertheless, von Neumann states thatthe content of the observer’s “subjective experience” corresponds to only onesingle term in the superposition.

So the distinction between collapses and unitary evolution for von Neumann isnot a distinction between two competing and potentially conflicting physicalinteraction mechanisms on the same level of description, but rather concerns whatcan be said in relation to two different points of view � an idea taken up anddeveloped by London and Bauer (discussed later).

Niels Bohr also took the position that the standard rules of quantum mechanicsapply even to measuring devices and other macroscopic objects, so that strictlyspeaking these cannot be characterized by sets of precise values of classicalquantities (even though folklore has it that Bohr assumed that quantum mechanicsdoes not apply to the macroscopic world, see Dieks 2017 for an extensive discus-sion). Thus, in 1935 Bohr wrote that “a purely classical account of the measuringapparatus still implies the necessity of latitudes corresponding to the uncertaintyrelations. If spatial dimensions and time intervals are sufficiently large, thisinvolves no limitation” (Bohr 1935). And in 1948 he commented in the same vein:“We may to a very high degree of approximation disregard the quantum characterof the measuring instruments if they are sufficiently heavy” (Bohr 1948). Althoughfor Bohr there is thus no difference of principle between macro and micro objects,he does assign a special role to the observer and to the “conditions of measure-ment” (the specific experimental setup, chosen by the observer). However, he doesnot acknowledge a sui generis measurement dynamics but rather refers to thespecific epistemological vantage point of the observer, who can only communicatewhat he finds by using definite values of classical quantities (the paradigm casebeing the assignment of a definite value of either position or momentum,depending on the chosen kind of measuring device).

According to Bohr, the object that is being measured and the measuring deviceform, in each individual case, one insoluble whole so that “an independent realityin the ordinary physical sense can not be ascribed to the phenomena” (Bohr 1928).

52 Dennis Dieks

The properties of a quantum system according to Bohr only become well defined inthe context of the system’s coupling to a measuring device � which points in thedirection of a relational nature of physical properties.

A more formal analysis of quantum measurements, close to von Neumann’saccount, was given by London and Bauer in their 1939 booklet on the Theory ofObservation in Quantum Mechanics. London and Bauer consider three interactingsystems: x, the object system, y, a measuring device, and z, the observer. As aresult of the unitary evolution of the combined object-device system, an entangledstate will result:

Pkck xj ik yj ik. When the observer reads off the result of the

measurement, a similar unitary evolution of the x,y,z system takes place, so thatthe final state becomes: Ψj i ¼ P

kck xj ik yj ik zj ik. London and Bauer (1939: 41–42)comment:

“Objectively” � that is, for us who consider as “object” the combined system x,y,z � thesituation seems little changed compared to what we just met when we were onlyconsidering apparatus and object. . . . The observer has a completely different viewpoint:for him it is only the object x and the apparatus y which belong to the external world, tothat what he calls “objective.” By contrast, he has with himself relations of a very specialcharacter: he has at his disposal a characteristic and quite familiar faculty which we can callthe “faculty of introspection.” For he can immediately give an account of his own state. Byvirtue of this “immanent knowledge” he attributes to himself the right to create his ownobjectivity, namely, to cut the chain of statistical correlations expressed byP

kck xj ik yj ik zj ik by stating “I am in the state zj ik,” or more simply “I see yk” or evendirectly “X ¼ yk .” [Here X stands for the observable whose value is measured by theapparatus.]

It is clear from this quote and the further context that London and Bauerbelieved that there is a role for human consciousness in bringing about a definitemeasurement outcome � even though they also assumed, like von Neumann, that“from the outside” the observer, including his consciousness, can be described in aphysicalist way, by unitary quantum evolution (see Jammer 1974). The appeal toconsciousness can hardly be considered satisfactory, though: It appears to invoke adeus ex machina, devised for the express purpose of reconciling unitary evolutionwith definite measurement results. More generally, the hypothesis that the definite-ness of the physical world only arises as the result of the intervention of (human?)consciousness does not sit well with the method of physics.

Although certain elements of London and Bauer’s solution are therefore hard toaccept, the suggestion that it should somehow be possible to reconcile universalunitary evolution and the resulting omnipresence of entangled states, with theoccurrence of definite values of physical quantities, appears plausible. Indeed,the theoretical framework of quantum mechanics itself (as opposed to modifica-tions of the quantum formalism, as in the Ghirardi–Rimini–Weber [GRW] theory)

Quantum Mechanics and Perspectivalism 53

does not in a natural way leave room for another dynamical process beside unitaryevolution; e.g., there is no time scale or scale of complexity at which this alterna-tive evolution could set in. As already mentioned, empirical results support thisverdict. Accordingly, in the next sections we shall investigate whether the earlyintuitions about the universality of unitary evolution, excluding collapse as adynamical process, can be salvaged in a purely physicalist way. We shall arguethat “perspectival” noncollapse interpretations capture the intuitions behind theLondon and Bauer and von Neumann analyses, without an appeal to consciousnessor human observers.

3.2 Relational Aspects of Noncollapse Interpretations

The common feature of noncollapse interpretations is that they single out unitaryevolution (Schrödinger evolution or one of its relativistic generalizations) as theonly way that quantum states evolve in time. Consequently, entangled quantumstates generally result after interactions, even after interactions with macroscopicobjects like measuring devices. Of course, the task is to reconcile this with thedefinite states of affairs encountered in experience.

There are several proposals for such a reconciliation. The best known isprobably the many-worlds interpretation, according to which each individual termin a superposition that results from a measurement-like interaction represents anactual state of affairs, characterized by definite values of some set of observables.In this approach there are many actual states of affairs, worlds, or “branches,”living together in a “super-universe.” The experience of any individual observer isrestricted to one single branch within this super-universe. In other words, theexperienced world is the part of the super-universe that is accessible from theobserver’s perspective (a relational aspect of the scheme, which is the reason thatEverett in 1957 first introduced it as the “relative state” formulation of quantummechanics).

A second category of interpretations, modal interpretations, holds that there isonly one actual reality, so that all except one of the “branches” of the totalentangled state do not correspond to actual worlds but rather to unrealized possi-bilities � “modalities.” Some of these modal interpretations make the assumptionthat there is one a priori preferred observable (or set of commuting observables)that is always definite-valued in each physical system, others assume that the set ofdefinite-valued quantities depends on the form of the quantum state and cantherefore change over time (see Bub 1997, Dieks and Vermaas 1998, Lombardiand Dieks 2017 for overviews). To the first category belongs the Bohm (1952)interpretation, in which position is always definite, and the modal-Hamiltonianinterpretation (Lombardi and Castagnino 2008), according to which energy plays a

54 Dennis Dieks

privileged role. An example of the second category is the proposal according towhich the bi-orthogonal (Schmidt) decomposition of the total state determines thedefinite quantities of partial systems (namely, the quantities represented by theprojection operators projecting on the basis vectors that diagonalize the partialsystem’s density matrix; see Vermaas and Dieks 1995); another proposal is tomake decoherence responsible for the selection of definite quantities, in the sameway as is now standard in many-worlds accounts.

Other noncollapse approaches are the consistent-histories interpretation(Griffiths 2017) and Rovelli’s relational interpretation (Laudisa and Rovelli2013). Interestingly, the latter interpretation posits from the outset that the dynam-ical properties of any physical system are purely relational and only becomedefinite with respect to some other system when an interaction between the twosystems (in the formalism described by unitary quantum evolution) correlates thesystems (so that there is an “exchange of information” between them).

However, relational features also have a natural place in most of the other just-mentioned noncollapse interpretations (although not in all of them), as can beillustrated by further considering the situation discussed by London and Bauer(1939: section 1) � which essentially is the well-known “Wigner’s Friend”thought experiment.

Suppose that an experimentalist (our friend, who is a perfect observer) performsa quantum measurement within a hermetically sealed room. Let us say that the spinof a spin-½ particle is measured in a previously fixed direction, and that theexperimentalist notes the outcome (either +½ or �½). After some time we, whoare outside the room, will be sure that the experiment is over and that our friendwill have observed a definite result. Yet, we possess no certainty about theoutcome. In a classical context we would therefore represent the state of the roomand its contents by an ignorance mixture over states: There are two possibilities(“up” and “down”), both with probability ½.

However, in unitary quantum mechanics the situation is different in an importantrespect. According to the von Neumann measurement scheme, the final situation ofthe room after the experiment, including a record of the friend’s observation, willbe given by a linear superposition of terms, each containing a definite spin state ofthe particle coupled to a state of our experimentalist in which he is aware of thespin value he found. For us outside, this superposition is the correct theoreticaldescription of the room and its contents, and this (coherent) superposition isdifferent from an (incoherent) ignorance mixture over different possible states.As mentioned in the previous section, experience supports the ascription of thissuperposed state: Experiments with Schrödinger cat states demonstrate that weneed the superposition to do justice to the experimental facts. For example, if weare going to measure the projection operator Ψj i Ψh j (where Ψj i stands for the

Quantum Mechanics and Perspectivalism 55

superposed state of the room and its contents), the formalism tells us that we shallfind the result “1” with certainty; this is different from what a mixed state wouldpredict. Experiment confirms predictions of this kind.

But we also possess robust experience about what happens when we watch anexperiment while finding ourselves inside a closed laboratory room: There will bea definite outcome. It therefore seems inevitable to accept that during the experi-ment our friend becomes aware of exactly one spin value. As stated by London andBauer, our friend will be justified in saying either “the spin is up” or “the spin isdown” after the experiment. The dilemma is that we, on the outside, can onlyderive an “improper mixture” as a state for the particle spin and that well-knownarguments forbid us to think that this mixture represents our ignorance about theactually realized spin-eigenstate (indeed, if the spin state actually was one of the upor down eigenstates, it would follow that the total system of room and its contentshad to be an ignorance mixture as well, which conflicts with the premise �supported both theoretically and empirically � that the total state is asuperposition).

Our proposed perspectival way out of this dilemma is to ascribemore than one stateto the same physical system. In the case under discussion, with respect to us,representing the outside point of view, the contents of the laboratory room arecorrectly described by an entangled pure state so that we should ascribe impropermixtures (obtained by “partial tracing”) to the inside observer, the measuring deviceand the spin particle. But with respect to the inside observer (or with respect to themeasuring device in the room) the particle spin is definite-valued. So the insideobserver assigns a state to his environment that appropriately reflects this definiteness.

This line of thought leads to the idea of assigning relational or perspectivalstates, i.e., states of a physical system A from the perspective of a physical systemB. This step creates room for the possibility that the state and physical properties ofa system A are different in relation to different “reference systems” B. As suggestedby the examples, this move may make it possible to reconcile the unitary evolutionduring a quantum measurement with the occurrence of definite outcomes. Theproperties associated with the superposition and the definite outcomes, respect-ively, would relate to two different perspectives � the idea already suggested byvon Neumann and by London and Bauer. Of course, we should avoid the earlierproblems associated with consciousness. The different perspectives, and differentrelational states, should therefore be defined in purely physical terms.

The idea as just formulated was tentative: We spoke in a loose way of “states,”thinking of wave functions (or vectors in Hilbert space) without specifying whatthe attribution of quantum states to physical systems means on the level of physicalquantities, i.e., in terms of physical properties of the systems concerned. In fact,this physical meaning is interpretation-dependent.

56 Dennis Dieks

In the many-worlds interpretation the perspectival character of quantum states,for the Wigner’s Friend–type of scenario that we just discussed, translates into thefollowing physical account. When the measurement interactions within the her-metically sealed room have completely ended, the contents in the room have splitinto two copies: one in which the outcome +½ has been realized and observed andone with the outcome �½. However, we as external observers can still verify thesuperposed state by measuring an observable like Ψj i Ψh j, so that for us the two“worlds” inside the room still form one whole. Apparently, the splitting (branch-ing) of worlds that happens in measurements cannot be a global process, extendingover the whole universe at once, but must be a local splitting that propagates withthe further physical interactions that take place (see Bacciagaluppi 2002). There-fore, although we know (if we reason in terms of the many-worlds interpretation)that there are two copies of our friend inside, each having observed one particularoutcome, we still consider the room plus its contents as represented by the coherentsuperposition that corresponds to the definite value “1” of the physical quantityrepresented by the observable Ψj i Ψh j. So here we encounter a perspectivalism onthe level of physical properties: There exists a definite spin value for the internalobserver but not for his external colleague.

The same type of story can be told in those modal interpretations in which thedefinite-valued physical properties of systems are defined by their quantum states(one detailed proposal for how to define physical properties from the quantum statecan be found in Bene and Dieks 2002 and Dieks 2009). The main difference withthe many-worlds account is that now the interactions within the room do not leadto two worlds but to only one, with either spin up or spin down. Additionally, inthis case there is a definite spin value for the internal observer whereas from theoutside it is rather the observable Ψj i Ψh j (and observables commuting with it) thatis definite-valued, which conflicts with the attribution of a value to the spin � eventhough outside observers may be aware that for their inside counterpart there issuch a value.

Rovelli’s relational interpretation, which takes part of its inspiration fromHeisenberg’s heuristics in the early days of quantum mechanics, says that aquantity of system B only becomes definite for A when an interaction (a measure-ment) occurs between A and B (Rovelli 1996). In our Wigner’s Friend scenario,this again leads to the verdict that the internal interactions in the laboratory roomlead to a situation in which the spin is definite with respect to an internal observerbut not for an external one. Only when (and if ) external observers enter the roomand interact with the spin system does the spin become definite for them as well.

In all these cases we obtain accounts that are similar to the London and Baueranalysis, but with the important distinction that nonphysical features do not enterthe story. It should be noted that the relational properties introduced here are

Quantum Mechanics and Perspectivalism 57

intended to possess an ontological status: It is not the case that for an outsideobserver the internal spin values are definite but unknown. The proposal is that thespin really is indeterminate with respect to the world outside the laboratory room.

This perspectivalism with respect to properties does not seem an inevitablefeature of all noncollapse interpretations, however. In particular, those interpret-ations of quantum mechanics in which it is assumed that there exists an a priorigiven set of preferred observables that is always definite � in all physical systems,at all times, and in all circumstances � are by construction at odds with theintroduction of a definiteness that is merely relative. The Bohm interpretation isa case in point. According to this interpretation all physical systems are composedof particles that always possess a definite position, as a monadic attribute inde-pendent of any perspective. So in our sealed-room experiment the instantaneoussituation inside is characterized by the positions of all particles in the room, andthis description is also valid with respect to the outside world � even though anoutside observer will usually lack information about the exact values of thepositions. Thus for an external observer there exists one definite outcome of theexperiment inside, corresponding to one definite particle configuration. The out-come of any measurement on the room as a whole that the outside observer mightperform again corresponds to a definite configuration of particles with well-definedpositions. The fact that this value is not what we would classically expect (forexample, when we measure Ψj i Ψh j) is explained by the Bohm theory via thenonclassical measurement interaction between the external observer’s measuringdevice and the room. The quantum states that in perspectival schemes encodeinformation about which physical properties are definite, in the Bohm types ofinterpretations only play a role in the dynamics of a fixed set of quantities, so thatthe possibility of relational properties or perspectivalism does not suggest itself.

However, it has recently been argued that all interpretations of this unitary kind,characterized by definite and unique (i.e., one-world) outcomes at the end of eachsuccessful experiment even though the total quantum state always evolves unitar-ily, cannot be consistent (Araújo 2016, Frauchiger and Renner 2016). This argu-ment is relevant for our theme, and we shall discuss it in some detail.

3.3 Unitarity and Consistency

In a recent paper, Frauchiger and Renner (2016) consider a sophisticated version ofthe Wigner’s Friend experiment in which there are two friends, each in her ownroom, with a private information channel between them. Outside the two rooms areWigner and an Assistant. The experiment consists of a series of four measure-ments, performed by the individual friends, the Assistant and Wigner, respectively.In the room of Friend 1 a quantum coin has been prepared in a superposition of

58 Dennis Dieks

“heads” and “tails”: 1=ffiffiffi3

p� �hj i þ ffiffiffiffiffiffiffiffi

2=3p

tj i. The experiment starts when Friend1 measures her coin and finds either heads (probability 1/3) or tails (probability 2/3).Friend 1 then prepares a qubit in the state 0j i if her outcome was h, and in the state1=

ffiffiffi2

p� �0j i þ 1j ið Þ if the outcome was t, and sends this qubit via the private channel

between the rooms to Friend 2. When Friend 2 receives the qubit, he subjects it to ameasurement of an observable that has the eigenstates 0j i and 1j i. As in the thoughtexperiment of Section 3.2, the external observers subsequently measure “global”observables on the respective rooms; this is first done by the Assistant (on the roomof Friend 1) and then by Wigner (on the room of Friend 2).

Frauchiger and Renner claim, via a rather complicated line of reasoning (Araújo2016 has given a concise version of the argument), that any interpretation ofquantum mechanics that assigns unique outcomes to these measurements “in onesingle world,” while using only unitary evolution for the dynamics of the quantumstate (also during the measurements), will lead to an inconsistent assignment ofvalues to the measurement outcomes. If this conclusion is correct, there aresignificant implications for the question of which unitary interpretations of quan-tum mechanics are possible. The theories that are excluded according to Frauchigerand Renner are theories that “rule out the occurrence of more than one singleoutcome if an experimenter measures a system once” (2016: 2). If this is right,accepting many worlds would seem inevitable. In fact, Frauchiger and Rennerthemselves conclude that “the result proved here forces us to reject a single-worlddescription of physical reality” (2016: 3).

However, we should not be too quick when we interpret this statement. AsFrauchiger and Renner make clear, they use their “single-world assumption” toensure that all measurement outcomes are context-independent. In particular, whatthey use in their proof is a compatibility condition between different “stories” of ameasurement: If one experimenter’s story is that an experiment has outcome t,every other experimenter’s story of the same event must also contain this sameoutcome t (Frauchiger and Renner 2016: 7). This is, first of all, a denial of thepossibility of perspectivalism. As we shall further discuss in a moment, perspec-tival interpretations will be able to escape the conclusion of the Frauchiger-Renner(F-R) argument. Therefore, we claim that the F-R argument can be taken to lendsupport to perspectivalism as one of the remaining consistent possibilities.

The details and the domain of validity of the F-R proof are not completelytransparent and uncontroversial. Indeed, there is at least one nonperspectivalsingle-world interpretation, namely the Bohm interpretation, whose consistencyis usually taken for granted. This consistency is confirmed by a result of Sudbery(2017), who has concretely constructed a series of outcomes for the F-R thoughtexperiment as predicted by a modal interpretation of the Bohm type. According toSudbery, there is an unjustified step in Frauchiger and Renner’s reasoning, because

Quantum Mechanics and Perspectivalism 59

they do not fully take into account that in unitary interpretations only the total(noncollapsed) state can be used for predicting the probabilities of results obtainedby the Assistant and Wigner (The bone of contention is statement 4 in Araújo’s(2016) reconstruction of the F-R inconsistency, in which Friend 1 argues that hercoin measurement result is only compatible with one single later result obtained byWigner in the final measurement. However, in unitary interpretations previousmeasurement results do not always play a role in the computation of probabilitiesfor future events. Indeed, a calculation on the basis of the total uncollapsedquantum state, as given by Araújo (2016: 4), indicates that Wigner may find eitherone of two possible outcomes, with equal probabilities, even given the previousresult of Friend 1 � this contradicts the assumption made by Frauchiger andRenner).

The situation becomes more transparent when we make use of an elegantversion of the F-R thought experiment recently proposed by Bub (2017). Bubreplaces Friend 1 by Alice and Friend 2 by Bob; Alice and Bob find themselves at agreat distance from each other. Alice has a quantum coin which she subjects to ameasurement of the observable A with eigenstates hj iA, tj iA; the coin has beenprepared in the initial state 1=

ffiffiffi3

p� �hj iA þ

ffiffiffiffiffiffiffiffi2=3

ptj iA. Alice then prepares a qubit in

the state 0j iB if her outcome is h and in the state 1=ffiffiffi2

p� �0j iB þ 1j iB

� �if her

outcome is t. She subsequently sends this qubit to Bob � this is the only“interaction” between Alice and Bob. After Bob has received the qubit, he subjectsit to a measurement of the observable B with eigenstates 0j iB, 1j iB.

In accordance with the philosophy of noncollapse interpretations, we assumethat Alice and Bob obtain definite outcomes for their measurements, but thatthe total system of Alice, Bob, their devices and environments, and the coinand the qubit, can nevertheless be described by the uncollapsed quantum state,namely:

Ψj i ¼ 1ffiffiffi3

p hj iA 0j iB þ tj iA 0j iB þ tj iA 1j iB� �

(3.1)

For ease of notation, the quantum states of Alice and Bob themselves, plus themeasuring devices used by them, and even the states of the environments that havebecome correlated to them, have here all been included in the stateshj iA, tj iA, 0j iB, 1j iB (so that these states no longer simply refer to the coin and thequbit, respectively, but to extremely complicated many-particles systems!).

Now we consider two external observers, also at a great distance from each other,who take over the roles of Wigner and his Assistant, and are going to performmeasurements on Alice and Bob (and their entire experimental setups), respectively.The external observer who focuses on Alice measures an observable X with eigen-states failj iA ¼ 1=

ffiffiffi2

p� �hj iA þ tj iA

� �and okj iA ¼ 1=

ffiffiffi2

p� �hj iA � tj iA

� �, and the

60 Dennis Dieks

observer dealing with Bob and Bob’s entire experiment measures the observable Ywith eigenstates failj iB ¼ 1=

ffiffiffi2

p� �0j iB þ 1j iB

� �and okj iB ¼ 1=

ffiffiffi2

p� �0j iB � 1j iB

� �.

A F-R contradiction now arises in the following manner (Bub 2017: 3). Thestate Ψj i can alternatively be expressed in the following forms:

Ψj i ¼ 1ffiffiffiffiffi12

p okj iA okj iB �1ffiffiffiffiffi12

p okj iA failj iBþ

þ 1ffiffiffiffiffi12

p failj iA okj iB þffiffiffi34

rfailj iA failj iB

(3.2)

Ψj i ¼ffiffiffi23

rfailj iA 0j iB þ

1ffiffiffi3

p tj iA 1j iB (3.3)

Ψj i ¼ 1ffiffiffi3

p hj iA 0j iB þffiffiffi23

rtj iA failj iB (3.4)

From Eq. (3.2), we see that the outcome ok; okf g in a joint measurement of X andY has a nonzero probability: This outcome will be realized in roughly 1/12 th of allcases if the experiments are repeated many times. From Eq. (3.3) we calculate thatthe pair ok; 0f g has zero probability as a measurement outcome, so ok; 1f g is theonly possible pair of values for the observables X, B in the cases in which X has thevalue ok. However, from Eq. (3.4) we conclude that the pair t; okf g has zeroprobability, so h is the only possible value for the observable A if Y has the valueok and A and Y are measured together. So this would apparently lead to the pair ofvalues h; 1f g as the only possibility for the observables A and B, if X and Y arejointly measured with the result ok; okf g. But this pair of values has zero prob-ability in the state Ψj i so it is not a possible pair of measurement outcomes forAlice and Bob in that state. So although the outcome ok; okf g for X and Y iscertainly possible, the (seemingly) necessarily associated outcome h; 1f g for A andB is not � this seems an inconsistency.

In this inconsistency argument there is a silent use of nonperspectivalismconditions. For example, if Bob’s measurement outcome is 1 from the perspectiveof the X measurement, it is assumed that this outcome also has to be 1 as judgedfrom the perspective of the Y observer. However this assumption does not sit wellwith what the quantum formulas show us: The relative state of Bob with respect tothe Y outcome “ok” is not 1j iB, but okj iB (see Eq. (3.5)).

To see what is wrong with the inconsistency argument from a perspectival pointof view that closely follows the quantum formalism, it is helpful to note that thestates in Eqs. (3.1), (3.2), (3.3), and (3.4) are all states of Alice and Bob, includingtheir devices and environments, but without the external observers. In a consistentnoncollapse interpretation we must also include the external observer states in the

Quantum Mechanics and Perspectivalism 61

total state if we want to discuss the measurements of the observables X and Y. If wedenote by oj i and fj i the external states corresponding to the measurement results“ok” and “fail,” respectively, we find for the final state, in obvious notation:

1ffiffiffiffiffi12

p okj iA okj iB oj iX oj iY � 1ffiffiffiffiffi12

p okj iA failj iB oj iX fj iYþ

1ffiffiffiffiffi12

p failj iA okj iB fj iX oj iY þffiffiffi34

rfailj iA failj iB fj iX fj iY

(3.5)

From this equation, and its counterparts for when only X or Y is measured, we readoff that the relative state of Alice and Bob with respect to Alice’s external observerin state oj iX is okj iA 1j iB; with respect to Bob’s external observer in state oj iY it ishj iA okj iB. Both these state assignments refer to the situation in which only X or Y ismeasured.

However, the state of Alice and Bob relative to the combined external observersstate oj iX oj iY is:

okj iA okj iB ¼ 12

hj iA � tj iA� �

0j iB � 1j iB� �

(3.6)

This is an entangled state in which neither the coin toss nor the qubit measurementhas a definite result � it is not the state hj i 1j i that was argued to be present inBub’s version of the F-R argument. This illustrates the fact that in the case of anentangled state between two systems, the perspectives of an external observer whomeasures one system and an observer who measures the other can generally not beglued together to give us the perspective of the system that consists of bothobservers. In fact, as we see from Eq. (3.5), in the final quantum state not onlyAlice and Bob, but also the external observers have become entangled with eachother � this should already make us suspicious of combining partial viewpointsinto a whole, as it is well known that entanglement may entail nonclassical holisticfeatures.

So, perspectival views, which make the assignment of properties dependent onthe relative quantum state, are able to escape the inconsistency argument justdiscussed by denying that the X-perspective and the Y-perspective can be simplyjuxtaposed to form the XY-perspective.

A further point to note is that the quantities measured by the external observersdo not commute with A and B, respectively. So the measurement by the observernear to Alice introduces a “context” that is different from Alice’s one, and similarlyfor the external observation near Bob. This reinforces the notion that the combinedXY perspective need not agree with the measurement outcomes initially found byAlice and Bob (compare Fortin and Lombardi 2019). Indeed, the correct “Aliceand Bob state” from the XY-perspective, given by Eq. (3.6), does not show one

62 Dennis Dieks

definite combination of results of Alice’s A and Bob’s B measurement but containsall of them as possibilities.

The Frauchiger and Renner argument, in Bub’s formulation, therefore does notthreaten perspectival one-world interpretations with unitary dynamics. However,we should wonder whether nonperspectival unitary schemes, like the Bohmtheory, will also be able to escape inconsistency, and if so, exactly how they doso. When we again follow the steps in the measurement procedure of the thoughtexperiment, we can conclude that the X result “ok” can only occur if Bob hadmeasured “1”. In the context of the Bohm interpretation, this means that theparticle configuration of X ends up at a point of configuration space that iscompatible with the state oj iX only if Bob’s configuration is in a part of configur-ation space compatible with 1j iB. The same conclusion can be drawn with respectto Y and A for a measurement series in which Y is measured before X: If the Ymeasurement is performed first and the result “ok” is registered, A must have seen“heads”. Now, X and Y are at space-like separation from each other, and this mightseem to imply that it cannot make a difference to the state of Alice and Bob whatthe order in time is of the X and Y measurements. If the X measurement with result“ok” is the earlier one, Bob must have been in state 1j iB before the externalmeasurements started; if Y is measured first, Alice must have been in state hj iAbefore the start of the external measurements. Therefore, if the time order isimmaterial, Alice and Bob together will with certainty have been in the stateh; 1f g in the cases in which the X,Y measurements have ended with the resultok; okf g. But this is in contradiction with what the unitary formalism predicts: Eq.(3.1) shows that the pair of outcomes h; 1f g is impossible. So we have aninconsistency, and the Bohm interpretation and other nonperspectival unitaryinterpretations seem to be in trouble.

However, in the Bohm theory the existence of a preferred reference frame thatdefines a universal time is assumed (see for more on the justification of thisassumption in Section 3.4). This makes it possible to discuss the stages of theexperiment in their objectively correct temporal order. Assume that after Alice’sand Bob’s “in-the-room” measurements (the first by Alice, the second by Bob), theexternal observer near to Alice measures first, after which the observer close toBob performs the second measurement. The first external measurement will disturbthe configuration of particles making up Alice and her environment, so Alice’sstate will be changed. If the result of the X measurement is “ok”, Bob’s internalresult must have been “1” (Bob is far away, but the possibility of this inference isnot strange, because the total state is entangled, which entails correlations betweenAlice and Bob). This “1” will remain unchanged until the second external meas-urement, of Y. This second measurement will change Bob’s result. Now, in thisstory it is not true that the outcome “ok” of Y is only compatible with Alice’s initial

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outcome “heads”, as used in the inconsistency argument: the previous measure-ment of X has blocked this conclusion, as inspection of the initial state given in Eq.(3.1), and its evolution under the external measurements, shows. In fact, the initialoutcomes combination {t, 1} will give rise to the later outcomes {ok, ok} in 1/4 thof the cases, which is exactly what is needed to achieve complete consistency:since {t, 1} will occur in 1/3 rd of the cases, {ok, ok} will be found in 1/12 th of allcases.

So also Bohm, and possibly other nonperspectival schemes, are able toescape the inconsistency argument. In the perspectival schemes the key wasthat two perspectives cannot always be simply combined into one globalperspective; because of this, we were allowed to speak about the X perspectiveand the Y perspective without specifying the temporal order of the X and Ymeasurements. The threat of inconsistency was avoided by blocking thecomposition of the two perspectives into one whole. In the nonperspectivalscheme the existence of a preferred frame of reference comes to the rescue andprotects us against inconsistency: We can follow the interactions and thechanges produced by them step by step in their unique real-time order, sothat no ambiguity arises about which measurement comes first and about whatthe actual configuration is at each instant. The issue of combining descriptionsfrom different perspectives accordingly does not arise, and this is enough totell one consistent story.

The difference between the perspectival and nonperspectival unitary accounts,and the apparent connection between perspectivalism and Lorentz invariance,suggests that there is a link between perspectivalism and relativity. Perspectivalismseems able to avoid inconsistencies without introducing a privileged frame ofreference. On the other hand, the introduction of such a privileged frame inBohm-like interpretations now appears as a ploy to eliminate the threat of incon-sistencies without adopting perspectivalism.

3.4 Relativity

The diagnosis of the previous section is confirmed when we directly study theconsequences of relativity for interpretations of quantum mechanics. In particular,when we attempt to combine special relativity with unitary interpretationalschemes, new hints of perspectivalism emerge. As mentioned, the Bohm interpret-ation has difficulties in accommodating Lorentz invariance. Bohmians have there-fore generally accepted the existence of a preferred inertial frame in which theequations assume their standard form � a frame resembling the ether frame ofprerelativistic electrodynamics. Accepting such a privileged frame in the context ofwhat we know about special relativity and Minkowski spacetime is, of course, not

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something to be done lightly; it must be a response to a problem of principle.Indeed, it can be mathematically proved that no unitary interpretation scheme thatattributes always definite positions to particles (as in the Bohm theory) can satisfythe requirement that the same probability rules apply equally to all hyperplanes inMinkowski spacetime (Berndl, Dür, Goldstein, and Zanghì 1996).

The idea of this theorem (and of similar proofs) is that intersecting hyper-planes should carry properties and probabilities in a coherent way, which meansthat they should give agreeing verdicts about the physical conditions at thespacetime points where they (the hyperplanes) intersect. The proofs demonstratethat this meshing of hyperplanes is impossible to achieve with properties that arehyperplane independent. The no-go results can be generalized to encompassnonperspectival unitary interpretations that attribute other definite propertiesthan position, and to unitary interpretations that work with sets of propertiesthat change in time (Dickson and Clifton 1998). A general proof along theselines was given by Myrvold (2002a). Myrvold shows, for the case of twosystems that are (approximately) localized during some time interval, that it isimpossible to have a joint probability distribution of definite properties alongfour intersecting hyperplanes such that this joint distribution returns the Bornprobabilities on each hyperplane. An essential assumption in the proof (Myrvold2002a: 1777) is that the properties of the considered systems are what Myrvoldcalls local: The value of quantity A of system S at spacetime point p (a pointlying on more than one hyperplane) must be well defined regardless of thehyperplane to which p is taken to belong and regardless of which other systemsare present in the universe. It turns out that such local properties cannot obey theBorn probability rule on each and every hyperplane. The assumption that theBorn rule only holds in a preferred frame of reference is one way of respondingto this no-go result.

The argument has been given a new twist by Leegwater (2018), who argues that“unitary single-outcome quantum mechanics” cannot be “relativistic,” where atheory is called relativistic if all inertial systems have the same status with respectto the formulation of the dynamic equations of the theory (i.e., what usually iscalled Lorentz or relativistic invariance). Like Frauchiger and Renner, Leegwaterconsiders a variation on the Wigner’s friend thought experiment: There are threelaboratory rooms, at spacelike distances from each other, each with a friend insideand a Wigner-like observer outside. In each of the lab rooms there is also a spin-½particle, and the experiment starts in a state in which the three particles (one in eachroom) have been prepared in a so-called GHZ-state (Greenberger, Horne, Shi-mony, and Zeilinger 1990). The description of the thought experiment in aninitially chosen inertial rest frame is assumed to be as follows: At a certain instantthe three friends inside their respective rooms simultaneously measure the spins of

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their particles, in a certain direction; thereafter, at a second instant, each of the threeoutside observers performs a measurement on his or her room. This measurementis of a “whole-room” observable, like in the Frauchiger-Renner thought experi-ment discussed in the previous section. As a result of the internal measurements bythe three friends the whole system, consisting of the rooms and their contents, hasended up in an entangled GHZ-state. Leegwater is able to show that this entails thatthe assumption that the standard rules of quantum mechanics apply to each of threedifferently chosen sets of simultaneity hyperplanes, gives rise to a GHZ-contradiction: The different possible measurement outcomes (all +1 or �1) cannotbe consistently chosen such that each measurement has the same outcome irre-spective of the simultaneity hyperplane on which it is considered to be situated(and so that all hyperplanes mesh). As in the original GHZ-argument (Greenbergeret al. 1990), the contradiction is algebraic and does not involve the violation ofprobabilistic (Bell) inequalities.

One way of responding to these results is the introduction of a preferred inertialsystem (a privileged perspective!), corresponding to a state of absolute rest,perhaps defined with respect to an ether. This response is certainly against thespirit of special relativity, in particular because the macroscopic predictions ofquantum mechanics are such that they make the preferred frame undetectable.Although this violation of relativistic invariance does not constitute an inconsist-ency, it certainly is attractive to investigate whether there exist other routes toescape the no-go theorems. Now, as we have seen, a crucial assumption in thesetheorems is that properties of systems are monadic, independent of the presence ofother systems and independent of the hyperplane on which they are considered.This suggests that a transition to relational or perspectival properties offers analternative way out.

In fact, that unitary evolution in Minkowski spacetime leads naturally to ahyperplane-dependent account of quantum states if one describes measurementsby effective collapses has been noted in the literature before (see e.g. Dieks 1985,Fleming 1996, Myrvold 2002b). The new light that we propose to cast on these andsimilar results comes from not thinking in terms of collapses, and of a dependenceon hyperplanes or foliations of Minkowski spacetime as such, but instead ofinterpreting them as consequences of the perspectival character of physical prop-erties: that the properties of a system are defined with respect to other systems.What we take the considerations in the previous and present sections to suggest isthat it makes a difference whether we view the physical properties of a system fromone or another system – or from one or another temporal stage in the evolution of asystem. In the case of the (more-or-less) localized systems that figure in therelativistic no-go theorems that we briefly discussed, this automatically leads toproperty ascriptions that are different on the various hyperplanes that are

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considered. As a result, the meshing conditions on which the theorems hinge nolonger apply.

3.5 Concluding Remarks

If unitary evolution is accepted as basic in quantum mechanics and is combinedwith the requirement that results of experiments are to be definite and situated inone single world, this naturally leads to a picture in which physical systems haveproperties that are relational or perspective dependent. As we have seen in Section3.3, perspectivalism makes it possible to escape arguments saying that interpret-ations of unitary quantum mechanics in terms of one single world are inconsistent.Moreover, perspectivalism removes obstacles to the possibility of formulatinginterpretational schemes that respect Lorentz invariance by making the introduc-tion of a preferred inertial frame of reference superfluous (Section 3.4).

The single world that results from perspectivalism is evidently much morecomplicated than the world we are used to in classical physics: There are morethan one valid descriptions of what we usually think of as one physical situation.This reminds of the many-worlds interpretation. There are important differences,though, between a multiplicity of worlds and the multiplicity of descriptions inperspectivalism. According to the single-world perspectivalism that we havesketched, only one of the initially possible results of a measurement becomesactual from the perspective of the observer, whereas in the many-worlds interpret-ation all possibilities are equally realized. So the multiplication of realities thattakes place in many-worlds is avoided in perspectivalism. It is of course true thatperspectivalism sports a multiplicity of its own, namely of different points of viewwithin a single world. But this multiplicity seems unavoidable in the many-worldsinterpretation as well, in each individual branch. For example, in the relativisticmeshing argument of Myrvold (2002a), a situation is discussed in which nomeasurements occur: The argument is about two freely evolving localized systemsas described from a number of different inertial frames. Since no measurements aretaking place during the considered process, the inconsistency argument goesthrough in exactly the same way in every single branch of the many-worldssuper-universe: There is no splitting during the time interval considered in theproof of the theorem. So even in the many-worlds interpretation the introduction ofperspectival properties (in each single branch) seems unavoidable in order to avoidinconsistencies. Another case to be considered is the Wigner’s friend experiment:When the measurement in the hermetically sealed room has been performed, anoutside observer will still have to work with the superposition of the two branches.So the splitting of worlds assumed by the many-worlds interpretation must remainconfined to the interior of the room, as mentioned in Section 3.2. In this situation it

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is natural to make the description of the measurement and its result perspectivedependent: For the two friend-branches inside the room there is a definite outcome,but this is not so for the external observer. So perspectivalism as a consequence ofholding fast to unitarity and Lorentz invariance seems more basic than the furtherchoice of interpreting measurements in terms of many worlds; even the many-worlds interpretation must be committed to perspectivalism. But perspectivalismon its own is already sufficient to evade the anti-single-world arguments of Section3.3, so for this purpose we do not need the further assumption of many worlds.

Finally, the introduction of perspectivalism opens the door to several newquestions. In everyday circumstances we do not notice consequences ofperspectivalism, so we need an account of how perspectival effects are washedout in the classical limit. It is to be expected that decoherence plays an importantrole here, as alluded to in the Introduction� however, this has to be further workedout (compare Bene and Dieks 2002). Further, there is the question of how thedifferent perspectives hang together; for example, in Section 3.3 it was shown thatperspectives of distant observers cannot be simply combined in the case ofentanglement, which may be seen as a nonlocal aspect of perspectivalism. Bycontrast, it has been suggested in the literature that perspectivalism makes itpossible to give a purely local description of events in situations of the Einstein-Podolsky-Rosen type, and several tentative proposals have been made in order tosubstantiate this (Rovelli 1996, Bene and Dieks 2002, Smerlak and Rovelli 2007,Dieks 2009, Laudisa and Rovelli 2013). These and other questions constitutelargely uncharted territory that needs further exploration.

References

Araújo, M. (2016). “If your interpretation of quantum mechanics has a single world but nocollapse, you have a problem,” http://mateusaraujo.info/2016/06/20/if-your-interpretation-of-quantum-mechanics-has-a-single-world-but-no-collapse-you-have-a-problem/

Bacciagaluppi, G. (2002). “Remarks on space-time and locality in Everett’s interpretation,”pp. 105–122 in T. Placek and J. Butterfield (eds.), Non-Locality and Modality.Dordrecht: Springer.

Bene, G. and Dieks, D. (2002). “A perspectival version of the modal interpretation ofquantum mechanics and the origin of macroscopic behavior,” Foundations of Physics,32: 645–671.

Berndl, K., Dür, D., Goldstein, S., and Zanghì, N. (1996). “Nonlocality, Lorentz invari-ance, and Bohmian quantum theory,” Physical Review A, 53: 2062–2073.

Bohm, D. (1952). “A suggested interpretation of the quantum theory in terms of ‘hidden’variables, I and II,” Physical Review, 85: 166–193.

Bohr, N. (1928). “The quantum postulate and the recent development of atomic theory,”Nature, 121: 580–590.

Bohr, N. (1935). “Can quantum-mechanical description of physical reality be consideredcomplete?,” Physical Review, 48: 696–702.

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Bohr, N. (1948). “On the notions of causality and complementarity,” Dialectica, 2:312–319.

Bub, J. (1997). Interpreting the Quantum World. Cambridge: Cambridge University Press.Bub, J. (2017). “Why Bohr was (mostly) right,” arXiv:1711.01604v1 [quant-ph].Dickson, M. and Clifton, R. (1998). “Lorentz invariance in modal interpretations,”

pp. 9–47 in D. Dieks and P. Vermaas (eds.), The Modal Interpretation of QuantumMechanics. Dordrecht: Kluwer Academic Publishers.

Dieks, D. (1985). “On the covariant description of wave function collapse,” Physics LettersA, 108: 379–383.

Dieks, D. (2009). “Objectivity in perspective: relationism in the interpretation of quantummechanics,” Foundations of Physics, 39: 760–775.

Dieks, D. (2017). “Niels Bohr and the formalism of quantum mechanics,” pp. 303–333 inJ. Faye and H. J. Folse (eds.), Niels Bohr and the Philosophy of Physics � Twenty-First-Century Perspectives. London and New York: Bloomsbury Academic.

Dieks, D. and Vermaas, P. (eds.). (1998). The Modal Interpretation of Quantum Mechan-ics. Dordrecht: Kluwer Academic Publishers.

Everett, H. (1957). “‘Relative state’ formulation of quantum mechanics,” Reviews ofModern Physics, 29: 454–462.

Fleming, G. (1996). “Just how radical is hyperplane dependence?,” pp. 11–28 in R. Clifton(ed.), Perspectives on Quantum Reality. Dordrecht: Kluwer Academic Publishers.

Fortin, S. and Lombardi, O. (2019). “Wigner and his many friends: a new no-go result?,”http://philsci-archive.pitt.edu/id/eprint/15552.

Frauchiger, D. and Renner, R. (2016). “Single-world interpretations of quantum theorycannot be self-consistent,” arXiv:1604.07422v1 [quant-ph].

Greenberger, D., Horne, M., Shimony, A., and Zeilinger, A. (1990). “Bell’s theoremwithout inequalities,” American Journal of Physics, 58: 1131–1143.

Griffiths, R. (2017). “The consistent histories approach to quantum mechanics,” in E. N.Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2017 Edition). https://plato.stanford.edu/archives/spr2017/entries/qm-consistent-histories/

Jammer, M. (1974). The Philosophy of Quantum Mechanics. New York: Wiley & Sons.Johnson, K., Wong-Campos, J., Neyenhuis, B., Mizrahi, J., and Monroe, C. (2017).

“Ultrafast creation of large Schrödinger cat states of an atom,” Nature Communi-cations, 8: Article 697. doi:10.1038/s41467-017-00682-6.

Laudisa, F. and Rovelli, C. (2013). “Relational quantum mechanics,” in E. N. Zalta (ed.),The Stanford Encyclopedia of Philosophy (Summer 2013 Edition). https://plato.stanford.edu/archives/sum2013/entries/qm-relational/

Leegwater, G. (2018). “When Greenberger, Horne and Zeilinger meet Wigner’s Friend,”arXiv:1811.02442 [quant-ph].

Lombardi, O. and Castagnino, M. (2008). “A modal-Hamiltonian interpretation ofquantum mechanics,” Studies in History and Philosophy of Modern Physics, 39:380–443.

Lombardi, O. and Dieks, D. (2017). “Modal interpretations of quantum mechanics,” in E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2017 Edition).https://plato.stanford.edu/archives/spr2017/entries/qm-modal/

London, F. and Bauer, E. (1939). La Théorie de l’Observation en Mécanique Quantique.Paris: Hermann. English translation, pp. 217–259 in J. A. Wheeler and W. H. Zurek(eds.), 1983, Quantum Theory and Measurement. Princeton: Princeton UniversityPress.

Myrvold, W. (2002a). “Modal interpretations and relativity,” Foundations of Physics, 32:1773–1784.

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Myrvold, W. (2002b). “On peaceful coexistence: is the collapse postulate incompatiblewith relativity?,” Studies in History and Philosophy of Modern Physics, 33: 435–466.

Rovelli, C. (1996). “Relational quantum mechanics,” International Journal of TheoreticalPhysics, 35: 1637–1678.

Smerlak, M. and Rovelli, C. (2007). “Relational EPR,” Foundations of Physics, 37:427–445.

Sudbery, A. (2017). “Single-world theory of the extended Wigner’s friend experiment,”Foundations of Physics, 47: 658–669.

Vermaas, P. and Dieks, D. (1995). “The modal interpretation of quantum mechanics and itsgeneralization to density operators,” Foundations of Physics, 25: 145–158.

von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin:Springer.

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4

Quantum Physics Grounded on Bohmian Mechanics

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4.1 Copenhagen and the Measurement Problem

Quantum mechanics is one of the greatest intellectual achievements of the twenti-eth century. Its laws govern the atomic and subatomic world and reverberate on amyriad of phenomena, from the formation of crystals to superconductivity, fromthe properties of low-temperature fluids to the emission spectrum of a burningcandle. However, as it is usually presented in textbooks, quantum mechanics isbasically a set of rules for calculating probability distributions of the results of anyexperiment (in the domain of validity of quantum mechanics). As such, it does notdirectly provide us with a description of reality. A description of reality should tellus what there is in the world and how it behaves.

Whereas there is an almost general agreement on the correctness of the formal-ism, the description of the reality that emerges from it remains controversial. It haseven been doubted whether such a description of reality should conform to therules of ordinary logic – and if any description is, after all, truly desirable. It hasalso been argued that quantum theory forces us to abandon the reality of anexternal world that exists objectively and independently of the human mind.

It is widely believed that between the end of the nineteenth and the beginning ofthe twentieth century, physics underwent a radical change. Experimental know-ledge about the atomic and subatomic world that was accumulating during thatperiod challenged the overall conceptual framework that physics had developedfrom Galileo and Newton onward. Thus the idea was born that not only would it benecessary to develop new theories that would replace classical mechanics andclassical electromagnetism, but that it was also necessary to abandon the classicalideal according to which the laws of physics govern an external worldobjectively given.

In the nineteen-twenties, the Danish physicist Niels Bohr, at that time probablythe most authoritative and influential quantum physicist, began to defend the idea

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that traditional scientific realism was childish and nonscientific, and he proposedwhat it is still called the Copenhagen interpretation of quantum mechanics. On thebasis of this doctrine, the physical laws no longer have to do with the question ofhow the world is made, but with our ability to know it, which is intrinsicallylimited: The quantum mechanics of Copenhagen refuses in principle to provide aconsistent history of what happens to microscopic objects. From the point of viewof Copenhagen, reality is divided into two worlds, the microscopic and themacroscopic, the classical and the quantum, the world regulated by classicallogic and the one regulated by quantum logic. Although it is not clear where theboundary between these two worlds lies and how this duality can be compatiblewith the fact that apples and chairs consist of electrons and other particles, theCopenhagen doctrine has become orthodoxy. That is to say, it has become not onlythe majority viewpoint among physicists, but also the dogma.

In more recent years, a version of quantum mechanics based on informationtheory has grown in popularity. It is a dress that seems new, packed on the wave ofthe theoretical and experimental development of quantum information and quan-tum computation, but in reality it is a used dress, which was already tailored inCopenhagen. Already in 1952, Schrödinger warned against the idea of reducingquantum mechanics to a simple representation of our knowledge (Schrödinger1995).

In spite of the pragmatic tribute reserved to the dogma, the peculiar role of theobserver in the formulation of the theory has always puzzled many physicists, ascan be seen for example, from the following considerations by Richard Feynman:

This is all very confusing, especially when we consider that even though we mayconsistently consider ourselves to be the outside observer when we look at the rest ofthe world, the rest of the world is at the same time observing us, and that often we agree onwhat we see in each other. Does this then mean that my observations become real onlywhen I observe an observer observing something as it happens? This is a horribleviewpoint. Do you seriously entertain the idea that without the observer there is noreality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Wasthere no reality in the universe before 10⁹ B.C. when life began? Or are you the observer?Then there is no reality to the world after you are dead? I know a number of otherwiserespectable physicists who have bought life insurance.

(Feynman, Morinigo, and Wagner 2003: 14)

Feynman is putting his finger on the most commonly cited conceptual difficultiesthat plague quantum mechanics – the measurement problem, or what amounts tomore or less the same thing, the paradox of Schrödinger’s cat. The problem can berephrased as follows: Suppose that the wave function of any individual systemprovides a complete description of that system. When we analyze the process ofmeasurement in quantum mechanical terms, we find that the after-measurement

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wave function for system and apparatus arising from the Schrödinger equationtypically involves a superposition over terms corresponding to what we would liketo regard as the various possible results of the measurement, e.g., different pointerorientations. It is difficult to discern in this description of the after-measurementsituation the actual result of the measurement, e.g., some specific pointer orienta-tion. In brief, quantum mechanics does not account for the obvious fact thatmeasurements do have results.

Bohr considered that philosophy was very important to understand quantummechanics and introduced the notion of complementarity, a many-purpose notiongood for solving the wave-particle duality, the measurement problem, and indeed,all interpretative problems of quantum mechanics. This attitude sustained the ideathat with the problem of measurement we are facing a purely philosophicalproblem. This idea was then nurtured and nourished by a sort of naive realismabout the operators – the idea that in quantum mechanics the observables and theproperties acquire a radically new, highly nonclassical meaning, reflected in thenoncommutative structure of the algebra of quantum observables.

4.2 Noncommutativity

The Hilbert space of quantum states is a vector space with a scalar product rule,and it would be surprising if the operators on this space did not play an importantrole in the formulation of quantum theory. And indeed, it is obviously so: Thetemporal evolution of the states is given by a unitary operator that is generated by aself-adjoint operator, the Hamiltonian. Not only are the temporal translationsgoverned by a self-adjoint operator, but so also are all the other symmetries ofthe system. For example, the momentum operator is the generator of spatialtranslations and the angular momentum operators govern the change of states asa consequence of a rotation of the physical space. In quantum field theory, theoperators of creation and annihilation, operators that transform the state of a systemwith a certain number of particles into another with a different number of particles,play an extremely important role, as basic bricks of the Hamiltonian. In brief, linearoperators play an important role in quantum mechanics. And the main algebraicfeature of the operators is not to commute. So far, everything is clear and nothing ismysterious.

The mystery arises when it is argued that the association of quantum observ-ables with self-adjoint operators is to be considered a direct generalization of thenotion of classical observables and that quantum theory should not be conceptuallymore problematic than classical physics once this fact is appreciated.

The classical observables – for a particle system, their positions, their momentaand the functions of these variables, that is, functions on phase space – form a

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commutative algebra. It is generally taken to be the essence of quantization, theprocedure that converts a classical theory into a quantum theory, what makes thecorresponding operators to correspond to the positions and momenta and thereforeto all the functions of these variables. Thus the quantization leads to a noncom-mutative algebra of observables, of which the usual examples are provided bymatrices and linear operators. In this way it seems perfectly natural that classicalobservables are functions on phase space and quantum observables are self-adjointoperators.

But in all this there is much less than meets the eye. What does it mean tomeasure a quantum observable, a self-adjoint operator? It seems rather clear thatthis should be specified – without such a specification it cannot have any meaning.Therefore, one should be careful and use a more cautious terminology by sayingthat, in quantum mechanics, the observables are associated with self-adjointoperators, as it is difficult to see what more can be understood than an association.What could it mean, an identification of the observables – considered as having, insome way, an independent meaning as regards observation or measurement – witha mathematical abstraction such as that of self-adjoint operators?

Note that it is important to insist on association rather than identification inquantum theory, but not in classical theory, because in this case we begin with arather clear notion of observable (or property) which is well captured by the notionof a function on the phase space, the state space of complete descriptions. If thestate of the system were observed, the value of the observable would of course begiven by this function of the phase point, but the observable might be observed byitself, yielding only a partial specification of the state. In other words, measuring aclassical observable means determining to which level surface of the correspond-ing function the state of the system, the phase point – which is at any time definitethough probably unknow – belongs. In the quantum realm the analogous notioncould be that of function on Hilbert space, not that of self-adjoint operator. But wedon’t measure the wave function (the nonmeasurability of the wave function isrelated to the impossibility of copying the wave function. This question arisessometimes in the form, “Can one clone the wave function? [Ghirardi personalcommunication; see Wooters and Zurek 1982, Ghirardi and Weber 1983]) so thatfunctions on Hilbert space are not physically measurable, and thus do not define“observables.”

4.3 Contextuality

A milestone in the foundations of quantum mechanics is Bell’s nonlocality analy-sis (Bell 1964). It has a by-product that is interesting in itself: The incompatibilityof Bell’s inequality with the predictions of quantum mechanics is a demonstration

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of the impossibility that self-adjoint operators can be interpreted as a way ofrepresenting the properties of an object.

There are other demonstrations of this impossibility in addition to that of Bell in1964. The best known are due to von Neumann (1932), Gleason (1957), andKochen and Specker (1967). These theorems are usually called “impossibilitytheorems for hidden variables,” a name that seems to suggest that what thetheorems show would be the impossibility of completing the quantum mechanicsthrough additional variables that determine, together with the quantum state, theresult of the experiments. But this interpretation of the theorems is completelyerroneous and is clearly disproved by the existence of a theory like Bohmianmechanics (discussed later), in which the additional variables are simply thepositions of the particles that make up an object, and the results of the experimentsturn out to be determined by the complete specification of the quantum state andparticle positions (discussed later). A more appropriate term for these theoremswould perhaps be “theorems of impossibility of naive realism about self-adjointoperators” (Daumer, Dürr, Goldstein, and Zanghì 1996).

A substantial difference between Bell’s theorem and other impossibility the-orems is, that while Bell’s theorem stems from a question with a clear physicalmeaning – is the locality principle compatible with quantum mechanics? – it is notat all clear to what question the theorems of impossibility for hidden variablescorrespond. If Bohr’s warning that the fundamental lesson of quantum mechanicsis to recognize the impossibility of a sharp division between the behavior of atomsand the interaction with the measurement apparatus is kept in mind, it would not behard to recognize that the hypotheses on which the theorems are based are in clearcontrast with the experimental principles of quantum mechanics.

But the halo of mystery is hard to erase, and this has led to the formation of anew myth: contextuality. According to contextuality, in quantum mechanics prop-erties acquire a new, highly nonclassical meaning; quantum properties depend onother compatible properties – if they exist – that are measured at the same time. Allthis sounds very mysterious, but if one remembers the active role of quantummeasurements, the mystery will disappear. Let us consider, for example, theoperator A that commutes with the operators B and C which do not commuteamong them. What is often called the result of A in an experiment for themeasurement of A and B together, usually differs from the result of A in anexperiment for the measurement of A and C together, because these experimentsare different from each other in the sense that they act on the state of the system in avery different way. The misleading reference to measurement, with the associatednaive realism about operators, makes the context appear much more mysteriousthan it is. The same terminology is misleading and fails to transmit with due forcethe peremptory character of what it brings: Properties that are uniquely contextual

Quantum Physics Grounded on Bohmian Mechanics 75

are not properties at all, do not exist, and their inadequacy to carry out the role ofproperties is in the strongest sense possible.

In short, contextuality means nothing more than the fact that the result of anexperiment depends on the experiment itself, and this applies equally to bothclassical physics and quantum physics: For any experiment, be it classical orquantum, it would be a mistake to assume that any device involved in the experi-ment plays only a passive role, unless the experiment is not the genuine measure-ment of a property of the system, in which case the result is determined by theinitial condition of the system only. In classical physics, it is traditionally assumedthat it is in principle possible to measure any property without sensitively disturb-ing the measured object, but this is false in quantum mechanics – and should bequestioned in classical physics, too.

4.4 The Classical Variables of Bohr

It is useful to go back to Bohr’s solution of the measurement problem. For Bohr, itis in principle impossible to formulate the fundamental concepts of quantummechanics without using classical mechanics. To put it in the words of Landauand Lifshitz

quantum mechanics occupies a very particular position in the realm of physical theories: itcontains classical mechanics as a limiting case, and at the same time needs this limit casefor its foundation.

(Landau and Lifshitz 1958)

So, the orthodox vision ends up providing a response to the problem of measure-ment that many orthodoxy enthusiasts still struggle to accept: the wave functiondoes not provide a complete representation of the state of affairs of the world; inaddition, you need to specify the values of classic variables – which for conveni-ence will be named here “Z-variables.”

According to Bohr, the Z-variables are precisely those that establish quantummechanics and make the quantum formalism coherent and applicable to thestudy of the phenomena we observe in the laboratory or in nature. In otherwords, according to the orthodox interpretation, the complete description of astate of affairs of the world is given by the pair Ψ;Zð Þ, where Ψ is the wavefunction and the Z are in some sense macroscopic classical variables. So,although according to the orthodox view Ψ does not represent anything real –thus the famous motto “there is no quantum world, there is only an abstractquantum description” – the role of the wave function is to govern the statisticsof the Z-variables that indeed do represent what is to be considered real, or atleast “concrete.”

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As pointed out by John Bell (1990) in the original formulation of the Copen-hagen interpretation, the sense in which the complete description of a state ofaffairs of the world is given by the pair Ψ;Zð Þ is not clearly specified. In addition,the dynamics of the pair is not specified in a clear and unambiguous way:Sometimes the dynamics of the wave function is given by the Schrödingerequation and sometimes the dynamics of the macroscopic variables is that fixedby the laws of classical mechanics and classical electromagnetism. However, whenthe classical variables interact with the quantum variables, the dynamical lawschange: The wave function no longer evolves according to the Schrödingerequation, instead it evolves according to the collapse rule and the Z-variablesundergo random leaps that are statistically governed by the wave function.

The difficulty raised by Feynman in the passage quoted in Section 4.1 might beso rephrased: Where is the borderline between what is classical and what isquantum? When can we treat an object as classical and when must we treat it asquantum? In other words, the distinction between microscopic and macroscopic, aswell as that between the classical world and the quantum world, lacks a precisedefinition and introduces a fundamental ambiguity that cannot have any place inany theory that claims physical accuracy.

4.5 The Classical Variables of Bohm

If we must anyhow pay the price of incompleteness, why should we also pay thesurcharge of fuzziness and ambiguity? Why not supplement the quantum descrip-tion provided by the wave function with variables that are well defined on all scalesand not simply on the macroscopic one? This was the conviction of AlbertEinstein:

I am, in fact, rather firmly convinced that the essentially statistical character ofcontemporary quantum theory is solely to be ascribed to the fact that this (theory)operates with an incomplete description of physical systems.

(Einstein, in Schilpp 1949: 666)

Indeed, if the completion is achieved in what is really the most obvious way – bysimply including the positions of the particles of a quantum system as part of thestate description of that system, allowing these positions to evolve in the mostnatural way – one arrives at the theory developed by David Bohm (1952). Thistheory is nowadays known as Bohmian mechanics, de Broglie-Bohm’s theory, thewave-pilot theory, or the causal interpretation of quantum mechanics.

In the theory proposed by Bohm, a particle system is described in part by itswave function Ψ, which evolves, as usual, according to the Schrödinger equation.However, the wave function only provides a partial description of the system. This

Quantum Physics Grounded on Bohmian Mechanics 77

description is complemented by the specification of the real positions of theparticles Q ¼ Q1; . . . ;QNð Þ. Their evolution is governed by a guiding equationthat expresses the velocities of the particles in terms of the wave function (dis-cussed later). Thus, in Bohmian mechanics the configuration Q of a particle systemevolves according to a movement that is somehow “choreographed” by the wavefunction. In Bohmian mechanics, the complete state description of a system isprovided by the pair Ψ;Qð Þ. The entire quantum formalism, including the uncer-tainty principle, quantum randomness, the quantum statistics for identical particles,and the role of operators as observables, emerges from an analysis of the dynam-ical system Ψ;Qð Þ.

With a theory such as the Bohmian mechanics, in which the description of thesituation after a measurement includes, in addition to the wave function, the valuesof the variables that record the result, there is no problem of measurement. InBohmian mechanics the pointers of the measurement devices always have a well-defined orientation: The particles that form the pointer of an apparatus alwayshave, according to this theory, a well-defined configuration, and the macroscopicappearance of such a configuration is precisely that of a pointer that points in awell-defined direction.

As such, Bohmian mechanics is a counterexample to the claim that quantummechanics forces us to abandon the idea of an objective external world, whichexists independently of the human mind. It is a “realistic” quantum theory, andsince in its formulation no reference is made to “observers,” it is also a “quantumtheory without observers.” For historical reasons it has been called a “hiddenvariables theory.” The existence of Bohmian mechanics shows that many of theradical epistemological consequences, which usually many physicists and philoso-phers have drawn from quantum mechanics, are groundless. It shows that there isno need for contradictory notions such as “complementarity,” that there is no needto imagine an object as if it were simultaneously in different places, or that thephysical quantities have values that are somehow “blurred,” that is, intrinsicallyundefined. Finally, it shows that there is no need for human consciousness tointervene in physical processes, for example, to collapse the wave function.Bohmian mechanics solves all the paradoxes of quantum mechanics, eliminatingoddities and mysteries. It is important to stress that the Q-variables of a Bohmiantheory need not be configurations of particles. These variables may representgeometry, field or string configurations, or whatever is needed to describenature best.

Starting from 1952, Bohm’s theory has been investigated and refined. Variousways have been proposed to extend Bohmian mechanics to quantum field theory.One of them (Bohm 1952), for bosons (i.e., for the quantum fields of force), isbased on field configurations on the three-dimensional physical space that evolve

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over time guided by a wave functional according to a similar dimensional infinityof the guiding equation. Another proposal (Dürr, Goldstein, Tumulka, and Zanghì2004) is based on Bell’s seminal work (1986) and assigns trajectories to electronsand to any sort of particles that intervene in a given field quantum theory; however,in contrast to the original Bohmian mechanics, this proposal involves a stochasticdynamics according to which particles can be created and destroyed.

4.6 The Equations of Motion

For a nonrelativistic system of N particles, Bohmian mechanics is defined by twoequations of motion: the Schrödinger equation

iℏ∂Ψ∂t

¼ HΨ (4.1)

for the wave function Ψ, and the guiding equation

dQk

dt¼ ℏ

mkIm

Ψ∗rkΨΨ∗Ψ

Q1; . . . ;QNð Þ, k ¼ 1, . . . ,N (4.2)

for the configuration Q ¼ Q1; . . . ;QNð Þ of the particles. The guiding equation isindeed the simplest law of evolution of the first order for the position of theparticles, compatible with the Galilean symmetry of the Schrödinger evolutionand with time reversal (Dürr, Goldstein, and Zanghì 1992).

A few clarifications on the symbols that appear in Eqs. (4.1) and (4.2): ℏdenotes, as usual, the Planck constant divided by 2π; H is the usual nonrelativistic(Schrödinger) Hamiltonian, which contains the particle masses mk, k ¼ 1, . . . ,N,and for a system of spinless particles has the simple form

H ¼ �XNk¼1

ℏ2

2mkr2

k þ U (4.3)

with U the corresponding classical potential energy;rk is the gradient with respectto the coordinates of the k-th particle; for a complex-valued wave function Ψ, Ψ∗

denotes its complex conjugate; if Ψ has spinorial values, the products in thenumerator and in the denominator of the guiding equation should be understoodas scalar products in the spinor space; if external magnetic fields are present, thegradient should be understood as a covariant derivative, involving the potentialvector; “Im” means taking the imaginary part of a complex number.

To gain familiarity with the guiding equation, it is useful to consider theextremely simple case of a single free particle of mass m guided by a monochro-matic wave with wave vector k (and thus with wavelength λ ¼ 2π=k), ideally

Quantum Physics Grounded on Bohmian Mechanics 79

approximated by the plane wave Ψ q; tð Þ / e�i k:q�ωtð Þ, ω ¼ ℏk2= 2mð Þ. The l.h.s. ofthe guiding equation is the velocity v of the particle. A simple calculation showsthat the right-hand side r.h.s. of the guiding equation is ℏ=mð Þk. Thus the guidingequation of Bohmian mechanics turns out to be precisely the relation p ¼ ℏkwhich de Broglie proposed in late 1923 and which quickly led Schrödinger, duringthe end of 1925 and the beginning of 1926, to the discovery of his wave equation.

Now consider the guiding equation in the case of two particles. The wavefunction Ψ ¼ Ψ q1; q2; tð Þ generates the possible speeds of the two particles, i.e.,

v1 ¼ ℏm1

ImΨ∗ q1; q2; tð Þr1Ψ q1; q2; tð ÞΨ∗ q1; q2; tð ÞΨ q1; q2; tð Þ (4.4)

for particle 1 and

v2 ¼ ℏm2

ImΨ∗ q1; q2; tð Þr2Ψ q1; q2; tð ÞΨ∗ q1; q2; tð ÞΨ q1; q2; tð Þ (4.5)

for particle 2. These formulas show that the velocity of a particle, at a certaininstant of time, depends in general on where, at the same time, the other particleis. An exception is the case in which the wave function factorizes, i.e., is of theform Ψ ¼ Ψ q1; tð ÞΨ q2; tð Þ. In this case, the velocity of particle 1 does notdepend on the position q2 of particle 2, and vice versa. However, for a generalwave function, i.e., for an entangled quantum state, there is a correspondingentanglement of the velocities which persists, without attenuating in any way,even when the distance between the two particles is very large. This importantproperty of the guiding law shows that the velocity of a particle can depend onthe position of the other even when the module of the wave function is verysmall (which, in fact, is what happens when the distance between the twoparticles is very large).

Thus, in Bohmian mechanics, the law that governs the motion of particles inphysical space makes manifest the most dramatic effect of quantum mechanics,quantum non locality – the fact that physical events can mutually influence eachother more quickly than the speed of light, even at arbitrarily large distances,without this mutual influence being mediated by physical fields (such as, forexample, the electromagnetic or gravitational field) or by particles (or energy orsignals) that can somehow travel from one event to another. Far from being adefect, it is a remarkable merit of the theory, since, as Bell has shown,nonlocality is a basic property of nature. One could say that Bohmian mechanicsdoes nothing but to inherit and to make explicit the nonlocal character implicit inthe notion, common to all the formulations and interpretations of quantummechanics, of a wave function on the configuration space of a system with manyparticles.

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Many objections have been and continue to be raised against Bohmianmechanics. Some of them arise from the lack of understanding that the structureof Bohmian mechanics cannot be interpreted in classical terms, for example, interms of force or in terms of conservation laws, such as energy or momentum,or in terms of fields that propagate in physical space. Moreover, the wavefunction, which is part of the description of the state of a Bohmian universe(and therefore part of the reality of this universe) is not the sort of field on thephysical space to which we are accustomed in classical physics, but a field onthe space of all possible configurations, and the role of the wave function inBohmian mechanics is that of determining the velocities of the particles, andtherefore, of making possible the formulation of a law of motion for stuff inphysical space.

4.7 Bohm’s Original Formulation

Unfortunately, Bohm chose the conceptual framework of classical mechanics toformulate his theory, which contributed, at least in part, to some of the objectionswe have just mentioned. Let us see how Bohmian mechanics can be dressed inclassical clothes.

If the two sides of the guiding Eq. (4.2) are derived with respect to time (and incalculating the derivatives with respect to the time of the r.h.s., one uses theSchrödinger equation), one obtains the equation

mkd2Qk

dt2¼ �rkU �rkW (4.6)

where W is a certain function of the positions of all the particles, which isdetermined uniquely by the square of the wave function in a rather complicatedway that, for the purposes of the present discussion, it is not necessary to makeexplicit; we just mention an important property of this function: If the wavefunction is multiplied by a constant, the value of W does not change. Bohm calledthe function W quantum potential (energy).

Eq. (4.4) has a Newtonian form: Its left-hand side (l.h.s.) is the second derivativewith respect to the time of the position of the k-th particle, i.e., its acceleration,multiplied by the mass of the particle. Therefore, it has the structure of the firstmember of the classical Newton equation (mass � acceleration). Now consider ther.h.s. of the equation: U is the potential energy of N interacting particles thatappears in the Schrödinger equation, a function that has exactly the same form as inthe corresponding classical situation. Thus the first term on the right is actually theforce, derived from the potential energy U, which in the corresponding classicalsituation would act on the k-th particle. If we interpret the second term as a force

Quantum Physics Grounded on Bohmian Mechanics 81

derived from the quantum potentialW, the r.h.s. of Eq. (4.6) can be interpreted as aforce – a force of classical origin added to a force of quantum origin.

However, this formulation of Bohmian mechanics in classical clothes has a cost.The most obvious cost is an increase in complexity: The quantum potential isneither simple nor natural. Moreover, it does not seem very satisfactory to thinkthat the quantum revolution is reduced to the understanding that, after all, nature isclassical, with an additional force term, all in all fairly ad hoc, the term thatcorresponds to the quantum potential.

Furthermore, Bohmian mechanics is not simply a reformulation of quantummechanics in classical terms with an additional force term. In Bohmian mechan-ics, the velocities are not independent of the positions, as in the classical case, butare constrained by the guiding equation. Despite the apparent form of the secondorder of Newton’s quantum equation, its solutions are not characterized bypositions and velocities at some initial time, but only by the initial positions,because the initial speeds are determined by the guiding law at the initial time. Inother words, having derived the members of the guiding equation with respect totime, and thus obtained the quantum Newton equation, does not change thenature of the theory; at most it makes it more complicated and less transparent.It is as if in classical mechanics we derived both members of the Newtonequation and decreed that the equation of the third order thus obtained shouldbe considered the classical law of motion. In this way we would not obtain agenuinely different theory, that is, a third-order theory, with positions, velocities,and accelerations as initial independent conditions, because the initial acceler-ations would in any case be bound by the Newton equation to be functions ofinitial positions and velocities.

Because the dynamics of Bohmian mechanics is completely defined by theSchrödinger equation and the guiding equation, there is no need for further axiomsinvolving a quantum potential. Therefore the quantum potential, together with thequantum Newton equation in which it appears, should not be considered funda-mental. The correct way to look at Bohmian mechanics is as a first-order theory, inwhich the fundamental quantities are the positions of the particles, whose dynam-ics is specified by the guiding equation. Second-order concepts, such as acceler-ation and force, work and energy, play no fundamental role in the theory. Theartificiality suggested by the quantum potential is the price you pay when dressinga highly nonclassical theory with classical clothes. This does not mean that thesesecond-order concepts cannot play any role in Bohmian mechanics. These areemerging notions, which are fundamental for the theory to which Bohmianmechanics converges in the classical limit – that is, Newtonian mechanics: Whenthe quantum force is negligible, one has in effect classical behavior (Allori, Dürr,Goldstein, and Zanghì 2002).

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4.8 The Classical Variables of Bohr out of the Classical Variables of Bohm

The statistical character of quantum theory was first fully acknowledged in1926 by Max Born, shortly after Schrödinger discovered his famous equation.Born interpreted Schrödinger’s function in a statistical sense and postulated thatthe configuration Q of a quantum system is random, with probability distributiongiven by the density Ψ qð Þj j2. Under the influence of the growing consensus infavor of the Copenhagen interpretation, the density Ψ qð Þj j2 began to be consideredthe probability of finding the configuration Q, whatever it was measured, ratherthan the probability that the configuration was really Q, a notion that was believedto be meaningless. In accordance with these quantum probabilities, measurementsperformed on a quantum system with a defined wave function typically providerandom results.

The density Ψ qð Þj j2 takes on a particular importance in Bohmian mechanics. Asan elementary consequence of the Schrödinger equation and the guiding equation,it is equivariant, in the sense that these equations are compatible with respect to thedistribution Ψ qð Þj j2. More precisely, this means that if the configuration Q of asystem is random, with distribution Ψ qð Þj j2 at some time, then this will also be truefor any other time. This distribution is therefore called the quantum equilibriumdistribution.

A Bohmian universe, although deterministic, evolves in such a way that anappearance of randomness emerges, precisely as described by quantum formalism.In order to understand how this comes about, one should realize that, in a worldgoverned by Bohmian mechanics, the measuring apparatuses are also made ofparticles. In a Bohmian universe, the measurement apparatus, tables, chairs, cats,and other objects of our daily experience are simply agglomerations of particles,described by the positions they occupy in physical space, and whose evolution isgoverned by Bohmian mechanics.

The following theorem is crucial:

Theorem 1 Bohmian mechanics provides the same predictions of quantum mech-anics (assuming the same Schrödinger equation for both) for the results of anyexperiment carried out on a system with a wave function ψ, if at the beginning ofthe experiment the configurationQ of the largest system required for the analysis of theexperiment is random, with probability density given by the quantum equilibriumdistribution, i.e., the probability distribution that quantummechanics prescribes forQ.

The “largest system required for the analysis of the experiment” means thecomposite system that includes the system on which the experiment is performed,as well as the measuring apparatus and all the other instruments used in theexecution of the experiment, together with all the other systems that have signifi-cant interaction with them during the experiment.

Quantum Physics Grounded on Bohmian Mechanics 83

If we denote the configuration of the particles forming the measured system byX, and the configuration of all the rest with Y, we have the natural decompositionQ ¼ X;Yð Þ. For example, we can think that X represents the positions of theelectrons, protons, and neutrons that form a silver atom, while Y represents thepositions of the electrons, protons, and neutrons that make up the magnets andthe screen of a Stern-Gerlach apparatus.

The assumption that the measured system has a well-defined initial wavefunction ψ xð Þ can be guaranteed by assuming that the initial wave function ofthe large system is of the product form Ψ qð Þ ¼ ψ xð ÞΦ0 yð Þ. This condition guaran-tees the initial independence of the system to be measured and the apparatuses andinstruments used in the experiment. In particular, we can think that the functionΦ0 yð Þ represents the initial READY state of the apparatus. The assumption that Qis initially in quantum equilibrium means that the initial randomness of Q ¼ X; Yð Þis governed by the probability density ψ xð Þj j2 Φ0 yð Þj j2, that is, the probabilitydistribution prescribed by quantum mechanics for the initial configuration.

Precisely because of this last assumption, the proof of the theorem is trivial,being nothing but an immediate consequence of the equivariance property. In fact,the initial configuration is transformed, by means of the guiding equation of thelarge system, in the final configuration at the end of the experiment. But, on thebasis of the hypothesis of quantum equilibrium – that the initial configuration ofthe large system is random, with distribution given by Ψ qð Þj j2 – the final configur-ation of the large system, which includes, in particular, the orientation of themeasuring devices, will be distributed according to the distribution Ψ qð Þj j2 at thefinal time. Thus, the macroscopic variable Z describing the result of the experi-ment, which is a function Z ¼ f Qð Þ of the final configuration Q of the system andapparatus, have in Bohmian mechanics and in quantum mechanics exactly thesame statistical distribution. So, whenever the predictions of quantum mechanicsfor the results of a certain experiment are unambiguous – ambiguity in orthodoxformulation arises, inter alia, from the absence of a well-defined microscopicdynamics – these predictions must necessarily coincide with those of Bohmianmechanics. It is important to be clear that, as far as the measured system isconcerned, we are not just talking about position measurements, but about anymeasurement or experiment, for example, a spin measurement. Theorem 1 estab-lishes that Bohmian mechanics provides the same predictions of quantum mech-anics for the measurement of any quantum observable.

Both the statement of the theorem and its proof contain some implicit assump-tions that we want to bring to light and comment. First, we assumed that the resultof any measurement or experiment can be, at least potentially, registered in termsof a macroscopic configuration Z ¼ f Qð Þ. However, this is in complete agreementwith the experimental practice and with any interpretation of quantum mechanics,

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including Bohr’s interpretation, in which the macroscopic configurations of objects(and their classical behavior) are assigned a privileged status. A more delicateassumption concerns the initial assignment of a wave function ψ ¼ ψ xð Þ to thesystem. Whereas in ordinary quantum mechanics one can always appeal to anexternal observer who somehow “prepares” the initial wave function, in a theory inwhich observers play no privileged role, and in particular, in a strictly deterministictheory such as Bohmian mechanics, the situation seems more problematic.

Nevertheless, this problem has been addressed and solved (Dürr et al. 1992).The main ingredients for its resolution are substantially three. The first ingredient,of a technical nature, concerns the clarification of the notion of random system in adeterministic theory (a clarification of the conditions according to which theidentity of a particular subsystem and the time in which it is identified are reflectedin the external environment of the system, and therefore, in the last instance, in theinitial conditions). The second ingredient concerns the ways in which, in Bohmianmechanics, it is possible to assign a wave function to a subsystem of a largersystem – what in Dürr et al. (1992) has been called the conditional wave function.Finally, the third ingredient concerns precisely the clarification of one of the crucialassumptions of the theorem: The hypothesis of quantum equilibrium, i.e., theassumption that, if the wave function is Ψ ¼ ψ qð Þ, then the configuration Q israndom with probability distribution Ψ qð Þj j2. Justifying this hypothesis of quantumequilibrium is in fact a rather delicate matter, a problem that has been investigatedin a very detailed manner (see Dürr et al. 1992) by showing that the probabilitiesfor the positions given by the quantum equilibrium distribution Ψ qð Þj j2 emergenaturally from an equilibrium analysis of the deterministic dynamical systemdefined by Bohmian mechanics, in roughly the same way that the Maxwellianvelocity distribution emerges from the equilibrium analysis of a gas.

4.9 Classical Variables and Operators

Since orthodox quantum theory provides us with a vast class of observables andproperties in addition to particle positions, it would seem, at first glance, that thistheory is much richer than Bohmian mechanics, which seems to relate exclusivelyto positions. It turns out, however, that the quantum observables, represented byself-adjoint operators, arise from an analysis of quantum experiments. The keytheorem is the following (Dürr, Goldstein, and Zanghì 2004):

Theorem 2 Consider an experiment carried out on a system with a wave function,and in addition to the hypotheses of Theorem 1, assume that the experiment isreproducible. Then, under minimal technical conditions, there exists a self-adjointoperator A that regulates the statistics of the results Z ¼ f Qð Þ of the experiment in the

Quantum Physics Grounded on Bohmian Mechanics 85

manner prescribed by quantum mechanics. In particular, the average value of theresults is given by the usual formula Zh i ¼ ψ;Aψh i.

The condition of reproducibility means that the quantum experiment should givethe same result if immediately repeated (after having brought the apparatus back toits READY state). It should be noted that if it is not assumed that the experiment isreproducible, the statistics of the experiment is given by a more general operatorstructure than that of the self-adjoint operators, namely that of generalized quantumobservables that has been introduced to describe quantum measurements in par-ticular situations encountered in quantum optics, and more generally, in the theoryof open quantum systems (these are the positive-operator-valued measures[POVM], also used in quantum information). Thus, Bohmian mechanics providesan immediate understanding of this generalization of the notion of quantumobservable and a clarification of the type of idealization involved in the notion of“operator as observable.”

The Stern-Gerlach experiment is particularly illuminating to clarify the contentof Theorem 2. In the Stern-Gerach experiment, as a consequence of the interactionwith the magnetic field, the parts of the wave function that are in the differentautospaces of the relevant spin operator (for example, the component along z)become spatially separated and the particle (the silver atom), which movesaccording to the guiding equation, ends up being in the support of only one ofthese parts. The end result (“up” or “down”) is therefore a function of the finalposition of the particle, which is revealed on the screen. Of this position, what wecan predict is only that it is random and distributed according to Ψ qð Þj j2 at the finaltime. By calibrating the results of the experiment with numerical values, forexample +1 for a top detection and �1 for a bottom detection, it is not difficultto show (by solving the Pauli-Schrödinger equation and the guiding equation forthis situation) that the probability distribution of these values is expressed in termsof the usual quantum spin operators – the Pauli matrices.

It is important to observe that, since the results of a Stern-Gerlach experimentdepend not only on the position and initial wave function of the particle, but alsoon the choice of the different possible magnetic fields that could be used tomeasure the same spin operator, this experiment is not a genuine measurement inthe literal sense; that is, it does not reveal a preexisting value associated with thespin operator. Indeed, there is nothing mysterious or nonclassical about the non-existence of such values associated with the operators. Bell said that (for Bohmianmechanics) spin is not real. Perhaps he should have said, “even spin is not real,”not simply because among all the quantum observables, the spin is considered tobe the paradigmatic quantum observable, but also because spin is treated inorthodox quantum mechanics in a fairly similar way to position, as a “degree of

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freedom,” a discrete index that completes the continuous degrees of freedomcorresponding to the position. Be that as it may, its fundamental significance isthis: Unlike the position, spin is not primitive, i.e., it should not be added to thestate description as a real degree of freedom, analogous to the real positions of theparticles. Speaking roughly, spin is simply in the wave function. At the same time,as we have just shown, spin measurements are completely clear and simply reflectthe role played by the spin wave functions in the particle motion description.

Moreover, Theorem 2 provides a clarification of the content of the impossibilitytheorems for hidden variables. Let us look at the obvious correspondence betweenthe experiments and the Z-variables that represent the results of the experimentsthemselves (assuming, as for the spin, that the measurement instrument does notintroduce any external randomness and therefore that the Z-statistics is given by aself-adjoint operator on the Hilbert space of the system on which the experiment isperformed): EXPERIMENT!Z. Moreover, since Theorem 2 establishes that a(reproducible) experiment corresponds to a self-adjoint operator that describes thestatistics of the results, there is a correspondence between these variables and theself-adjoint operators: Z!OPERATOR. But there is no reason to expect thiscorrespondence to be invertible, that is, that the correspondence OPERATOR!Zexists, which is the premise of the theorems of nonexistence of hidden variables.

Therefore, the widespread idea that in a realist quantum theory all quantumobservables should possess real and preexisting values was ab initio an idea not asreasonable as it might seem, but rather the fruit of the prejudice of having taken theoperators too seriously – an attitude almost implied in the words ‘observable’ or‘property.’

4.10 The Collapse of the Wave Function

According to the quantum formalism, an ideal quantum measurement performedon a quantum system causes a jump or “collapse” of its wave function into one ofthe eigenstates of the measured observable. But while in orthodox quantummechanics collapse is simply superimposed on the unitary evolution of the wavefunction, without a specification of the circumstances in which it can legitimatelybe invoked – and this ambiguity is nothing but another facet of the problem ofmeasurement – Bohmian mechanics incorporates both the unitary evolution andthe collapse of the wave function as appropriate.

This claim might seem puzzling, since in Bohmian mechanics the wave functionevolves unitarily according to the Schrödinger equation. However, since anyobservation implies interaction, a system under observation cannot be an isolatedsystem, but must be a subsystem of a larger system that is isolated (for example,

Quantum Physics Grounded on Bohmian Mechanics 87

the entire universe). Additionally, there is no a priori reason that a subsystem of aBohmian universe is itself a Bohmian system – a system governed by its wavefunction that evolves according to its Schrödinger equation – even if the system isto some extent isolated. In truth, it is not even clear a priori what should beinterpreted by the wave function of a subsystem of a Bohmian universe.

Recall the comments after Theorem 1: The configuration Q of the largestsystem – the Bohmian universe – naturally separates into X, the configuration ofthe subsystem, and Y, the configuration of his environment. LetΨ ¼ Ψ qð Þ ¼ Ψ x; yð Þ be the wave function of the universe. According to Bohmianmechanics, this universe is completely described by Ψ, which evolves according tothe Schrödinger equation, and by the X and Y configurations. There is therefore afairly natural choice of what should be considered the wave function of thesubsystem: It is the conditional wave function ψ xð Þ ¼ Ψ x; Yð Þ, which is obtainedby inserting the actual configuration of the environment in the wave function of theuniverse. (Note that orthodox quantum mechanics lacks the resources necessary todefine the conditional wave function. From an orthodox point of view it is not at allclear what should be understood by wave function of a subsystem). Consider nowhow the conditional wave function depends on time, ψ x; tð Þ ¼ Ψ x;Y tð Þ; tð Þ. It isnot difficult to show that ψ x; tð Þ obeys the Schrödinger equation for the subsystemwhen this is appropriately decoupled from its environment, and that using thehypothesis of quantum equilibrium, it undergoes, when the appropriate conditionsare realized, processes of collapse, that is, random transformations in completeagreement with the rules of the quantum formalism. In Bohmian mechanics, therule of the collapse of the wave function is a theorem and not an axiom as inorthodox quantum mechanics.

4.11 The Paradox of Schrödinger’s Cat

In Bohmian mechanics the cat is alive or dead at any time, regardless of whether itis observed or not. Even if its wave function is in a superposition

ψ ¼ 1ffiffiffi2

p ψdead xð Þ þ ψalive xð Þ½ � (4.7)

the cat is an agglomeration of particles, described by their actual configuration X,which determines its microscopic and macroscopic properties at each moment –first of all, its property of being alive or being dead. The two wave functions of thesuperposition discussed earlier have macroscopically disjoined supports, sayMalive

and Mdead, which create a natural partition of the configuration space of the catM ¼ Malive [Mdead; the actual configuration X of the cat is either in Malive or inMdead, i.e., the cat is alive or is dead.

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If at certain time we have that X 2 Mdead (as it results from appropriate condi-tions of the external environment Y, for example, from the temperature value of thecat measured with a thermometer), the relevant wave function for the subsequenttemporal evolution of the configuration of the cat is ψdead xð Þ. In other words, theguiding law provides the same temporal evolution for the cat configurationwhether we use the whole wave function ψ, or just its ψdead xð Þ. Using only thelatter, that is, applying the rule of collapse, is therefore, in Bohmian mechanics,only a practical matter that does not change anything of what actually happens inthe world – the actual history of the cat and, more generally, of the physicalsystems that, like cats, populate the world.

The following question arises: Can a dead cat become alive in the future? Whatis it that forbids the X configuration, originally in Mdead, to enter at a certainmoment of the future in the region Malive? Strictly speaking, nothing. In Bohmianmechanics, such an event is possible, although the probability of its occurrence isfrighteningly small, ridiculously small. However, the mere possibility that such anevent is realized does not involve any interpretative problem for the theory. On thecontrary, it only emphasizes that the explanatory structure of Bohmian mechanicsis quite analogous to that of classical statistical mechanics.

In fact, an event of this type, extraordinary and highly improbable, is alsopossible in the dynamical scheme provided by classical mechanics, as it is possiblethat all the air in this room spontaneously leaves the window and I die fromsuffocation, and if it were possible to imagine a cat such that the particles thatcompose it were governed by the laws of classical mechanics, this hypotheticaldead cat could return alive. Events of this type are possible, but highly improbablebecause they would entail a decrease in the entropy of the universe. The explan-ation of the impossibility for the cat to return alive is, in Bohmian mechanics,exactly of the same type: It is guaranteed by simple entropic reasons, that is, by thetypical behavior of the physical systems according to the second law ofthermodynamics.

The difference with the classical case concerns only how the macroscopicthermodynamic laws are stabilized by the underlying microscopic dynamics. Inthe quantum case, quantum entanglement plays a very important role in the processof stabilizing macroscopic properties of physical systems and makes (if everneeded) it extremely unlikely that dead cats can resurrect. In fact, even if theregions Malive and Mdead were not macroscopically disjoint, but only sufficientlydisjoint, the interaction of the system with the external environment and theconsequent formation of entangled states in superpositions containing many terms,would produce a partition of the configuration space of the larger system (whichincludes the starting system and its external environment) in distinct regions, eachin a two-way correspondence with a term of the superposition of the wave function

Quantum Physics Grounded on Bohmian Mechanics 89

of the largest system. These regions would be macroscopically disjoint, and thegreater the number of external environment systems that come into play, thegreater the macroscopic separation that is achieved, and thus the greater the degreeof irreversibility of the process.

4.12 Relativity

Many of the objections to Bohmian mechanics – if not the overwhelming major-ity – are very weak and often arise from a gross misunderstanding of the theory.Some of them arise from the lack of understanding that Bohmian mechanics shouldbe considered, mathematically and conceptually, as a theory profoundly differentfrom Newtonian mechanics.

The most serious objection raised against Bohmian mechanics is that this theorydoes not account for those phenomena, such as the creation and destruction ofparticles, which are characteristic of the quantum theory of fields. Actually, this isnot, in itself, an objection to the Bohmian mechanics, but simply the observationthat quantum field theory, for what concerns what happens in the physical world,explains much more than the nonrelativistic theory, be it in the orthodox or theBohmian form. However, this objection has the merit of highlighting how import-ant and necessary finding an adequate Bohmian version of quantum field theory is.We have already mentioned proposals in this direction, involving a stochasticdynamics according to which particles can be created and destroyed (Dürr, Gold-stein, Tumulka, and Zanghì 2004).

An objection in some way connected to the previous one is that Bohmianmechanics cannot be made invariant under Lorentz transformations, with what,presumably, it is meant that it is not possible to find any Bohmian theory – a theorythat could be considered as a natural extension of Bohmian mechanics – that isinvariant under Lorentz transformations. If it were correct, this objection should beseriously considered. However, it is not supported by any argument that makesplausible the alleged impossibility of finding an extension of the Bohmian mech-anics invariant under Lorentz transformations. The reason for this widespreadbelief is the manifest nonlocality of Bohmian mechanics. But, as Bell has shown,nonlocality is a fact established by the experiments.

Moreover, as regards the equally widespread belief that conventional quantumtheories would have no difficulty in incorporating Einstein’s relativity, whereasBohmian mechanics would have it, there is, even in this case, much less than meetsthe eye. One should always bear in mind that the empirical content of conventionalquantum mechanics is based on (1) the unitary evolution of the state vector (or theequivalent unitary evolution in the Heisenberg representation) and (2) the collapseor reduction of the wave function (or any other equivalent artifact that allows the

90 Nino Zanghì

effect of observations and measurements to be incorporated into the theory). Butthe Lorentz invariance of the latter is rarely considered – most of the empiricalcontent of conventional relativistic quantum mechanics concerns the scatteringregime. If this were done, the tension between Lorentz invariance and quantumnonlocality would be immediately apparent. On the other hand, various approachesfor constructing an invariant Bohmian theory for Lorentz transformations havebeen proposed, some models have been formulated and discussed (Berndl, Dürr,Goldstein, and Zanghì 1996), and the subtleties involved in the task of constructinga relativistic version of Bohmian mechanics have been thoroughly analyzed (Dürr,Goldstein, Norsen, Struyve, and Zanghì 2014).

4.13 What Is a Bohmian Theory?

In the structure of a Bohmian theory, one can see some very general features thatare, in fact, common to all the precise and serious formulations of quantummechanics that are not based on vague and imprecise concepts such as measureor observer:

1. The theory must be based on a clear ontology describing what the theory isfundamentally about – what has been named the primitive ontology (Allori,Goldstein, Tumulka, and Zanghì 2008) – that is, the type of basic entities (likethe particles in Bohmian mechanics) that are the building blocks for anythingelse in the world, including tables, chairs, cats, and measuring devices. Theprimitive ontology may include geometry, particle, field, or string configur-ations, or whatever is needed out to best describe nature.

2. There must be a quantum state vector, a wave function, which evolves (at leastapproximately) unitarily and whose role is to generate the temporal evolution(which does not need to be deterministic) of the variables that describe theprimitive ontology.

3. The empirical relevance of the theory should emerge from its providing a notionof typical spatiotemporal histories, presumably specified by a measure of typic-ality on the set of all possible histories of the primitive ontology of the theory.

4. The predictions must agree (at least approximately) with those of orthodoxquantum theory – in the cases that the latter ones are unambiguous.

In brief, a Bohmian theory is simply a quantum theory with a coherent ontology.

References

Allori, V., Dürr, D., Goldstein, S., and Zanghì, N. (2002). “Seven steps towards theclassical world,” Journal of Optics B, 4: 482–488.

Quantum Physics Grounded on Bohmian Mechanics 91

Allori, V., Goldstein, S., Tumulka, R., and Zanghì, N. (2008). “On the common structureof Bohmian mechanics and the Ghirardi-Rimini-Weber theory,” British Journal forthe Philosophy of Science, 59: 353–389.

Bell, J. S. (1964). “On the Einstein-Podolsky-Rosen paradox,” Physics, 1: 195–200.Reprinted in Bell (1987).

Bell. J. S. (1986). “Beables for quantum field theory,” Physics Reports, 137: 49–54.Reprinted in Bell (1987).

Bell. J. S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge:Cambridge University Press.

Bell. J. S. (1990). “Against measurement,” Physics World, 3: 33–40.Berndl, K., Dürr, S., Goldstein, S., and Zanghì, N. (1996). “Nonlocality, Lorentz invari-

ance, and Bohmian quantum theory,” Physical Review A, 53: 2062–2073.Bohm, D. (1952). “A suggested interpretation of the quantum theory in terms of ‘hidden’

variables: Parts I and II,” Physical Review, 85: 166–193.Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N. (1996). “Naive realism about

operators,” Erkenntnis, 45: 379–397.Dürr, D., Goldstein, S., Norsen, T., Struyve, W., and Zanghì, N. (2014). “Can Bohmian

mechanics be made relativistic?,” Proceedings of the Royal Society A, 470: 20130699.Dürr, D., Goldstein, S., Tumulka, R., and Zanghì, N. (2004). “Bohmian mechanics and

quantum field theory,” Physical Review Letters, 93: 090402.Dürr, D., Goldstein, S., and Zanghì, N. (1992). “Quantum equilibrium and the origin of

absolute uncertainty,” Journal of Statistical Physics, 67: 843–907.Dürr, D., Goldstein, S., and Zanghì, N. (2004). “Quantum equilibrium and the role of

operators as observables in quantum theory,” Journal of Statistical Physics, 116:959–1055.

Feynman, R., Morinigo, F., and Wagner, W. (2003). Feynman Lectures On Gravitation.Boca Raton: CRC Press.

Ghirardi, G. C. and Weber, T. (1983). “Quantum mechanics and faster-than-light commu-nication: Methodological considerations,” Nuovo Cimento, 78B: 9–20.

Gleason, A. M. (1957). “Measures on the closed subspaces of a Hilbert space,” Journal ofMathematics and Mechanics, 6: 885–893.

Kochen, S. and Specker, E. P. (1967). “The problem of hidden variables in quantummechanics,” Journal of Mathematics and Mechanics, 17: 59–87.

Landau, L. D. and Lifshitz, E. M. (1958). Quantum Mechanics: Non-relativistic Theory.J. B. Sykes and J. S. Bell (trans.). Oxford and New York: Pergamon Press.

Schilpp, P. A. (ed.). (1949). Albert Einstein, Philosopher-Scientist. Evanston, IL: TheLibrary of Living Philosophers.

Schrödinger, E. (1995). The Interpretation of Quantum Mechanics. Dublin Seminars(1949–1955) and Other Unpublished Essays. M. Bitbol (ed.). Woodbridge: OxBow Press.

von Neumann, J. (1932/1955). Mathematische Grundlagen der Quantenmechanik. Berlin:Springer Verlag. R. T. Beyer (trans.), Mathematical Foundations of Quantum Mech-anics. Princeton, NJ: Princeton University Press.

Wooters, W. K. and Zurek, W. H. (1982). “A single quantum cannot be cloned,” Nature,299: 802–803.

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5

Ontology of the Wave Function and theMany-Worlds Interpretation

lev vaidman

5.1 Introduction

Quantum theory is about a century old, but as the existence of this volume shows, weare far from consensus about its interpretation. Science does not develop in a straightline. For a decade, quantum theory had no real basis, only phenomenologicalequations found by Bohr, who made many of us believe until today that quantummechanics cannot be understood. The relativistic generalizations of the Schrödingerequation, however, provide a complete, elegant physical theory that is fully consist-ent with experimental data with precision of up to 10 significant digits. The situationtoday is much better than at the time of Lord Kelvin’s speech in 1900, in which heargued that physics is almost finished except for solving “two clouds,” which werelater to become the theory of relativity and quantum theory. This error, and the factthat quantum equations describe several outcomes for quantum measurementsalthough we always see just one, are probably the main reasons why contemporaryphysicists are reluctant to state that physics is close to being finished.

To deal with this second problem, we either have to add something to the waveequation, but no proposal attractive enough to reach consensus has been found, orto admit that what we see is only a tiny part of what is and that there are multipleparallel worlds similar to ours. I find that this last option is the only reasonable one,and I hope it will reach consensus in a foreseeable future.

5.2 A Toy Model of Classical Mechanics

Before discussing quantum mechanics, which is clearly not simple since we arevery far from consensus about its interpretation, I would like to discuss a toy modelof classical mechanics. It is an implementation of Laplace Universe:

We ought to regard the present state of the universe as the effect of its antecedent state andas the cause of the state that is to follow. An intelligence knowing all the forces acting in

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nature at a given instant, as well as the momentary positions of all things in the universe,would be able to comprehend in one single formula the motions of the largest bodies aswell as the lightest atoms in the world, provided that its intellect were sufficiently powerfulto subject all data to analysis; to it nothing would be uncertain, the future as well as the pastwould be present to its eyes. The perfection that the human mind has been able to give toastronomy affords but a feeble outline of such an intelligence.

(Laplace 1820/1951)

I assume that the clouds Lord Kelvin talked about do not exist. Newton’s laws andMaxwell’s equations are somewhat different, such that they provide a consistenttheory for microscopic particles moving on continuous trajectories, which, usingthe methods of statistical mechanics, explains well all experimental data. It is alsoan assumption about different experimental data, because we know that actuallyobserved data cannot be explained by a classical model.

In the model we consider, the three-dimensional space is given. There areparticles moving in well-defined trajectories and fields spread out in space. Par-ticles create fields that propagate in space and change the motion of other particlespresent in locations with nonvanishing fields. The laws of creation and propagationof fields explain the existence of stable rigid objects and everything else (includingourselves) that we experience with our senses. The behavior of objects is deter-ministic; free will is an illusion.

Like in actual physics, our model can be presented in a different way. There is apoint in the configuration space of all particles and the configuration of fieldsfulfilling some global equation of extremal action. The global laws provide thesame solution for trajectories. Both explanations are acceptable, but I feel that it isthe first presentation, with fields locally acting on particles moving on trajectoriesin three-dimensional space, that is a more convincing explanation of the world.The existence of a global mathematical representation is important, but it hides thelocal causal story, which is what is considered as an explanation of motion ofmicro systems as well as our behavior.

It seems to me that in a counterfactual universe with successful classical physicsas described earlier, there will be no philosophical controversy in how to describereality. Particle trajectories governed by local forces through fields in three dimen-sions would be a clear consensus.

5.3 Ontology of Collapsed Wave Function

Our world is not classical. Numerous experiments (e.g., particle interference)contradict this picture. Moreover, Bell-type correlations show that no classical-type local theory, i.e., a theory that locally predicts a single outcome for eachpossible experiment, can reproduce observed correlations. We have a new theory,

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quantum mechanics. It is true that the community of physicists working in the fieldof foundations of quantum mechanics has not reached any consensus over itsinterpretation. Many of them (like me) are certain that their favorite view is asatisfactory (or even an excellent) solution, but each separate group is a smallminority, so the message of the community as a whole is that currently there is nogood solution. There are many, also outside the foundations community, who feelthat we need the correct interpretation, but that it is different from all currentproposals. However, the majority of physicists really think that the problem doesnot exist and that textbook quantum mechanics is satisfactory. It tells us that everytime we perform a quantum measurement there is a collapse of the quantum wavefunction and that the collapsed wave function well describes all that we see aroundus. Von Neumann proved that the tough question of when exactly collapse occursneeds not to be answered, because wherever we put the cut between classical andquantum, we will observe no contradiction with our experience. A vague statementaccording to which all “macroscopic” objects are “well” localized provides asatisfactory criterion.

Von Neumann collapse is ad hoc without any concrete mechanism. In physicalcollapse theories, such as those of Pearle (1976), Ghirardi, Rimini, and Weber(1986), Diosi (1987), and Penrose (1996), the collapsed wave function is notcompletely identical to that of von Neumann, but it is very close; so proponentsof physical collapse theories also consider the collapsed wave function as asatisfactory description of what we see. Apparently it is the promotion of the wavefunction ontology in configuration space by Albert (2013) that led to strongcriticism. Maudlin (2013) understandably complained: How can a mathematicalobject in high dimensional space represent our experience in three dimensions?

The key to answering this question is the understanding that our experiencesupervenes on macroscopic objects. We do not directly experience the electronwave function in the atoms of our body. Parts of our body and our neurons aremacroscopic bodies. We needed configuration space because of entanglement.Whereas for describing classical particles we had a choice between one point in3N dimensional space or N points in three-dimensional space, in the quantum casewith entanglement the second option does not exist, entanglement requires multi-dimensional space. Complete descriptions of all particles separately do not providea complete description of entangled particles. But entanglement of quantumsystems does not exist for macroscopic systems. In von Neumann’s approach, itis absent by definition, and in physical collapse theories, the mechanism removesentanglement of macroscopic objects very quickly. Without entanglement, we candescribe all objects in three dimensions. To summarize, the analysis of the processof obtaining experience from our senses when the universe is described by acollapsed wave function of a textbook or by the collapsed wave function of a

Ontology of the Wave Function and the Many-Worlds Interpretation 95

physical collapse proposal provides an explanation similar to that of classicalphysics: (Macroscopic) objects moving in three-dimensional space locally triggersensory organs of observers living in this three-dimensional space.

Current (quantum) physical theory based on minimal action does not alwaysprovide a simple local explanation of the behavior of particles as in our gedankenclassical model. The basic concept of quantum theory is that referring to potentials,which are not measurable locally. Classical physics also uses potentials, but theyare just auxiliary tools, helping to solve various problems more efficiently. Inclassical physics, the operational meaning of potentials is that their derivativesprovide observable forces. In quantum mechanics, the Aharonov-Bohm effectteaches us that potentials provide a direct physical effect: A particle moving in afield-free region behaves differently depending on the potentials present there(Aharonov and Bohm 1959). At least for myself, I resolved the difficulty byfinding a local explanation of the Aharonov-Bohm effect based on consideringthe source of the potential to be quantum and taking into account entanglementbetween the electron and the source (Vaidman 2012a). This explanation removedthe proof that quantum theory based on direct action of local fields cannot exist.I opened the way to such a theory, which, if constructed, will be conceptually moresatisfactory than the present one.

5.4 Against Collapse

The nonlocality of the Aharonov-Bohm effect makes a physical explanation moresophisticated, but the main ugly scar of quantum mechanics is collapse. I want tobelieve that today’s physics by and large correctly explains everything we seearound us. For this belief I need two ingredients. First, that predictions of thetheory correspond to what we see, i.e., to be confirmed by experimental results.Second, that the theory is elegant enough to believe that this is the description ofnature. We have had incredible success with the first part. In all cases we cancalculate and measure, there is a complete agreement, and sometimes with astro-nomical precision of more than 10 digits. There are no “clouds” similar to thoseseen by Lord Kelvin in classical physics. There is also incredible success of thetheory. All basic laws of physics can be written on a t-shirt. Today’s quantummechanics (even with potentials) is a good theory. It is deterministic and it does nothave action at a distance. Only the collapse spoils it. It is the only random processin physics.

Collapse is sometimes considered to be in “peaceful coexistence” with specialrelativity. Indeed, we cannot send superluminal signals using the collapse process.But it is an action at a distance. A particle described by a superposition of wavepackets in two separate locations leads to action at a distance when the presence of

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a particle in one location is measured. The complete quantum description of aregion of space with the second wave packet before the measurement is a mixedstate. It has the following operational meaning: Everyone, everywhere, has a well-defined betting strategy on the result of a measurement testing the presence of theparticle in this place. Measurement in the location of the first wave packet changesthe situation immediately. Obviously so if the wave function is an ontic entity,because it is changed by a remote action, but also if it is only an epistemic concept.There is a change in operational meaning. If I and you are in the location of the firstwave packet, you are ready to bet with me on the result of the remote measurement,but this changes immediately after you observe that I made a measurement on thefirst wave packet.

Contrary to a widespread belief, there is no evidence for collapse of the wavefunction. It is true that for every quantum experiment we observe only oneoutcome, while a theory without collapse has a wave function corresponding toseveral outcomes. But to be considered evidence, we need for the theory to predicta different experience in case the collapse takes place or not. The theory does notpredict this. In fact, looking on a wave function of quantum mechanics withoutcollapse it is not easy to make any prediction about experience. We need to addsome postulates to connect the ontology with our experience.

5.5 Connecting Ontology with our Experience

Let us start the analysis with an interpretation of quantum mechanics where theconnection is simple. In Bohmian mechanics (see, e.g., Goldstein 2017), thepostulate is that experience supervenes on Bohmian positions of particles andthe wave function is only a pilot wave of these particles. The Bohmian picture ofpositions of all particles is similar to the gedanken classical theory I sketchedearlier. All particles provide a familiar picture of what we see around us drawnin three dimensions in a pointillist style. Bohmian particles, according toBohmian mechanics postulates, are distributed in the locations of a nonvanish-ing wave function according to the Born Rule. But the size of the atoms, andthus the size of the wave packets of elementary particles, is so small that thedifference between locations of Bohmian points and centers of particle’s wavepackets is inessential. Thus, considering the centers of the wave functions ofparticles in the collapsed wave function also provides a familiar picture. The factthat most of the elementary particles are entangled and do not have a pure statedoes not spoil this picture. Their mixed states are still well localized on theatomic scale, so the expectation values of their positions are not far from thosecorresponding to Bohmian positions.

Ontology of the Wave Function and the Many-Worlds Interpretation 97

Bohmian particles describe a world that looks like the one we observe and soalso does the collapsed wave function. Because the wave functions of macroscopicobjects are well localized, the picture drawn by the expectation values of theposition vectors of all particles of these objects also provides a familiar picture.The particles are very close to each other, so we do not observe the points. Weobserve the smoothed picture of everyday objects: tables, cats, people, etc. – thecollapsed wave function provides a familiar picture of the world. The theory has atacit assumption, a postulate: Our everyday experience supervenes on the collapsedwave function. Since the picture drawn by the particles and the picture we drawbased on our experience are so similar, we usually forget that we make anassumption connecting the formalism with experience. The theory is supposed todescribe our experience. In this theory there is only one picture that looks like theworld we see, we assume that there is only one world, so naturally we connectthem. But the postulate is needed, because there can be other options: We can alsoimagine the presence of Bohmian particles in a theory that makes collapses, andattaches experience to Bohmian particles. It is a different rule, although it seems toprovide essentially the same experience.

5.6 Connecting Ontology with Experience in the Framework of the MWI

In quantum mechanics without collapse we must add a postulate to connect to ourexperience, because mathematics does not provide a (unique) picture correspond-ing to what we see around us.

My postulate will be as follows. The universe with a noncollapsing wavefunction corresponds to multiple experiences. Each experience should correspondto at least one world, the definition of the concept of a world does not allowmultiple simultaneous experiences of a person in a world. To connect the wavefunction of the universe to our experience, we first need to decompose the wavefunction of the universe into a superposition of wave functions of worlds.

Ψj iUNIVERSE ¼Xi

αi ψij iWORLD (5.1)

The wave function of a world ψij i must be of a particular type. Probably the mostinformative definition is that this is the type of wave function that might appear inthe textbook as “collapsed wave function of the universe.” Without relying on atextbook definition, this type of wave function can be defined by the property thatall macroscopic objects must be well localized. Both definitions are vague. Weusually require rigor and precision in our theories. However, it must be so when weconsider exact sciences. The many-worlds interpretation (MWI) has two clearlyseparated parts: (i) a precise physical theory of evolution of the wave function of

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the universe Ψj iUNIVERSE and (ii) the connection of this universal wave function toour experience(s) (see Vaidman 2002). In classical theory, in Bohmian theory, andin quantum mechanics with collapse, the separation between the two parts of thetheory was not emphasized because the second part was very simple: The connec-tion is natural and obvious in contrast to the MWI, where the second part issignificant. But I believe that the connection implicitly taken in single-worldtheories can be directly adopted to the MWI, and vagueness of the splitting ofthe worlds is much less problematic than the vagueness of the answer to thequestion: When does collapse occur? Our experiences cannot be described in termsof exact science. So, it is understandable and acceptable that the conceptsbelonging to the second part of the theory are not rigorously defined. The wavefunction of a world has the following form

ψij iWORLD ¼ ψ1CM r!CM

1

� �φ1rel r!1i � r!1j

� �ψ2CM r!CM

2

� �φ2rel r!2i � r!2j

� �. . .

ψMCM r!CM

M

� �φMrel r!Mi � r!Mj

� �ΦREST (5.2)

It is a product state of wave functions ψkCM r!CM

k

� �of centers of mass of macro-

scopic objects, times entangled states of relative coordinates of these objectsφkrel r!ki � r!kj

� �, times the wave function ΦRESTof the remaining particles that are

not part of macroscopic objects. The terms “macroscopic” and “well localized”might be chosen as more or less “fine grained,” so the decomposition Eq. (5.1) isonly approximately defined.

It seems to me that I can stop here. Textbook quantum theory is a well-established and well-tested theory that well explains everything we see aroundus. However, it includes the unphysical process of collapse, which makes it verydifficult to believe that it is true. I remove the collapse and use the same postulateof connection to our experience. Now, the theory is a good physical theory(deterministic, no action at a distance, and no ad hoc rules). The postulate of thecorrespondence of the experience with the wave function of the type of theuniversal wave function of a collapsed world makes the experience, by fiat,identical to that of an experience of an observer living in a universe with collapsingwave functions. However, I know that this picture is (still) not in a consensus.I need to persuade the community that it is consistent.

5.7 Naïve Criticism

The most naïve criticism is that we do not have experimental evidence for theMWI. How can all these numerous worlds be present in the same space as ourworld without us noticing their presence? Equations tell us that there is nointeraction between objects in different worlds. Moreover, we see in the laboratory

Ontology of the Wave Function and the Many-Worlds Interpretation 99

that, for micro systems, there is no scattering between wave packets of the sameelectron. Also note that we have no experimental evidence for the existence ofcollapse. Testing MWI versus collapse theories will require a quantum experimentup to a stage that the collapse proponents accept that collapse must have happenedand then undoing the experiment (Deutsch 1986, Vaidman 1998). If we get theoriginal state every time we perform this procedure, it is a proof that collapse didnot take place. Until today there is no sign of a collapse, but we are very, very farfrom a decisive experiment. If collapse proponents will claim that collapse happensonly after we write the result of a quantum experiment in a notebook, it can neverbe tested. Of course, it is the MWI that cannot be proved. If collapse exists at anearly stage, it will be observed (although it will not be easy to persuade opponentsthat the explanation of the signal is not a failure of experimentalists to preventdecoherence).

Slightly less naïve criticism is the problem of preferred basis. Mathematically,one can decompose the wave function of the universe into a superposition oforthogonal components, not just as in Eq. (5.1), but in many other ways that willnot provide a familiar world’s picture in every branch. So, critics might say that theproposal is circular: I define by fiat what I want to explain. First, a simple definitionthat is confirmed by observation sounds to me like a legitimate strategy. But thereis also a more specific answer. The basis of the decomposition is indeed preferred.Until now I have not mentioned time evolution. Everything was considered at aparticular moment. But we cannot experience anything at zero time. We need anorder of 0.1 seconds to identify our experience. Thus, the world needs some finitetime to be defined. The world has to be stable, at least on the scale of seconds.Locality of interactions in nature ensures that only the decomposition of wavefunctions corresponding to well-localized macroscopic objects can be stable.A quantum state describing the superposition of a macroscopic object in separatelocations with a particular phase evolves almost immediately into a mixture thathas a large component with a different phase. This obvious fact is analyzed innumerous papers using the buzzword ‘decoherence.’

5.8 The Probability Problem

The most difficult issue is probability. The idea of parallel worlds is relatively newfor humanity, so language and philosophy describing this situation is not welldeveloped. Lewis (1986) provided some insight, but the philosophical issuesrelated to plurality of worlds are far from clear.

The MWI that I advocate has no intrinsic uncertainty, randomness, or objectivechance, which makes the concept of probability difficult. In the collapse theory,probability has a clear meaning: The event happened while it could also not have

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happened. In the MWI, such meaning does not exist. The event happened in thisworld, but there is no alternative, it could not be otherwise. I want to say that thetraditional concepts of probability are not applicable in the framework of the MWIwhen we consider the outcomes of quantum experiments to be performed. Everytime we perform a quantum experiment and it seems to us that a single result isobtained, all possible outcomes are obtained, each in a different world. There is nomeaning to the question: In which world will I end up? In some sense I will be inall. In no sense will I be in a particular one.

Still, we have to explain our experience of apparent random behavior and thefrequency pattern of the results of quantum experiments. My claim remains thatthere is no difference between my experience if I live in one of the MWI worldsand my experience if I live in the only world of the universe with collapses onevery measurement. How do we reconcile the difference between the existence of aprobability concept and identity of experience? To avoid that difficulty, there areproposals to introduce uncertainty in the MWI and provide the meaning for theprobability that I will end up in a world with a particular outcome (Saunders andWallace 2008). In my view, adding uncertainty to the theory spoils it. The wavefunction of the universe is supposed to be the whole physical ontology, and it doesnot have any pointer moving from one world to another.

But do we really have a problem here? The fact that there is no meaning for theprobability of the result of a future measurement does not contradict the claim ofidentical experiences. The standard approach to probability is to consider eventsthat will happen, but testing probability claims relies on records of experiments inthe past – frequencies of the outcomes of repeated identical measurements per-formed in the past. So even in the framework of collapse theory, probabilityassignments are confirmed or refuted by our experiences in the past. Thus, thedifficulty of the MWI to introduce the concept of probability for future outcomes isnot relevant.

Assignment of probability for future experiment relies on an additional assump-tion, even in the framework of a collapse theory. We assume that nature will notchange its laws. Another way to make predictions about future measurement is toassume that, after performing the measurement, the frequency of the measurementresults fits our probability assignment. This approach may also be applied in theframework of the MWI. We expect that within the world, the frequencies of theresults of past measurements will correspond to the probability assignment.

5.9 Probability of Self-Location

The counterpart of the Born Rule in the framework of the MWI is sometimesnamed the Born-Vaidman Rule:

Ontology of the Wave Function and the Many-Worlds Interpretation 101

The probability of self-location of an observer in a particular world is proportionalto the measure of existence of that world.

The (somewhat controversial) term ‘measure of existence’ (Vaidman 1998, Grois-man, Hallakoun, and Vaidman 2013) is just the square of the amplitude of thecoefficient of the corresponding wave function in the decomposition of the univer-sal wave function Eq. (5.1). This rule explains the correspondence of experimentaldata of quantum experiments with the statistics predicted by the Born Rule.

Probability is a very controversial philosophical concept. Frequency of particularoutcomes in experimental records is just one approach. Another approach is readi-ness to put bets on the results of experiments according to their probability ofhappening. Since in the MWI all possible outcomes happen, it is difficult tounderstand this aspect of probability. I proposed a gedanken experiment whichallows sensible betting also in the framework of theMWI, tightening the connectionof the Born-Vaidman rule with the concept of probability. In my proposal, afterarranging a quantum experiment, the observer takes a sleeping pill (Vaidman 1998).During the observer’s sleep, the experiment is performed and the observer is movedto various rooms with identical interiors, according to the results of the experiment.When the observer is awakened, the observer understands that there are severalcopies of the observer in different rooms corresponding to the results of the experi-ment; however, the observer does not know who he or she is, and thus the observerdoes not know the result of the experiment in the observer’s world. The observermight be given the wave function of the universe; the observer still has ignorance,which allows the observer to bet. Awakened descendants of the experimentalistpreparing the experiment have a genuine ignorance concept of probability, similar tothat of an experimentalist in the collapsing universe. Only the questions are differ-ent. The latter asks: What will be the result of the experiment? The former asks: Inwhich world (defined by the result of the experiment) am I?

Note that all descendants have the same information and thus provide identicalassignments for probabilities of different outcomes. This allows them to define arational betting strategy for an experimentalist before the experiment. Instead of aprobability postulate, he or she has the caring principle (Vaidman 1998, Greaves2004):

The experimentalist cares about his descendants in proportion to their measure ofexistence.

The justification of this principle for our betting example is as follows. Everydescendant will have a genuine probability concept and would like to have a betaccording to a particular probability assignment. Since the descendants will get thereward of the bet, the experimentalist, naturally caring for his or her descendants,

102 Lev Vaidman

has a rational reason to put the bet for the results of the experiment. Tappenden(2011) suggested (and I think it is a reasonable approach) that there is no need toperform the complicated procedure with the sleeping pill. It is enough that one canimagine performing such a procedure to justify the betting assignment of theexperimentalist who believes in the MWI. So, identical experiences lead to identi-cal behavior, although the argumentation is different, which is not surprising inview of different world views.

5.10 Deriving the Born Rule

The betting assignments are according to the Born Rule or to the Born-VaidmanRule. The rules are postulates that are added to the formalism of quantummechanics of evolution of the wave function. In my view, the addition of apostulate is well justified through its confirmation by our observations. However,I do not want to leave unnoticed a large and increasing effort to derive the BornRule. Deutsch (1999) started a program (extended by Wallace 2012) to derive theproper betting behavior in the framework of the MWI based on some postulatesof Decision Theory. The program encountered criticisms of apparently circulardefinitions. The Deutsch-Wallace proof is complicated, and it is not simple tounderstand what exactly is proved and what is assumed. I also suggested a proof(Vaidman 2012b). My work was triggered by Deutsch’s (1999) paper, but theproof is apparently different. It is based on symmetry and relativistic causality.Apart from the fact that in the framework of collapse theory we have to postulaterelativistic causality, while in the framework of the MWI it is part of the physicaltheory, my proof is applicable to both cases (McQueen and Vaidman 2018). Veryrecently, Sebens and Carroll (2018) proposed yet another proof. They say itfollows my approach of self-locating uncertainty and relies on the assumptionthat actions on the environment cannot change outcomes of local measurements.This assumption is not very different from my relativistic causality; however, itseems to me that their proof does not hold. It is based on permutation symmetryand relies on a metaphysical approach to personal identity, the justification forwhich I cannot understand. Sebens and Carroll manipulate the concept of prob-ability of an observer to be in a particular world when splitting has alreadyhappened, but the wave function of the observer is identical in both worlds.Indeed, both the mental states and the (wave functions of the relative variables ofthe) bodies of the observers in the two worlds are identical. The worlds aredistinguished by macroscopic differences of other objects. In this case there is nomatter of fact about what is the world, out of the two worlds, in which theobserver is present, because the observer is present in both (see details in Kent2015, and in McQueen and Vaidman 2018).

Ontology of the Wave Function and the Many-Worlds Interpretation 103

Putting aside the correctness of the proofs of the Born Rule, I argue that none ofthem can be considered as unconditioned on any assumption. For the Born Rule inthe framework of collapse theories, we need an additional postulate about collapse.It is not part of the standard formalism. A priori it need not follow any laws ofstandard physical theory. We need to postulate some assumption – in my proof it isthe impossibility of superluminal signaling. In the framework of the MWI we alsoneed an assumption. The physics part, the evolution of the wave function of theuniverse, has to be supplemented by some law connecting it to our experience.Here, the assumption may be considered natural and minimal: Everything, includ-ing our experiences, supervenes on the wave function of the universe. Then,physical laws governing the evolution of the wave function are relevant to ourexperience, too; so we might claim that no additional assumption was made. Still,in all cases there is a tacit assumption that probability (or the illusion of probabil-ity) depends on the wave function.

5.11 Conclusions

Today’s physics is quantum theory and it enjoys unprecedented successexplaining all observed phenomena. There are questions that do not have goodanswers (yet). Quantum gravity, dark matter, dark energy. . . it might happenthat resolving these questions will require new revolutionary ideas. However,quantum mechanics apparently will remain the theory explaining electromag-netic interactions – the interactions that are responsible for almost everythingwe see in everyday life. Unphysical features of collapse are the main reasons fordoubts that this is the final theory of nature. But actually, there is no evidencefor collapse. Apparently, it is just the difficult philosophical consequences of nocollapse that prevents consensus about quantum theory without collapse andabout the existence of multiple worlds. It took time before people were ready toaccept that the Earth is not the center of the universe. We also need time toaccept that we are not unique and that there are many similar copies of us. Weneed time to establish the connection between the well-established mathemat-ical part of the theory and our experience. It is an unusual situation, which wedid not encounter in old physical theories. Philosophers should play an import-ant role in this project because it requires a dramatic change in our world view.Observing the rapidly increasing number of publications related to the MWI inthe philosophical literature makes me optimistic. I am not sure that the largeeffort to find a derivation of the Born Rule is justified (I doubt that the MWI hassignificant advantage here), but this activity leads to accepting the legitimacy ofthe MWI by physicists, and I believe that its advantage as a physical theory willbring it to consensus.

104 Lev Vaidman

Acknowledgments

I benefited greatly from numerous discussions with Kelvin McQueen, DavidAlbert, and the participants of the workshop Identity, indistinguishability andnon-locality in quantum physics (Buenos Aires, June 2017). This work has beensupported in part by the Israel Science Foundation Grant No. 1311/14.

References

Aharonov, Y. and Bohm, D. (1959). “Significance of electromagnetic potentials in thequantum theory,” Physical Review, 115: 485–491.

Albert, D. Z. (2013). “Wave function realism,” pp. 52–57 in A. Ney and D. Z. Albert(eds.), The Wave Function: Essays on the Metaphysics of Quantum Mechanics.Oxford: Oxford University Press.

Deutsch, D. (1986). “Three experimental implications of the Everett interpretation,”pp. 204–214 in R. Penrose and C. J. Isham (eds.), Quantum Concepts of Space andTime. Oxford: The Clarendon Press.

Deutsch, D. (1999). “Quantum theory of probability and decisions,” Proceedings of theRoyal Society of London A, 455: 3129–3137.

Diósi, L. (1987). “A universal master equation for the gravitational violation of quantummechanics,” Physics Letters A, 120: 377–381.

Ghirardi, G. C., Rimini, A., and Weber, T. (1986). “Unified dynamics for microscopic andmacroscopic systems,” Physical Review D, 34: 470–491.

Goldstein, S. (2017). “Bohmian mechanics,” in E. N. Zalta (ed.), The Stanford Encyclo-pedia of Philosophy (Summer 2017 Edition), https://plato.stanford.edu/archives/sum2017/entries/qm-bohm/

Greaves, H. (2004). “Understanding Deutsch’s probability in a deterministic multiverse,”Studies in History and Philosophy of Modern Physics, 35: 423–456.

Groisman, B., Hallakoun, N., and Vaidman, L. (2013). “The measure of existence of aquantum world and the sleeping beauty problem,” Analysis, 73: 695–706.

Kent, A. (2015). “Does it make sense to speak of self-location uncertainty in the universalwave-function? Remarks on Sebens and Carroll,” Foundations of Physics, 45:211–217.

Laplace, P. (1820/1951). “Essai Philosophique sur les Probabilités,” Introduction toThéorie Analytique des Probabilités. Paris: V Courcier. F. W. Truscott and F. L.Emory (trans.), A Philosophical Essay on Probabilities. New York: Dover.

Lewis, D. (1986). On the Plurality of Worlds. Oxford: Basil Blackwell.Maudlin, T. (2013). “The nature of the quantum state,” pp. 126–153 in A. Ney and D. Z.

Albert (eds.), The Wave Function: Essays on the Metaphysics of Quantum Mechanics.Oxford: Oxford University Press.

McQueen, K. and Vaidman, L. (2018). “In defence of the self-location uncertainty accountof probability in the many-worlds interpretation,” Studies in History and Philosophyof Modern Physics, in press, https://doi.org/10.1016/j.shpsb.2018.10.003.

Pearle, P. (1976). “Reduction of statevector by a nonlinear Schrödinger equation,” Phys-ical Review D, 13: 857–868.

Penrose, R. (1996). “On gravity’s role in quantum state reduction,” General Relativity andGravitation, 28: 581–600.

Saunders S. and Wallace D. (2008). “Branching and Uncertainty,” The British Journal forthe Philosophy of Science, 59: 293–305.

Ontology of the Wave Function and the Many-Worlds Interpretation 105

Sebens, C. T. and Carroll, C. M. (2018). “Self-locating uncertainty and the origin ofprobability in Everettian quantum mechanics,” The British Journal for the Philosophyof Science, 69: 25–74.

Tappenden, P. (2011). “Evidence and uncertainty in Everett’s multiverse,” The BritishJournal for the Philosophy of Science, 62: 99–123.

Vaidman, L. (1998). “On schizophrenic experiences of the neutron or why we shouldbelieve in the many-worlds interpretation of quantum theory,” International Studiesin the Philosophy of Science, 12: 245–261.

Vaidman, L. (2002). “Many-worlds interpretation of quantum mechanics,” in E. N. Zalta(ed.), The Stanford Encyclopedia of Philosophy (Fall 2016 Edition), https://plato.stanford.edu/archives/fall2016/entries/qm-manyworlds/

Vaidman, L. (2012a). “Role of potentials in the Aharonov-Bohm effect,” Physical ReviewA, R86: 040101.

Vaidman, L. (2012b). “Probability in the many-worlds interpretation of quantum mechan-ics,” pp. 299–311 in Y. Ben-Menahem and M. Hemmo (eds.), Probability in Physics,The Frontiers Collection. Berlin/Heidelberg: Springer-Verlag.

Wallace, D. (2012). The Emergent Multiverse: Quantum Theory According to the EverettInterpretation. Oxford: Oxford University Press.

106 Lev Vaidman

6

Generalized Contexts for Quantum Histories

marcelo losada, leonardo vanni, and roberto laura

6.1 Introduction

In the standard approach to quantum mechanics, there is no way to compute theprobability for expressions involving properties at different times. These probabil-ities can be useful to relate a property of a microscopic system, before themeasurement process, to the value of the pointer variable of the macroscopicapparatus after the measurement. Moreover, in the double-slit experiment, it isimportant to have a suitable language to describe through which slit the particledetected on a photographic plate has passed.

The theory of consistent histories has been introduced by Griffiths (1984),Omnès (1988), and Gell-Mann and Hartle (1990), defining the notion of historyas a sequence of properties at different times. The probability for a history wasdefined in this theory by an expression motivated by the path integral formalism,but with no direct relation to the usual Born Rule. For a valid description of aquantum system, the histories with well-defined probabilities should belong to afamily satisfying a state-dependent consistency condition.

In this theory, measurement is considered as a quantum interaction between themeasured microscopic system and the measuring macroscopic apparatus, and thereis no collapse postulate. Thus, the theory of consistent histories appeared to someas a strong candidate for the realization of a quantum theory in which the act ofmeasurement would not have the distinguished role assigned in ordinary quantummechanics.

However, an important problem of this theory is that it does not provide us witha single family of consistent histories, and the choice of different families may givedifferent descriptions for the time evolution of the same physical system. More-over, different families of consistent histories can provide the prediction or theretrodiction of contrary properties. If this is the case, it seems that the future or the

107

past of the quantum system could depend on the choice of the universe of discourse(see Dowker and Kent 1996, Laloë 2001, Okon and Sudarsky 2014).

In this chapter we present a summary of our formalism of generalized contextsfor quantum histories (Laura and Vanni 2009, Losada, Vanni, and Laura 2013), inwhich the ordinary contexts of properties for each different time should satisfycompatibility conditions given by commutation relations in the Heisenbergrepresentation. A family of histories satisfying these compatibility conditions isorganized in a distributive lattice having well-defined probabilities obtained by anatural generalization of the Born Rule of ordinary quantum mechanics.

In Section 6.2, our formalism is introduced for the case of histories in classicalmechanics. In Section 6.3, the formalism of generalized contexts for quantumhistories is presented in Section 6.4, it is applied to quantum measurements and tothe description of the double-slit experiment, with and without measurementapparatuses. The conclusions are presented in Section 6.5.

6.2 Classical Histories

6.2.1 States and Properties in Classical Mechanics

In classical mechanics, the states of a physical system are represented by points inthe phase space Γ (i.e., the space of generalized coordinates and momenta). Eachproperty p of the system is represented by a subset Cp of the phase space (Cp � Γ).A classical system in a state represented by a point x 2 Γ has a property prepresented by the set Cp � Γ if x 2 Cp. A partition of the phase space is obtainedby considering a collection of subsets Cj of the phase space, where j 2 σ, and σ is aset of indexes. This is a disjoint collection of subsets covering the phase space, i.e.,Ci \ Cj ¼ ϕ for i 6¼ j and [j2σCj ¼ Γ.

The system’s properties, represented by all possible unions of the sets of thepartition, with the order relation represented by the inclusion (�), form a Booleanlattice (i.e., it is orthocomplemented and distributive). The conjunction (∧) and thedisjunction (∨) of two properties are represented respectively by the intersectionand the union of the corresponding subsets. The null element of the lattice ofproperties is represented by the empty set, and the universal element by the phasespace Γ. The complement �p of a property p, which is represented by a subset Cp, isrepresented by the set C�p ¼ Γ� Cp. The set of classical properties pj correspond-ing to the partition Cj (j 2 σ) of the phase space Γ are the atomic properties of thelattice.

Anticipating what is going to be done in quantum mechanics, we use the termclassical context to denote the Boolean lattice generated by the atomic propertiespj (j 2 σ) of the system. It is always possible to combine different classical contexts

108 Marcelo Losada, Leonardo Vanni, and Roberto Laura

in a single, more refined one. Let us consider two sets of atomic properties, pjrepresented by the sets Cj, (j 2 σ), and pμ, represented by the sets Cμ and (μ 2 σ0),generating two different classical contexts. A more refined partition of the phasespace is obtained with the sets Cj \ Cμ, representing a new set of atomic properties.This new set generates a classical context containing the two previous ones. As animportant consequence, there is no restriction for the properties that can beincluded in a classical context. This is not the case for quantum properties, as wewill discuss in Section 6.3.

6.2.2 Probabilities for Properties at a Single Time

In some cases there is no precision about the point representing the state at a giventime t. Therefore, there is no certainty about the system having or not having aproperty, but only about the probability of having it. In these cases it is necessaryto appeal to what is known as a probability distribution. It is represented by afunction ρt : Γ⟶R, nonnegative and normalized (

ÐΓ ρt xð Þdx ¼ 1).

By means of the density function ρt, it is possible to define a probability on theset of all properties. Classical statistical mechanics gives the following expressionfor the probability of a property p represented by the set Cp at time t:

Prt pð Þ ¼ðCp

ρt xð Þdx, (6.1)

which satisfies the Kolmogorov axioms: (i) Prt Cð Þ ¼ ÐC ρt xð Þdx � 0 for any

property represented by C, (ii) Prt Γð Þ ¼ ÐΓ ρt xð Þdx ¼ 1, and (iii) if C \ C0 ¼ ϕ,

then Prt C [ C0ð Þ ¼ Prt Cð Þ þ Prt C0ð Þ.The time evolution of the state is determined by the Hamilton equations. A state

represented by the point x 2 Γ at the time t evolves into a state represented by thepoint x0 ¼ St0tx, where St0t : Γ ! Γ is invertible (S�1

t0t ¼ Stt0) and volume preserv-ing. If ρt xð Þ is the state probability density at time t, the probability density attime t0 is given by

ρt0 xð Þ ¼ ρt S�1t0t x

� �: (6.2)

This last equation can be used to obtain the probability Prt pð Þ of property p at timet given in Eq. (6.1) in terms of the probability density at a reference time t0. Byconsidering Eq. (6.2) with t0 equal to an arbitrary fixed time t0 in Eq. (6.1), weobtain

PrtðpÞ ¼ðCp

ρtðxÞdx ¼ðCp,0�St0 tCp

ρt0ðxÞdx: (6.3)

Generalized Contexts for Quantum Histories 109

We notice that the probability for the property p at time t can be expressed in twodifferent forms: either with a time-dependent probability density ρt xð Þ togetherwith a time-independent set Cp representing the property p, or with a time-independent density ρt0 xð Þ together with a time-dependent set Cp, 0 � St0tCp repre-senting the property p. By anticipating what is found in quantum mechanics, wewill use the terms Schrödinger and Heisenberg representations to name the firstand the second forms of expressing the same probability for property p at time t.

6.2.3 Probabilities for Properties at Different Times

In what follows, we will present a formalism suitable for including properties atdifferent times in a probabilistic description of a classical system (Losada et al.2013). As we showed in the previous subsection, the probabilities Pr p1; t1ð Þ for aproperty p1 at time t1 and Pr p2; t2ð Þ for a property p2 at a different time t2 can bothbe written in the Heisenberg representation corresponding to a single fixed time t0:

Prðp1, t1Þ ¼ðC1,0�St0 t1C1

ρt0ðxÞdx , (6.4a)

Prðp2, t2Þ ¼ðC2,0�St0 t2C2

ρt0ðxÞdx, (6.4b)

where C1 and C2 (C1,0 and C2,0) are the subsets of the phase space correspondingto the properties p1 and p2 in the Schrödinger (Heisenberg) representation.

The previous equations strongly suggest the following definitions for the prob-abilities corresponding to the conjunction and the disjunction of the properties p1 attime t1 and p2 at time t2:

Pr½ðp1, t1Þ∧ðp2, t2Þ� �ðC1,0\C2,0

ρt0ðxÞdx, (6.5)

Pr½ðp1, t1Þ∨ðp2, t2Þ� �ðC1,0[C2,0

ρt0ðxÞdx: (6.6)

In these expressions, C1,0 \ C2,0 can be interpreted as the Heisenbergrepresentation for the conjunction p1; t1ð Þ∧ p2; t2ð Þ, while C1,0 [ C2,0 can beconsidered as the Heisenberg representation for the disjunction p1; t1ð Þ∨ p2; t2ð Þ.

Classical histories involving properties at two times can be obtained startingfrom two classical contexts of properties at times t1 and t2. A classical context ofproperties at time t1, generated by atomic properties pj1 (j 2 σ), with Schrödinger(Heisenberg) representations given by the partition Cj

1 (Cj1,0 � St0t1C

j1) of the

phase space, can be combined with a classical context of properties at time t2,

110 Marcelo Losada, Leonardo Vanni, and Roberto Laura

generated by atomic properties pμ2 (μ 2 σ∗), with Schrödinger (Heisenberg) repre-sentations given by the partition Cμ

2 (Cμ2,0 � St0t2C

μ2) of the phase space. The sets

Cjμ0 � Cj

1,0 \ Cμ2,0, with j; μð Þ 2 σ � σ∗, form a partition of the phase space (i.e.,

they satisfy Cjμ0 \ Cj0μ0

0 ¼ ϕ if j; μð Þ 6¼ j0; μ0ð Þ and Sjμ C

jμ0 ¼ Γ). They represent the

histories “property pj1 at time t1 and property pμ2 at time t2,” and they are the atomic

elements generating a distributive and orthocomplemented lattice of classicalhistories for the two times t1 and t2, with well-defined probabilities defined byEq (6.5) and Eq. (6.6). The generalization to a lattice of classical histories at ntimes t1 < t2 < . . . < tn can easily be obtained.

6.3 Quantum Histories

6.3.1 States, Properties, Probabilities, and Contexts in Quantum Mechanics

In quantum mechanics, states are represented by vectors of a Hilbert space H or,more generally, by density operators. Each property p of the system is representedby a subspace H p of the Hilbert space H or, equivalently, by a projector Πp,such that H p ¼ ΠpH . The time evolution between times t0 and t of a densityoperator ρ, representing a state is given by ρt ¼ U t; t0ð Þρt0U�1 t; t0ð Þ, whereU t; t0ð Þ ¼ exp � i

ħH t � t0ð Þ� �, being H the Hamiltonian operator for the quantum

system. In the Schrödinger representation, the probability at time t for a property pin a state ρt is given by the Born Rule, Pr p; tð Þ ¼ Tr ρtΠp

� �. In the Heisenberg

representation, the same probability is given by Pr p; tð Þ ¼ Tr ρt0Πp, 0� �

, whereΠp, 0 ¼ U�1 t; t0ð ÞΠpU t; t0ð Þ.

The lattice structure for the properties of a quantum system is built from theorder relation (�) corresponding to the inclusion of the Hilbert subspaces.For two properties p and p0, p � p0 if H p H p0 . The conjunction p∧p0 of twoproperties is represented by H p∧p0 ¼ Inf H p;H p0

� � ¼ H p \ H p0 . The disjunction

p∨p0 is represented by H p∨p0 ¼ Sup H p;H p0� � ¼ H p þ H p0 . The complement �p of

a property p is represented by H �p ¼ H p

� �⊥. With this structure, the set of all

possible quantum properties forms an orthocomplemented and nondistributivelattice, and the Born Rule does not yield, in general, well-defined probabilities.An orthocomplemented and distributive lattice with well-defined probabilities isobtained if the universe of discourse is restricted to a single quantum context.A quantum context is generated by a projective decomposition of the Hilbert space

into mutually orthogonal subspaces H i ¼ ΠiH (i 2 σ). The projectors Πi satisfyΠiΠj ¼ δijΠj for all i, j 2 σ, and

Pj2σ Π

j ¼ I, where I is the identity operator. Any

property p of the quantum context is represented by a projector of the formΠp ¼ P

j2σp�σ Πj.

Generalized Contexts for Quantum Histories 111

In the previous section, we have shown that it is always possible to includetwo different classical contexts in a single, more refined one. This is not, ingeneral, the case for quantum contexts. Two different quantum contexts gener-ated by atomic properties represented by projective decompositions Πj (j 2 σ) andΠμ (μ 2 σ∗) are said to be compatible if

�Πj;Πμ

� ¼ 0 for all j; uð Þ 2 σ � σ∗. Ifthis is the case, the properties of the two contexts are included in a single contextgenerated by the properties represented by the projectors Πjμ � ΠjΠμ withj; μð Þ 2 σ � σ∗.

6.3.2 Generalized Contexts for Quantum Histories

Now we will extend the notion of quantum contexts to consider the probabilitiesfor expressions involving properties at different times.

Let us consider at time t0 a system in a state represented by the densityoperator ρt0 , a context of properties C1 at the time t1, and a context of propertiesC2 at the time t2. The atomic properties pk11 for the context C1 are represented bythe projective decomposition with projectors Πk1

1 (k1 2 σ1), while the atomicproperties pk22 for the context C2 are represented by projectors Πk2

2 (k2 2 σ2).The Heisenberg representation with reference time t0 of these projectors isgiven by

Πk11,0 � U�1 t1; t0ð ÞΠk1

1 U t1; t0ð Þ, Πk22,0 � U�1 t2; t0ð ÞΠk2

2 U t2; t0ð Þ: (6.7)

By definition, C1 and C2 are said to be compatible quantum contexts (Laura andVanni 2009, Losada et al. 2013), if the corresponding projectors commute in theHeisenberg representation, i.e., if

Πk11,0;Π

k22,0

h i¼ 0 for all k1; k2ð Þ 2 σ1 � σ2: (6.8)

If C1 at t1 and C2 at t2 are compatible contexts, they generate what we call ageneralized context of histories at two times, with atomic histories “pk11 at time t1and pk22 at time t2,” having the Heisenberg representation given by the projectorsΠk1k2

0 � Πk11,0Π

k22,0, for k1; k2ð Þ 2 σ1 � σ2. These projectors provide a decompos-

ition of the Hilbert space H into mutually orthogonal subspaces and, therefore, theatomic histories generate an orthocomplemented and distributive lattice of twotimes histories. The Born Rule in Heisenberg representation provides well-definedprobabilities on this lattice. For the atomic histories we have

Pr pk11 at time t1 and pk22 at time t2� � � Tr ρt0Π

k1k20

� �: (6.9)

The formalism presented for the case of two times can be easily generalized toinclude histories for sequences of n times.

112 Marcelo Losada, Leonardo Vanni, and Roberto Laura

6.4 Results Obtained with the Formalism of Generalized Contexts

In axiomatic quantum theories, the states are considered functionals acting on thespace of observables and, therefore, they appear after the observables in a some-how subordinate position (Laura and Castagnino 1998, Castagnino, Id Betan,Laura, and Liotta 2002). Quantum histories play the role of the observables ofordinary quantum theory and, as a consequence, it seems reasonable that theallowed sets of histories satisfy state-independent conditions. The consistencyconditions of the theory of consistent histories produce state-dependent familiesof consistent histories (Griffiths 1984, Omnès 1988, Gell-Mann and Hartle 1990).On the contrary, the compatibility conditions of the generalized context formalismgiven in Eq. (6.8) are state independent.

The state-independent compatibility conditions of the generalized-contextformalism produce an important difference with respect to the theory ofconsistent histories. Each quantum history has a Heisenberg representationgiven by a projection operator and each valid set of quantum histories isgenerated by a projective decomposition of the Hilbert space. As a conse-quence, a generalized context of quantum histories has the logical structure ofa distributive orthocomplemented lattice of subspaces of the Hilbert space,i.e., the same logical structure of the quantum properties of an ordinarycontext. It is because of this logical structure that in our formalism there isno place for the retrodiction of contrary properties (Losada and Laura 2014b),which is a problem for the theory of consistent histories (Okon and Sudarsky2014).

In the opinion of some authors, the theory of consistent histories allows toomany histories and some of them are difficult to interpret (Dowker and Kent 1996,Laloë 2001). We have proven that our compatibility conditions, given by thecommutation of the projectors representing the properties translated to a commontime, are equivalent to the consistency conditions imposed on all possible states ofthe system (Losada and Laura 2014a). Therefore, the formalism of generalizedcontexts imposes more restrictions than the theory of consistent histories for thevalid families of quantum histories, and it allows fewer families of histories (seediscussion in Losada and Lombardi 2018). The fact that the universe of discourseabout a physical system depends on a choice of a family of histories is a problemfor quantum theory. The freedom of choice of our formalism is smaller than in thetheory of consistent histories, but not enough to single out a realist interpretation ofquantum mechanics. We are working in the direction of endowing the formalismwith interpretive content.

In what follows, we will introduce a brief description of two physically relevantapplications of our formalism, presented in our previous works (Losada et al. 2013,Vanni and Laura 2013).

Generalized Contexts for Quantum Histories 113

6.4.1 Quantum Measurements

A nonideal measurement of an observable Q of a system S is an interaction duringthe time interval t1; t2ð Þ of the measured system S with the measuring apparatus A.It is represented in the Hilbert space H S⊗H A by a unitary transformationU ¼ U t2; t1ð Þ satisfying

qiij ja0i!U jϕii jaii, (6.10)

where jqii is an eigenvector of the observable Q with eigenvalue qi, ja0i is theinitial reference state of the apparatus, and jaii is the state of the apparatus with thevalue ai of the pointer variable. The state of the composite system at time t1 isjψ1i ¼jφ1i ja0i, where jφ1i ¼

Pi ci jqii.

The formalism of generalized contexts can provide a description of the processinvolving the possible values qi of the observable Q of system S at time t1 and thepossible pointer values aj of the apparatus A at time t2. These properties arerepresented by the projectors

Πqi ¼ qiihqij j⊗IA, Πaj ¼ IS⊗ ajihaj�� ��, (6.11)

and satisfy the compatibility conditions when translated to the common time t1,i.e., U�1ΠajU;Πqi

� � ¼ 0. Therefore, the generalized context formalism allowscomputing the conditional probability

Pr qi; t1jaj; t2� � ¼ Pr qi; t1ð Þ∧ aj; t2

� �� �Pr aj; t2� � ¼ hψ1 jU�1ΠajUΠqi jψ1i

hψ1 jU�1ΠajU jψ1i¼ δij: (6.12)

For the composite system prepared in the state jψ1i ¼jφ1i j a0i, this result can beinterpreted by saying that if the apparatus’ pointer variable has the value aj after themeasurement, the system S had the property Q ¼ qj before the measurement. Moredetails of the application of this formalism to the logic of quantum measurementscan be found in our previous works (Vanni and Laura 2013, Losada, Vanni, andLaura 2016).

6.4.2 The Double-Slit Experiment

We also applied the generalized-context formalism to describe the double-slitexperiment (Losada et al. 2013). A particle in a state represented by a wave packet,coming from left to right, passes through a double slit at time t1. The particlereaches a vertical zone located to the right of the double slit at a later time t2.

We can attempt to give a description of the process involving through which slitthe particle has passed at time t1, that appears to be in some region of the verticalzone at a later time t2. As we assume no measurement apparatus, the Hilbert space

114 Marcelo Losada, Leonardo Vanni, and Roberto Laura

to describe this process is the Hilbert space of the particle (H ¼ H particle). Theprojectors representing the particle located in each slit at time t1 are

Πut1�

ðVu

d3r �rih�rj j, Πdt1�

ðVd

d3r �rih�rj j , (6.13)

where Vu (Vd) is the volume of the upper (lower) slit, and j�ri is a generalizedeigenvector of the position operator of the particle with generalized eigenvalue �r.For the later time t2, the projectors corresponding to the particle in small regions ofthe vertical zone to the right of the double slit are

Πnt2�

ðVnd3r �rih�rj j, (6.14)

where Vn is the volume of the small region of the vertical zone labelled by theindex n. We proved that the properties represented by the projectors Eq. (6.13) andEq. (6.14), translated to the common time t1, are represented by noncommutingprojectors, i.e.,

Πut1;U�1Πn

t2U

h i6¼ 0, Πd

t1;U�1Πn

t2U

h i6¼ 0, (6.15)

where U ¼ U t2; t1ð Þ ¼ e�iH0 t2�t1ð Þ=ℏ is the unitary evolution generated by the free-particle Hamiltonian H0 ¼ p2=2m. Therefore, our formalism shows the well-known fact that it is not possible to provide a description of the quantum processsuitable to specify through which slit the particle passed before reaching a regionof the vertical zone.

We also considered a modified double-slit experiment with an ideal measure-ment apparatus A located in the slits zone, interacting with the particle during theshort time interval t1; t1 þ Δ1½ �, and with its pointer variable indicating au (ad) ifthe particle is detected passing through the upper (lower) slit. A second idealmeasurement apparatus B is located in the vertical zone to the right of the doubleslit, interacting with the particle in the short time interval t2; t2 þ Δ2½ �, and with apointer variable indicating bn if the particle is detected in the small zone labelled bythe index n of the vertical zone. The Hilbert space for the description of thisprocess is the tensor product of the Hilbert space of the particle and the two Hilbertspaces of the detectors, i.e., H ¼ H particle⊗H A⊗H B. The unitary time evolution isassumed to be dominated by the interaction between the particle and apparatus A inthe short time interval t1; t1 þ Δ1½ �, by the free evolution in the time intervalt1 þ Δ1; t2½ �, and by the interaction of the particle and apparatus B in the timeinterval t2; t2 þ Δ2½ �.

The possible pointer indications of the apparatus A at time t1 þ Δ1 are repre-sented by the projectors

Generalized Contexts for Quantum Histories 115

Πaut1þΔ1

� Iparticle⊗ auihauj j⊗IB, (6.16a)

Πadt1þΔ1

� Iparticle⊗ adihadj j⊗IB, (6.16b)

and the possible indications of apparatus B at time t2 þ Δ2 are represented by theprojectors

Πbnt2þΔ2

� Iparticle⊗IA⊗ bnihbnj j: (6.17)

We also proved that the properties corresponding to the projectors Eq. (6.16) andEq. (6.17), translated to a common time, are represented by commuting projectors.Therefore, within the generalized-context formalism there is a generalized contextfor the composite system’s history that includes the fact that the particle ismeasured to pass through a definite slit at a certain time and the fact that theparticle is measured in a definite region of the vertical plane at a later time. Thecorresponding conditional probabilities give the expected noninterference pattern.

6.5 Conclusions

We have presented our formalism of generalized contexts for quantum histories. Itwas successfully applied to describe the logic of quantum measurements (Vanniand Laura 2013, Losada et al. 2016) and the results of the double-slit experimentwith and without measurement apparatuses (Losada et al. 2013). It was alsosuitable to give a discussion of the decay process (Losada and Laura 2013) andto provide a deduction of the complementarity principle for the case of the Mach-Zehnder interferometer (Vanni and Laura 2010).

The compatibility conditions of our formalism impose stronger conditions onthe allowed families of histories than the conditions imposed by the theory ofconsistent histories (Losada and Laura 2014a). Therefore, the number of universesof discourse (families of histories) allowed by our formalism is reduced. Twoimportant consequences of our formalism are that different families of historieswould not give retrodictions or predictions of contrary properties (Losada andLaura 2014b), and that any allowed family of histories is organized in a distributiveand orthocomplemented lattice.

However, our formalism is not in position to provide a full interpretation ofquantum mechanics. There are no different allowed families of histories predictingor retrodicting contrary properties, but there is still the freedom of choice ofdifferent generalized contexts. The formalism in itself gives no indication aboutwhich family should be privileged in a description of the time evolution of asystem. It seems that stronger conditions should be added to our formalism in orderto satisfy a realist perspective. In order to endow this formalism with realistinterpretive content, it is necessary to associate it with a specific interpretation that

116 Marcelo Losada, Leonardo Vanni, and Roberto Laura

is able to be consistently combined with the compatibility condition – but this isstill a work in progress.

Acknowledgments

This work was made possible through the support of Grant 57919 from the JohnTempleton Foundation and Grant PICT-2014–2812 from the National Agency ofScientific and Technological Promotion of Argentina.

References

Castagnino, M., Id Betan, R., Laura, R., Liotta, R. (2002). “Quantum decay processes andGamov states,” Journal of Physics A, 35: 6055–6074.

Dowker, F. and Kent, A. (1996). “On the consistent histories approach to quantummechanics,” Journal of Statistical Physics, 82: 1575–1646.

Gell-Mann, M. and Hartle, J. B. (1990). “Quantum mechanics in the light of quantumcosmology,” pp. 425–458 in W. Zurek (ed.), Complexity, Entropy and the Physics ofInformation. Reading: Addison-Wesley.

Griffiths, R. (1984). “Consistent histories and the interpretation of quantum mechanics,”Journal of Statistical Physics, 36: 219–272.

Laloë, F. (2001). “Do we really understand quantum mechanics? Strange correlations,paradoxes, and theorems,” American Journal of Physics, 69: 655–701.

Laura, R. and Castagnino, M. (1998). “Functional approach for quantum systems withcontinuous spectrum,” Physical Review E, 57: 3948–3961.

Laura, R. and Vanni, L. (2009). “Time translation of quantum properties,” Foundations ofPhysics, 39: 160–173.

Losada, M. and Laura, R. (2013). “The formalism of generalized contexts and decayprocesses,” International Journal of Theoretical Physics, 52: 1289–1299.

Losada, M. and Laura, L. (2014a). “Generalized contexts and consistent histories inquantum mechanics,” Annals of Physics, 344: 263–274.

Losada, M. and Laura, R. (2014b). “Quantum histories without contrary inferences,”Annals of Physics, 351: 418–425.

Losada, M. and Lombardi, O. (2018). “Histories in quantum mechanics: distinguishingbetween formalism and interpretation,” European Journal for Philosophy of Science,8: 367–394.

Losada, M., Vanni, L., and Laura, R. (2013). “Probabilities for time-dependent propertiesin classical and quantum mechanics,” Physical Review A, 87: 052128.

Losada, M., Vanni, L., and Laura, R. (2016). “The measurement process in the generalizedcontexts formalism for quantum histories,” International Journal of TheoreticalPhysics, 55: 817–824.

Okon, E. and Sudarsky, D. (2014). “On the consistency of the consistent histories approachto quantum mechanics,” Foundations of Physics, 44: 19–33.

Omnès, R. (1988). “Logical reformulation of quantum mechanics. I. Foundations,” Jour-nal of Statistical Physics, 53: 893–932.

Vanni, L. and Laura, R. (2010). “Deducción del principio de complementariedad en lateoría cuántica,” Epistemología e Historia de la Ciencia. Selección de trabajos de lasXX Jornadas, 16: 647–656.

Vanni, L. and Laura, R. (2013). “The logic of quantum measurements,” InternationalJournal of Theoretical Physics, 52: 2386–2394.

Generalized Contexts for Quantum Histories 117

Part II

Realism, Wave Function, and Primitive Ontology

7

What Is the Quantum Face of Realism?

james ladyman

7.1 Introduction: Realisms and Theories of the Quantum

The main argument of this paper is that there are many forms of realism and manyforms of quantum physics, and that the interaction of the two is apt to confuse (thissection begins to make this case below). The second argument (which reinforces thefirst and is presented in the next section) is that there is a considerable tension betweenthe arguments for scientific realism in the philosophy of science literature and theinvocation of realism as a reason for adopting revisionary interpretations of quantummechanics that are popular among philosophers. The third argument (in the thirdsection) is that scientific realists ought anyway to consider quantum physics as awhole,where this includes much more than nonrelativistic many-particle quantummechanics(NRMPQM) and that doing so does not lend any support to some of the most popularrealist interpretations of quantummechanics among philosophers. The fourth argument(made briefly in the last section) is that there is a form of scientific realism that iscompatible with accepting the revolution in our understanding of matter wrought byquantum physics, and that despite the objections of many philosophers and somephysicists, the revisions to our conception of matter ought not to be undone.

The history of quantum physics teaches us several important things aboutrealism and the quantum.

(1) There are many different forms of realism that are often run together indebates about the interpretation of quantum mechanics (see the exchange betweenDeutsch and Ladyman in Saunders et al. 2010). Often the triumph of the Copen-hagen interpretation over Einstein’s “common sense” realism is associated withgeneral and strong forms of antirealism (Maudlin 2018), and not without justifica-tion, given some of the weird things that Bohr and others have said about the natureof reality in the light of quantum physics (for example,Maudlin is among the authorsof Daumer et al. 2006 who strongly object to Wheeler’s and Zeilinger’s ideas aboutreality and information being somehow inseparable). However, as Folse (1985)

121

documents, Bohr often made realist claims to go alongside the much quoted anti-realist ones. For example, in his Nobel Prize lecture Bohr expressed his belief inatoms and our knowledge of their microscopic constituents. Similarly, presumablymost, if not all, the physicists who immediately applied quantum mechanics toproblems in chemistry and solid-state physics, believed in atoms and electrons. Thefact that many physicists learned to invoke Copenhagen as a way of avoidingworrying about philosophy does not imply their adherence to any detailed positiveinterpretation, and certainly not to anything but a very localized antirealism.

A much discussed survey of physicists, asking them to which interpretation ofquantum mechanics they subscribe, has results as follows for the most discussedinterpretations: 42% Copenhagen, 18% Everett, 0% de Broglie-Bohm, and 9%Objective Collapse (see Schlosshauer, Kofler, and Zeilinger 2013). Sean Carroll(2013) says that the lack of agreement makes this the most embarrassing graph inphysics. However, such lack of agreement is not unusual in the history of science.(It would be interesting to know what they would have said if asked first whetherthey accepted metaphysical and scientific realism as defined below.) The physicistswho expressed their support for the Copenhagen interpretation in the poll surelywould not deny that there is a supermassive black hole at the center of the MilkyWay. In this way, quantum orthodoxy sits alongside scientific realism, and it isplausible that many of those who opted for Copenhagen did so because for manypracticing physicists the measurement problem is irrelevant to their working lives,and their adherence to Copenhagen orthodoxy goes as far as instrumentalism aboutwave functions and collapse when pressed (the idea of wave function collapse as areal process has nothing to do with Bohr, but is largely due to von Neumann; seeHoward 1985). It is because of the confusion and conflation surrounding the ideaof the Copenhagen orthodoxy that pragmatic attitudes to the interpretation ofquantum mechanics, by which is meant solving the measurement problem, aretaken to be tantamount to idealism or other forms of antirealism about the world ingeneral and even physics in particular (there is more discussion below of thedifferent ideas associated with the Copenhagen interpretation).

To bring more precision to the discussion consider the following:

Metaphysical realism. The (physical) world is (largely) independent of ourbeliefs and desires.

Scientific realism. Our best scientific theories should be taken literally, astalking about unobservable entities and processes, and as such successfullyrefer, and are approximately true.

These definitions are representative of those in canonical discussions of scientificrealism (see Psillos 1999). Of course, this definition of scientific realism is very

122 James Ladyman

vague without an account of reference and truth, and much ink has been spilledabout these topics without much agreement, and different forms of scientificrealism have proliferated. However, even those who defend the strongest formsof scientific realism are not required to think that all theoretical terms successfullyrefer. For example, a scientific realist may think superstrings have not yet beenshown to exist. Structural realism arose from these discussions and various othervariants including, most relevantly for the present discussion, entity realism.Nancy Cartwright and Ian Hacking, among others, argued for this view, and it isthe fairly mainstream position in the scientific realism debate that seems to clearlytake the side of the scientific realist where it counts most for scientific practice.

Entity realism. Unobservable entities that are interacted with in the laboratoryshould be taken as real, even if the theories that describe them are not taken tobe true.

It should be noted that entity realists are skeptical about claims to completenessand fundamentality (which Section 7.3 argues are particularly dubious in the caseof quantum mechanics). Entity realism is clearly compatible with an instrumental-ist view of the wave function and the problem of collapse. The measurementproblem might be supposed to force the adherent of scientific realism to committo more, but it is argued that it does not in Section 7.3.

(2) No way was found in practice to distinguish in advance among physicalsituations to which quantum physics assigns only probabilities, but which havedifferent observable outcomes. Paradigmatically, radioactive decay is still con-sidered random for all intents and purposes in atomic and nuclear physics. JohnStuart Mill (1843) defined determinism as the claim that for any situation there is adescription of it, such that for any other situation satisfying the same description,the same future will unfold. This may sound overly epistemic to current ears but itis the way the notion of determinism is applied to the world in scientific practice.Classical phenomena pass Mill’s test insofar as there is a level of precision ofinitial conditions that makes it possible to specify in advance, for example, whethera fair coin will land heads or tails. Effective classical deterministic behavior ofmacroscopic bodies emerges somehow, even though quantum systems fail Mill’stest as far as anybody has been able to determine. This is a remarkable fact whenone considers the exponential advances and growth in measurement technologysince the discovery of radioactivity more than a century ago. When physicists firstdebated the uncertainty relation, they were interested in whether it was a constrainton practice, and Einstein argued it was not (Bohr 1949). The hidden variablestheorists lost the scientific battle long ago in this sense. Despite the unsoundness ofthe no-go theorems for any form of hidden variable, the idea of Bohr – that

What Is the Quantum Face of Realism? 123

quantum mechanics was complete for all practical purposes – has only beenvindicated by subsequent developments.

De Broglie, and later Bohm, showed that the idea that the world is fundamen-tally random is not logically required by NRMPQM. It is possible to attributeunderlying deterministic trajectories to quantum particles moving in accordancewith the evolution of the wave function. However, since such hidden variablesmust be nonlocal in the strong sense, namely that a preferred frame is required inwhich it is unambiguously true that the adjustment of a measurement settingchanges the trajectories of particles at spacelike separation, they require revision-ary physics for no immediate practical use just because they are hidden. These areunlikely to have been received well by those who had learned to accept relativity,even if their possibility had not been deliberately or mistakenly ignored by mostphysicists. Cushing (1998) speculates that Bell’s theorem could have becomeknown in the Twenties, and sees this in terms of people accepting that quantummechanics is nonlocal. This is highly tendentious, as discussed later, but evengranting it, he must admit that the hidden variables would not have been used inpractice for the problems people went on to solve in all the areas of physics towhich quantum mechanics was applied. He assumes that Dirac’s operator formal-ism would still have been available to apply quantum mechanics to problems.Scientific realist literature abounds with the call to take the practice of scienceseriously and not to only consider theoretical possibilities. Any form of scientificrealism that does not take quantum randomness at face value is at least out of stepwith the practice of science.

Cushing discusses the Forman thesis linking the repudiation of the law ofcausality by German quantum theorists with the neoromantic disdain for sciencein Weimar culture (see Forman 1971, 1984; for a critique, see Kragh 2002: chapter10). However, note that Peirce, in the nineteenth century, rejected determinism andaccepted brute randomness in nature within his philosophy, which was based onhis appreciation and study of science (see Ladyman and Ross 2013). Similarly,both Cushing and Forman offer no explanation of why British physicists such asDirac and Mott readily embraced the new quantum mechanics. More importantly,there is an equivocation on causality in this context, for while Kant and others tookit to be the same thing as determinism in the sense of there being necessary andsufficient antecedent events for everything that happens, in the twentieth century,probabilistic causality became the norm as a result of the extension of causalmodeling to the behavioral and social sciences.

(3) Quantum mechanics was successfully applied in chemistry and solid statephysics in the first months and years of its existence. It was also more or lessimmediately extended to relativistic physics and to field theory. All of its

124 James Ladyman

seemingly impossible implications that have been tested have been confirmed. It isapplied throughout the rest of physics, and quantum statistical mechanics andsemiclassical domains have been discovered. However, it has not been successfullyapplied to gravity itself (though quantum field theory [QFT] has been applied to theeffects of gravity in the form of Hawking radiation) and so is not a complete theoryof reality. In philosophical discussions of physics, people often consider models inwhich there is only a spacetime and some fields or particles of some particle kind.This generates the idea of the ontology of the theory, as if it purported to be anaccount of the whole of reality rather than just an aspect of it. For example,diffraction and Stern-Gerlach experiments on beams of particles or individual ones,and EPR-type experiments on entangled systems, are a tiny proportion of theapplications of quantum physics to reality. Any viable form of quantum realismmust not conflate NRMPQMwith quantum physics, and it must be apt for phenom-ena other than familiar, simple experiments that are not representative of quantumphysics, though they are legitimately used to understand the conceptual and math-ematical foundations of the theory and how it represents physical systems.

7.2 Scientific Realism and the Interpretation of NRMPQM:The Case of Bohm Theory

As yet, the only grounds for adopting Bohm theory are philosophical, namelybecause it is a causal and realist version of quantum theory that solves themeasurement problem and eliminates in principle puzzles about the weird behaviorof quantum systems. However, this section argues that to adopt Bohm theory is tocut the philosophical ground from under the feet of the realist. Much of what is saidis relevant to other examples of alternatives to quantum orthodoxy, but the case ofrelativistic dynamical collapse theories is in many ways very different, insofar asthey make predictions that differ from those of quantum mechanics.

In discussions in the philosophy of science, scientific realism is usuallyarticulated in terms of the theories that the relevant scientific community hasaccepted. The question is whether the scientific community adopting some theoryof unobservables compels belief in their existence. It is standardly assumed on allsides that theory choice is rational and that the methods scientists use to solve theunderdetermination problem in practice are appropriate. The mere theoreticalavailability of empirically equivalent alternatives to accepted theories is not takento be sufficient for skepticism about the latter. In philosophy of science, those whooffer sociological explanations for theory choice in science are usually antirealists,and the invocation of theoretical empirical equivalence of a rival research programas a reason for skepticism about the orthodoxy is commonly associated with

What Is the Quantum Face of Realism? 125

constructive empiricism (van Fraassen 1980), which is the most discussed alterna-tive to realism. It is ironic then that the positive case for Bohmian mechanicsinvolves the combination of an insistence on a realist interpretation of quantummechanics based on the theoretical empirical equivalence of the formalism, withthe claim that the reasons why Bohmian mechanics was not accepted by thescientific community are extrascientific. Hence, the aforementioned discussion ofthe Forman thesis by Cushing, and the emphasis on how discussion of themeasurement problem, and the work of de Broglie, Bohm, and Everett wassupressed. (Bohm’s ideas were not well known until after the work of Bell, andthe current relative prominence of many-worlds interpretations is due to the revivalof Everett’s ideas by others. Saunders et al. 2010 includes discussion of all this anda very comprehensive relevant citation.)

Realist responses to the underdetermination problem usually aim at rationallyreconstructing theory choice, for example, in the case of wave versus particletheories of light, special relativity versus Lorentzian contraction, and so on. Thetheoretical virtues of novel predictive success, non-ad hocness, explanatory power,simplicity, coherence with background metaphysics, and so on, are discussed toexplain why mere empirical equivalence is not sufficient to make the community’schoice arbitrary and why the chosen theory scientifically preferable. In this context,historical theories of confirmation and the idea of progressive versus degeneratingresearch programs are much discussed following the work of Popper, Lakatos, andMusgrave. Among the theoretical vices are the negation of each of the virtues justdescribed, such as ad hocness and ontological profligacy. Another theoretical vicethat is often discussed is parasitism, which is when a theory requires another theoryfor its formulation. An extreme example is the theory that says “the world is as ifQED”; it generates no predictions at all without the input of QED. The wave andparticle theories of light were not like this. Ray optics fits well with the mechanicsof particles, and diffraction fits well with waves.

The main virtue of Bohm theory for many of its current defenders is that it offersthe best prospects for realism about the quantum world. This is because Bohmtheory posits definite values for all physical quantities at all times, eliminating theapparent indeterminacy in the values of physical quantities prior to measurement.The Bohmian mechanics usually discussed is such that all particles have well-defined trajectories at all times and the evolution of all quantum systems is entirelydeterministic and causal. Thus particles can be individuated by their spatiotem-poral properties (although of course these will not be accessible to experiment). Itis also virtuous in cohering with background ideas of causality and determinism,and of the nature of particles. However, it is important to note that, as HarveyBrown and others (1996) argue, the “particles” of Bohm theory are not those ofclassical thought. The dynamics of the theory is such that the properties normally

126 James Ladyman

associated with particles like mass, charge, and so on, are in fact, all inherent in thequantum wave function and not in the particles. It seems that the particles only haveposition. Apart from any worries we might have about the intelligibility of thisnotion of particle, it seems that they have none of the features of classical particlesother than point position; hence, there would seem to be little referential continuityfor ‘particle’ available to the realist. Furthermore, if the trajectories are enough toindividuate particles in Bohm theory, what makes the difference between an “empty”trajectory and an “occupied” one? Since none of the physical properties ascribed tothe particle inheres in points of the trajectory, giving content to the claim that there isactually a “particle” there would seem to require some notion of the raw stuff of theparticle; in other words, haeccities seem to be needed to make the ontology of Bohmtheory intelligible after all. This is without even beginning the discussion of thenature of the wave function in Bohm theory (Brown et al. 1996 also argue that theaction–reaction principle is violated by Bohm theory).

Bohm theory is often criticized for its alleged ad hocness and lack of simplicityrelative to the standard formalism. These complaints aside, more problematic is that itis nonlocal in the strong sense that, if Bohm theory is correct, then physicists willhave to rewrite their textbooks so that a Newtonian or Galilean spacetime with aglobal time coordinate underlies the Lorentz invariant phenomena of electromagnet-ism. When such theories are proposed as alternatives to relativity, they are sometimeslinked to cosmic conspiracies because they propose far-fetched hidden machinations.Similarly, instantaneous action at a distance as a physical process has been widelyregarded as unscientific since the beginnings of field theory in the mid-nineteenthcentury, but Bohm theory makes it a pervasive feature of the world, and it requiresthat the change of setting on the apparatus on one side has a physical effect on thetrajectory of the particle on the other side. Unsurprisingly, Bohmians always arguethat quantum mechanics requires action at a distance anyway, but this is not correct.Bell’s theorem does not show that there is an objective causal asymmetry between thetwo wings of the Aspect experiment, only that the attribution of possessed values, orcounterfactual definiteness, or some other condition, is incompatible with locality.There is no proof of Bell’s theorem without what is often termed a “realist”assumption of some kind, but note that none of these realist assumptions are requiredby either metaphysical or scientific realism.

The so-called pessimistic meta-induction argument against scientific realismwould be reinforced if Bohm theory is adopted by most scientists for two reasons:

(a) It would support the arguments of those who argue that we cannot learnmetaphysical lessons from science, because the orthodoxy among physicistswould have been quite wrong about the world since at least 1935, insofar as ithas been widely held and taught that quantum phenomena are genuinely

What Is the Quantum Face of Realism? 127

indeterministic – that electrons and other “particles” are really neither wavesnor particles, that matter has new properties like spin, and so on.

(b) The acceptance of relativity theory as a theory of the nature of space and time,such that there is no absolute simultaneity or privileged coordinate system, andsuch that the Lorentz invariance of Maxwell’s equations has been taken to benot merely empirical but also reflective of fundamental symmetries in thenature of reality, would have to be regarded as quite mistaken. The more casesthere are of the ontological interpretation of scientific theories being laterabandoned, the more compelling the meta-induction becomes.

Overall, skepticism about scientific realism will be encouraged if Bohm theory isadopted, because it will have to be accepted that most scientists have been wrongabout most of the most important metaphysical implications of twentieth-centuryphysics for most of the twentieth century.

Furthermore, the positive arguments for realism will also be undermined ifBohm theory replaces standard quantum theory. Consider the most straightforwarddefense of realism by inference to the best explanation: according to the realist,since the quantum theory of spin has been so empirically successful, we ought tobelieve that the best explanation of this fact is that there really are subatomicparticles with spin states. By contrast, Bohm theory denies that spin is a funda-mental property, reducing it to some aspect of particle trajectories and theirinteraction with the wave function. Can we explain the importance of spin inphysics and chemistry if Bohmian mechanics is true? Is it plausible that quantumchemistry and quantum field theory would have been developed without realismabout spin and other quantum numbers? As argued previously, we are owed morethan an appeal to the in principle empirical equivalence of the theories.

Similarly, the no-miracles argument is undermined by the empirical success ofrelativity theory if we are to interpret its ascription of deep structure to spacetime asmerely empirically useful and not to be believed. The more cases there are of greatempirical success where a realistic construal of the theory is not available, theweaker the realist’s argument that the history and practice of science is onlyintelligible if scientific realism is adopted.

Bohm theory requires the plausibility of a counterfactual history. In the light ofthe discussion of underdetermination and how realists appeal to the role of theoriesin heuristics, the onus is on the defender of the alternative theory to give a detailedaccount of a plausible counterfactual history that leads to equally productivephysics. Cushing’s case depends only on the high-level theoretical equivalenceof Bohmian and quantum mechanics. The situation is even worse when it comes tointeracting QFT and the standard model, because no fully worked out Bohmianalternatives exist.

128 James Ladyman

According to van Fraassen, the big problem with scientific realism is that it addsmetaphysics to scientific theories for no empirical gain, and this seems to be true inthe case of Bohm theory. In this context we might ask: What has the interpretationof quantum mechanics ever done for us? To which the answer is a lot, mostobviously, it has more or less directly given us Bell’s theorem (Bohm), quantumcomputation and information processing (Deutsch), and weak values (Aharonov).

As with the theoretical virtues, there is no support for any one interpretationfrom consideration of fecundity for physics, except perhaps for the Copenhageninterpretation, insofar as the orthodoxy, such as it was and is, has undoubtedlyaccompanied the most extraordinary scientific success. Quantum physics has beenoutstandingly virtuous on all criteria other than that of cohering with backgroundmetaphysics. For all the theory’s indeterminism, when it is taken to the relativisticdomain, it is as predictively accurate in practice as any determinist could reason-ably demand. It does not tell us what the fundamental nature of being is, in terms ofsomething we can think of as objectively and separately located, with intrinsicproperties in space and time. However, it is not a complete theory of the world, asdiscussed in the next section.

7.3 Realism about What?

There is no lost age of ontological purity in science. Classical physics is not just pointparticle mechanics, and the existence of ethers and other weird kinds of matter wasposited throughout the history of physics to explain the phenomena of electricity, heat,light, and magnetism as well as in chemistry. No theory we have ever had in physicshas had a settled metaphysical interpretation or been a plausible theory of the wholeworld. As argued in Section 7.1, not being a realist about NRMPQM is completelycompatible with metaphysical realism, entity realism, and scientific realism. More-over, it has not been our best physical theory since very soon after its inception. Forthis reason it is bizarre to seek an interpretation of it as if it were complete. As pointedout in the last section, the true object of any kind of scientific realism that goes beyondeffective entity realism is our best physics as a whole. Yet realists arguing aboutNRMPQM often espouse instrumentalism about QFT because of its mathematicalinexactitudes and the fact that it does not present a clear conceptual framework(outside of algebraicQFT,which some philosophers think is apt to study accordingly).

Several points that are not always noted are relevant to the measurementproblem in this context:

(1) The time evolution of open systems is not unitary.(2) There is no empirical warrant for applying quantum theory to macroscopic

objects outside of very special circumstances.

What Is the Quantum Face of Realism? 129

(3) There many cases in physics and science generally in which “more isdifferent.”

For all these reasons it seems that the measurement problem does not compel achoice between Everett, Bohm, and dynamical collapse. Maudlin (2018) quotesLakatos, saying that Bohr and his associates brought about the “defeat of reasonwithin modern physics.” However, all the revisionist interpretations of NRMPQMare ultimately parasitic, in practice as well as in theory, on the great empiricalsuccess of standard quantum physics, which they have not yet matched, even inprinciple and with the benefit of hindsight. They may say there is no collapse butthey all have some notion of “effective collapse.” In the two-slit experiment, nomatter what the mechanism for detecting which slit a particle went through,determining the position always has the same effect on the determinacy of themomentum in accordance with the uncertainty relation.

7.4 Against Counterrevolutionary Conservativism

It is not exaggerating to describe the replacement of classical mechanics and fieldtheory by quantum physics as a revolution in physics (in the colloquial, not theKuhnian theoretical sense) because so much changed. In particular, the distinctionbetween matter and radiation broke down, and everyday matter is understood to bestable and apparently continuous and solid because of the quantum nature of theatom and how its parts interact electromagnetically. Very small things do notbehave like big things made small. This should not now be so hard to swallow.Major scientific advances almost always require conceptual innovation and theabandonment of the kind of assumptions we get from common sense and themanifest image. Invariably, things are more complicated than they first seem. Asargued in the previous sections, rejecting the realist rivals to standard quantumphysics does not require abandoning metaphysical or scientific realism. Indeed,forms of scientific realism based on the ideas of structure and pattern rather than ofmaterial particles (Ladyman and Ross 2007), which are compatible with acceptingthe revolution in our understanding of matter wrought by quantum physics, arespecifically motivated by our scientific understanding of what matter is like. Froma naturalistic point of view, revisionary physics should be a last resort.

In spite of the pressure to adopt a realist interpretation of NRMPQM, saying “aplague on all your houses” is compatible with both realism and reason. Wavefunctions do an excellent job of representing the physical state of systems that inpractice are sufficiently separable from the rest of the world to have an effectivestate of their own. There should be no ontology of the wave function because wavefunctions are representations of, for example, the momentum and spin of particles,

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not extraphysical properties of particles. Realists insist that the “beables” ofquantum mechanics be specified. Prima facie they are the pure states of the theory,which can only be attributed to systems separately if they are not entangled.Schrödinger time evolution only applies to systems that are effectively isolated,and does not apply to the whole universe or to open systems. As yet, nobodyknows where to make the final cut.

References

Bohr, N. (1949). “Discussions with Einstein on epistemological problems in atomicphysics,” pp. 200–241 in P. A. Schilpp, Albert Einstein: Philosopher-Scientist.Evanston: The Library of Living Philosophers.

Brown, H., Elby, A., and Weingard, R. (1996). “Cause and effect in the pilot-waveinterpetation of quantum mechanics,” pp. 309–319 in J. T. Cushing, A. Fine, andS. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal.Dordrecht: Kluwer Academic Publishers.

Carroll, S. (2013). “The most embarrassing graph in modern physics,” www.preposterousuniverse.com/blog/2013/01/17/the-most-embarrassing-graph-in-modern-physics/

Cushing, J. T. (1998). Philosophical Concepts in Physics: The Historical Relation betweenPhilosophy and Scientific Theories. New York: Cambridge University Press.

Daumer, M., Dürr, D., Goldstein, S., Maudlin, T., Tumulka, R., and Zanghì, N. (2006).“The message of the quantum?,” pp. 129–132 in A. Bassi, D. Dürr, T. Weber, andN. Zanghì (eds.), Quantum Mechanics: Are There Quantum Jumps? and On thePresent Status of Quantum Mechanics, AIP Conference Proceedings. College Park,Maryland: American Institute of Physics.

Folse, H. J. (1985). The Philosophy of Niels Bohr: The Framework of Complementarity.Amsterdam-Oxford: North-Holland.

Forman, P. (1971). “Weimar culture, causality, and quantum theory: Adaptation byGerman physicists and mathematicians to a hostile environment,” Historical Studiesin the Physical Sciences, 3: 1–115.

Forman, P. (1984). “Kausalität, Anschaullichkeit, and Individualität, or how culturalvalues prescribed the character and lessons ascribed to quantum mechanics,”pp. 333–347 in N. Stehr and V. Meja (eds.), Society and Knowledge. New Bruns-wick-London: Transaction Books.

Howard, D. (1985). “Einstein on locality and separability,” Studies in History and Phil-osophy of Science, 16: 171–201.

Kragh, H. (2002). Quantum Generations: A History of Physics in the Twentieth Century.Princeton: Princeton University Press.

Ladyman, J. and Ross, D. (2007). Every Thing Must Go. Metaphysics Naturalized. Oxford-New York: Oxford University Press.

Ladyman, J. and Ross, D. (2013). “The world in the data,” pp. 108–150 in D. Ross,J. Ladyman, and H. Kincaid (eds.), Scientific Metaphysics. Oxford: Oxford UniversityPress.

Maudlin, T. (2018). “The defeat of reason,” Boston Review, http://bostonreview.net/science-nature-philosophy-religion/tim-maudlin-defeat-reason

Mill, J. S. (1843). “System of logic,” pp. 1963–1991 in J. M. Robson (ed.), The CollectedWorks of John Stuart Mill. Toronto: University of Toronto Press, London: Routledgeand Kegan Paul.

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Psillos, S. (1999). Scientific Realism: How Science Tracks Truth. London: Routledge.Saunders, S., Barrett, J., Kent, A., and Wallace, D. (2010). Many Worlds? Everett,

Quantum Theory, and Reality. Oxford: Oxford University Press.Schlosshauer, M., Kofler, J., and Zeilinger, A. (2013). “A snapshot of foundational

attitudes toward quantum mechanics,” Studies in History and Philosophy of ModernPhysics, 44: 222–230.

van Fraassen, B. (1980). Scientific Image. Oxford: Oxford University Press.

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8

To Be a Realist about Quantum Theory

hans halvorson

8.1 Introduction

There is a story that some philosophers have been going around telling. It goessomething like this:

The pioneers of quantummechanics –Bohr, Heisenberg, Dirac, et al. – simply abandoned hopeof providing a realist theory of the microworld. Instead, these physicists settled for acalculational recipe, or statistical algorithm, for predicting the results of measurements. Inshort, Bohr et al. held an antirealist or operationalist or instrumentalist viewof quantum theory.

Implicit in this story is a contrast with the “traditional aspirations of science” todescribe an observer-independent reality. Having built up a sense of looming crisisfor science, the storyteller then introduces us to the heroes, those who would staytrue to the traditional aspirations of science.

As the 20th century moved into its second half, there arose a generation of renegadephysicists with the courage to stand up against antirealism and operationalism. Thesevaliant men – David Bohm, Hugh Everett, John Bell – renewed the call for a realisttheory of the microworld.

This kind of story can be very appealing. It is the age-old “good guys versus bad guys”or “us versus them”motif. And those “ist”words make it easy to distinguish the goodguys from the bad, sort of like the white and black hats of the classic westerns.

The story is brought into clearer focus by talking about the quantum wavefunction. What divides the realists from the antirealists, it is said, is their respectiveattitudes toward the wave function: Antirealists treat it as “just a bookkeepingdevice,” whereas realists believe it has “ontological status.” Witness the faux-historical account of Roger Penrose:

It was part of the Copenhagen interpretation of quantum mechanics to take this latterviewpoint, and according to various other schools of thought also, ψ is to be regarded as acalculational convenience with no ontological status other than to be part of the state of

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mind of the experimenter or theoretician, so that the actual results of observation can beprobabilistically assessed.

(Penrose 2016: 198)

I suppose that Penrose can be forgiven for oversimplifying matters, as well as forpropagating the myth of the “Copenhagen interpretation” (see Howard 2004).After all, there can be great value in simple fictional tales if they get readersinterested in the issues.

I also imagine that Sean Carroll is aiming to generate some heat – rather moreheat than light – when he poses the following dilemma about the wave function:

The simplest possibility is that the quantum wave function isn’t a bookkeeping device atall . . .; the wave function simply represents reality directly.

(Carroll 2017: 167)

This seemingly simple dilemma – ontological status: yes or no, – is a fine devicefor popular science writing, which should not demand too much from the reader.But is it really the right place to locate a pivot point? Is the question, “Ought I tocommit ontologically to the wave function?” the right one to be asking?

Popular science writers are not the only ones to have located a fulcrum at this point.In fact, some philosophers say that if you are a scientific realist, then you are logicallycompelled to accept the Everett interpretation. I am thinking of this kind of argument:

If you’re a realist about quantum theory, then you must grant ontological status to thequantum state. If you grant ontological status to the quantum state, and if quantummechanics is true, then unitary dynamics is universal. Under these conditions, realismand unitary dynamics, you have two options: either you accept the completeness ofquantum theory, or you don’t. And if you accept the completeness of quantum theory,then the Everett interpretation is true.

In short, we are told that the following implication holds:Realismþ Pure QM ) Everett

Notice how much work realism is supposed to do in this implication!You might accuse me of caricature, and I am sure I have left out much of the

nuance in this argument. And yet, Everettians regularly gesture in this direction. Forexample, Wallace (2013) claims that the Everett interpretation “is really just quan-tummechanics itself understood in a conventionally realist fashion,” and that “thereis one pure interpretation which purports to be realist in a completely conventionalsense: the Everett interpretation” (Wallace 2008). Similarly, Saunders claims that ifwe don’t think of the wave function as a measure of our ignorance, then

the only other serious alternative (to realists) is quantum state realism, the view that thequantum state is physically real, changing in time according to the unitary equations and,somehow, also in accordance with the measurement postulates.

(Saunders 2010: 4)

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In short, if you are a good realist, then you will say that the quantum state isphysically real, and from there it is a short step to the Everett interpretation.

There is something strange about this sort of argument. The notion of “realism”is doing so much of the work – and yet, nobody has told us what it means. Howcould the “if realism, then Everett” argument be valid when “realism” has not beendefined clearly? And how could the argument be convincing when realism has notbeen motivated, except through its undeniable emotional appeal?

In this chapter, I will take a closer look at the distinction between realist andantirealist views of the quantum state. I will argue that this binary classificationshould be reconceived as a continuum of different views about which properties ofthe quantum state are representationally significant. What is more, the extremecases – all or none – are simply absurd and should be rejected by all parties. Inother words, no sane person should advocate extreme realism or antirealism aboutthe quantum state. And if we focus on the reasonable views, it is no longer clearwho counts as a realist and who counts as an antirealist. Among those taking amore reasonable intermediate view, we find figures such as Bohr and Carnap – instark opposition to the stories we have been told.

8.2 Extremists

Suppose that you were asked to list historical figures on two sheets of paper: Onthe first sheet, you are supposed to list realists (about the quantum state), and onthe second sheet you are supposed to list antirealists (about the quantumstate). Suppose that you are asked to sort through all of the big names of quantumtheory – Bohr, Heisenberg, Dirac, Bohm, Everett, etc.

I imagine that this task would be difficult, and the outcome might be controver-sial. For almost none of these people ever explicitly said, “I am a realist” or “I aman antirealist” or “the wave function has ontological status” or anything like that.You would have to do quite a bit of interpretative work before you could justifyassigning a person to one of the lists. You would have to assess that person’sattitude toward the quantum state by studying their behavior and utterances withrespect to it. For example, if person X makes free use of the collapse postulate, withno proposed physical mechanism, then you might surmise that X is either a mind-body dualist, or an operationalist about the quantum state, or both. In other words,an operationalist stance might serve as the best explanation for X’s utterances andbehavior.

The task of sorting people into realist and antirealist would be simplerfor contemporary figures, who seem happy to embrace one of these two labels. Forexample, Sean Carroll and Lev Vaidman will tell you, with great passion, that thewave function is just as real as – in fact, more real than! – a rock, or a tree, or yourspouse. In contrast, Carlo Rovelli speaks of the wave function as Laplace spoke of

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God: Je n’avais pas besoin de cette hypothèse-là. These are just a few examplesamong the many philosophers and physicists who have openly labeled themselvesas realist or antirealist about the quantum state. Self-identified state realists includeEsfeld, Goldstein, Ney, Saunders, Wallace, Zanghì, etc. Self-identified state anti-realists include Bub, Fuchs, Healey, Peres, etc. The battle lines have been clearlydrawn, but what is at stake?

The right-wing extremists say: Quantum wave functions are things. That view issilly. The left-wing extremists say: Quantum wave functions are just bookkeepingdevices. That view is just as silly.

8.3 Right-Wing Extremists

One might think that the litmus test for realism about quantum theory could beposed as:

Do you believe that the wave function (more generally, the quantum state)exists?

Or, as Callender (2015) puts it,

Is the quantum state part of the furniture of the world?

So, when Carroll (2017: 142) says that, “the basic stuff of reality is a quantumwave function,” he is declaring his allegiance to wave function realism.

But what is this wave function thingy? Should I be thinking about it likeI think about chairs or tables? No, say the philosophers; you have to be a bitmore sophisticated about it. The preferred ontological reading of the wavefunction is as a field, on analogy to things like the magnetic field that surroundsthe earth. Thus, to push the ontological picture further, things are representedby points in the domain of that field, and the properties of those things are thevalues of that field. What then are the things according to this ontologicalview? Some philosophers say that the wave function is a field on configurationspace (Albert 1996, Ney 2012, North 2013), so that the things are points ofconfiguration space. Others say that the wave function is a multi-field onphysical space (Forrest 1988, Belot 2012, Chen 2017), so that the things arespacetime points.

These straightforwardly ontological views have been subjected to many criti-cisms (see Wallace and Timpson 2010, Belot 2012). Here I want to raise anotherkind of objection. Or rather, I want to make a request of the ψ-field theorists:Would you please describe your theory clearly, including its states, properties, andthe relationship between them? To my mind, the attraction of ψ-field theories isdue in large extent to the vague realist associations that they conjure up in our

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heads: The wave function is a thing with a definite shape! I wager that such theoriesare plausible only to the extent that it is unclear what they are really saying.

For starters, in quantum theory, the primary theoretical role of the wave functionψ is as a state. The ψ-field theorists ask us to change our point of view. Instead ofthinking of ψ as a state, we are to think of ψ as a field configuration. There arenumerous problems with this proposal.

In classical physical theories, the word “state” is shorthand for “a maximallyconsistent list of properties that could be possessed by the system simultaneously,”or equivalently, “an assignment of properties to objects.” In that case, there are twopossible things we could mean by the sentence “the state σ exists.” First, we couldmean that the list of properties exists. But this list is an abstract mathematicalobject, which would exist whether or not the corresponding theory is true. So, inthis first sense, “σ exists” is not interesting from the point of view of physics.Second, we could use “σ exists” as an obscure shorthand for “σ is the actual state,”which, in turn, is shorthand for saying that certain other objects have certainproperties. Thus, in this second case, “σ exists” is cashed out in terms that donot refer to σ at all. In philosophers’ lingo, “σ exists” is grounded in facts aboutother objects, and so is not really about σ at all.

Now, the defender of quantum state realism might simply say: “That wasclassical physics. In quantum physics, the state takes on a new role.” I certainlyaccept that quantum physics changes some of the ways we talk about the physicalworld. But I am not so sure that it makes sense to reify states. According to thenormal senses of “object” and “state,” we affirm that objects can be in states. Thus,if states are objects, then states themselves can be in states. But then, to beconsistent, we should reify the states of those states, and these new states willhave their own states, ad infinitum. In short, if you run roughshod over thegrammatical rules governing the word “state,” then you can expect some strangeresults.

To continue that line of thought, we assume that things can be counted. In otherwords, it makes sense to ask: How many things are there? But then, if states werethings, it would make sense to ask: How many states are there? But now I amcompletely puzzled. According to quantum theory, the universe has an infinitenumber of potential states, but only one actual state. What in the world wouldexplain the absence of all the intermediate possibilities? Why couldn’t there havebeen 17 states? And what’s more, why do physicists never raise as an empiricalquestion: How many states are there? The reason is simple: Physicists do not treatstates as they do things, not even in the extended sense where fields also count asthings.

I hope that by this stage you are at least partially convinced that it does violenceto the logic of physical theories to talk about states as if they were things. But then

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you should agree that the role of a wave function is not to denote an object.Moreover, if ψ does not denote a physical object, then the properties of ψ do notdirectly represent the properties of a physical object. Granted, we should be carefulwith this latter claim. Even in classical physics, the properties of a state canrepresent, albeit indirectly, the properties of a physical object. For example, for aclassical particle, “being in subset Δ of statespace” is a property of states thatrepresents a corresponding property of the relevant particle. Nonetheless, there aretwo different types of things here – the particle, which is a concrete physical object,and its state, which is an abstract mathematical object. The latter tells us about theformer, but should not be conflated with it.

In classical theories, there is also a sharp distinction between instantaneousconfigurations and states. If a configuration is represented by a point in themanifold M, then a state is represented by a point in the cotangent bundleT∗M. In many scenarios, T∗M looks like a Cartesian product M �M, wherethe first coordinate gives instantaneous configuration, and the second compon-ent gives momentum. In every case, there is a projection mappingπ : T∗M ! M, and the preimage of any particular configuration q 2 M is aninfinite subset of T∗M. But now, if ψ is both a state and a field configuration,then it is unclear where it lives. Does ψ live in the space M of configurations,or does it live in the space T∗M of states? How can it do both jobs at thesame time?

These considerations show that the ψ-field view stretches the logic of classicalphysics beyond the breaking point. To treat ψ as representing a field configurationis to disregard its primary theoretical role as a state. Or, at the very least, to treat itthus would obscure the difference, central to classical theories, between configur-ations and states.

If that is not enough trouble for ψ-field views, we can also ask them to give anaccount of the properties that are possessed by this thing, the ψ-field. Recall that ina classical theory with statespace S, properties are typically represented by subsetsof statespace S. (We might require that these subsets be measurable or somethinglike that. But that point will not matter in this discussion.) Then, we say that thesystem has property E � S just in case it is in state σ 2 E.

Now, ψ-field theorists would like us to think of quantum theory on the model ofa classical field theory. In this case, the statespace would be the space C∞ Xð Þ ofsmooth complex-valued solutions to some field equation, and subsets of C∞ Xð Þwould represent properties that the system can possess. (Let us ignore here the factthat classical field theories typically use the space of real-value functions.) Forexample, for any field state f 2 C∞ Xð Þ, the singleton set ff g represents theproperty of being in state f , and its complement C∞ Xð Þ∖ ff g represents the propertyof not being in state f .

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For the purpose of performing certain calculations, a classical field theoristmight complete C∞ Xð Þ relative to some norm, obtaining a Hilbert space, such asthe space L2 Xð Þ of (equivalence classes of ) square-integrable functions on X. Theelements in L2 Xð Þ are no longer smooth functions, and in fact, they are not reallyfunctions at all – they are equivalence classes of functions under the relation: f � gjust in case

Ð j f � g j dμ ¼ 0.In contrast, for a quantum theorist, L2 Xð Þ is simply one instance of, or one

representation of, a Hilbert space H of countably infinite dimension. Any twoHilbert spaces of the same dimension are isomorphic, so it does not matter (for thephysics) which one we choose. The states of the system are represented not bypoints in L2 Xð Þ, but by rays. And the properties of the system are represented notby subsets of L2 Xð Þ but by closed subspaces. Thus, in short, while L2 Xð Þ is used byboth the classical field theorist and the quantum mechanic, it is used in completelydifferent ways in the two cases. For those of us who believe that the Hilbert spaceformalism is intended to represent reality, we could say that it represents reality ina very different way than a classical field theory does.

The ψ-field views ask us to forget the differences between quantum mechanicsand classical field theories. But it will not be easy to forget these differenceswithout doing violence to the representational role of the various pieces of theHilbert space formalism. A classical theory comes with representatives (subsets ofstatespace) for many properties that are not represented in the Hilbert spaceformalism. The ψ-field theorist wants to lay claim to all these properties – for thisseems to provide the coveted “god’s-eye view” of reality. In effect, the ψ-fieldpicture is designed to make us feel like we have evaded the Kochen-Speckertheorem. For, if physical properties are represented by subsets of L2 Xð Þ, or bythe mathematical properties of a function ψ 2 L2 Xð Þ, then each such property iseither definitely possessed or definitely not possessed when the system is in state ψ.This view is intended to hide (or ignore? or deny?) the fact that a quantum statedoes not answer all questions about which properties are possessed.

What is more, the ψ-field view only follows classical physics as far as giving aninstantaneous snapshot of the possessed properties. As soon as it comes to drawinginferences about the system, it imposes ad hoc rules to block fallacious inferences.For example, in a classical field theory, if σ and σ0 are distinct field states, thenknowing that the system is in state σ permits you to assert that the system is not instate σ0. Or in probabilistic terms, the probability of σ0 conditional on σ is 0. If youcarry that inference rule over directly into quantum theory, then you will makefalse predictions. A Gaussian function ψ centered at 0 is a different field state thana Gaussian function ψ0 centered at 0:01. Thus, on a classical picture, the property Eof “being in state ψ” is inconsistent with the property E0 of “being in state ψ0 ,” andPr E0jEð Þ ¼ 0. But quantum theory says that Pr E0jEð Þ � 1.

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Of course, ψ-field theorists are too clever to fall into the trap of carryingclassical inference rules into the quantum domain. Although they purport to viewthe ψ-field classically, they stop short when it comes to reasoning about it. For thepurposes of reasoning and making predictions, they turn to the Hilbert spaceformalism to guide them. Thus, we might summarize the attitude of these ψ-fieldviews in a phrase: You can look at the world from the god’s-eye point of view; justdon’t reason about it as god would.

8.4 Left-Wing Extremists

At one extreme, we have people telling us that the wave function is part of thefurniture of reality. At the opposite extreme, we have people telling us that thewave function is “a mere calculational device” (Rovelli 2016: 1229), and that “it ismistaken to view the universal wave-function as a beable” (Healey 2015). Thissecond group of extremists is a curious bunch. They protest loudly against thewave function, producing elaborate (and interesting) arguments against its onto-logical status. And yet, they cannot seem to live without it. In their books andarticles, they accord a privileged role to the wave function. When they want to saysomething true about a quantum system, they consult the wave function beforeanything else. It makes one wonder: If they do not believe in the wave function,then why do they grant it a special role in their representations of reality?

The practice in physics, followed by realists and antirealists alike, is that eachclassically described “preparation” or “experimental setup” may be representedby a unique quantum state. In fact, the ability to associate quantum states toclassically described experiments is one of the skills that displays mastery ofquantum theory. Once an experiment has been adequately described, then there isno remaining latitude for idiosyncratic or subjective state assignment. There isjust one correct state, as will be borne out by checking the statistics of measure-ment outcomes. The fact that physicists have correctness standards for quantumstate assignments strongly suggests that they grant these states some sort ofrepresentational role.

Healey (2017) makes exactly this point, and he uses it to make an argument forquantum state objectivism, i.e., the belief that there are objectively correct ascrip-tions of quantum states to physical systems. But isn’t this sort of state objectivismstrictly inconsistent with state antirealism? If the quantum state is not real, thenhow could one be wrong about the quantum state? In order to answer this question,Healey engages in subtle reasoning about how objective correctness can be disen-tangled from the correspondence theory of truth and about how the meaning of thequantum state can be accounted for in an inferential theory of content. This justgoes to show that matters are not as simple as they initially appeared to be.

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Healey’s subtlety is laudable, but sometimes it verges on doublespeak. Forexample, Healey hangs much on the distinction between ascribing a state ψ to athing X, and describing X with ψ.

Pragmatists agree with QBists [quantum Bayesians] that quantum theory should not bethought to offer a description or representation of physical reality: in particular, to ascribe aquantum state is not to describe physical reality.

(Healey 2016: on line, emphasis added)

What are we supposed to be doing when we “ascribe” a quantum state? If ψ has norepresentational role whatsoever, then why speak of “ascribing” it to a physicalobject or situation? Why not just speak of “using” the wave function – as one usesa computer or a hammer – to get a job done?

In the English language, the word “ascribe” involves a subject postulating arelation between two objects: S ascribes Y to X. More is true: In normal conversa-tion, to ascribe Y to X involves judging that there is a preexisting relation betweenY and X. For example, “He ascribed Jane’s short temper to her upset stomach.” Inthis way, ascribing is different than using: I can use Y to do something to X withoutmaking any judgment about the relation between Y and X. These considerationsshow that the word “ascribe” is tantalizingly close to other words – such as“describe” – that connote the existence of a representational relation, exactly thesort of thing that Healey wishes to deny. To consistently carry out his pragmatistprogram, Healey should use a different word than “ascribe.”

Here is what I think is really going on here. The phrase “Y describes X” is rathervague; and being vague, it can be thought to license all sorts of inferences about therelationship between Y and X. When people say that “Y describes X,” they tend toimport a lot of baggage that goes far beyond the simple existence of a representa-tional relation between Y and X. In fact, it seems all too easy to fall into the mistakeof thinking that Y describes X only if Y is similar to X. Of course that is not true:The phrase “is over 6 feet tall” describes Goliath, but this phrase is not similar toGoliath.

That temptation to assume similarity is all the more difficult to resist when thefirst argument of “Y describes X” is a geometrical object, such as a wave function.The reason we fall into this trap, I assume, is because we do frequently usegeometric objects as pictorial representations. For example, I might draw a rect-angle on a piece of a paper and say, “This rectangle describes the shape of mydesk.” In this case, the rectangle on the paper is indeed similar to the desk in a well-defined mathematical sense.

Healey, Rovelli, and other self-proclaimed antirealists have surrendered toomuch to their opponents. They have allowed their opponents to define words like“ontological status” and “describes.” Then, because Healey and Rovelli reject the

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implications that come along with this particular definition of “describes,” they areforced to say that the quantum state does not describe at all. Thus, Healey andRovelli lay themselves open to the charge of antirealism – which, of course, carrieshighly negative connotations. To be an antirealist implies a sort of failure ofcourage – it implies a sort of retreat. Ergo, Healey and Rovelli are seen as makingless bold assertions about reality than their realist counterparts are making.

8.5 The State as Directly Representing

Are you a realist about the quantum state? We have already seen that this questioncannot be paraphrased as: Do you believe that the quantum state exists? So whatcould the question mean? According to Wallace and Timpson,

Traditionally realist interpretations . . . read the quantum state literally, as itself standingdirectly for a part of the ontology of the theory.

(Wallace and Timpson 2010: 703)

In fact, Wallace (2018b) locates the crucial divide between “representational” and“non-representational” views of the quantum state. Thus the shift is signaled fromthe material mode of speech (does the state exist?) to the formal mode of speech(does the state represent?). In particular, Wallace and Timpson claim that realisminvolves commitment to both literal and direct representation. Thus, Carroll uttersthe shibboleth when he says, “the wave function simply represents reality directly”(Carroll 2017: 167). But what work is the word “directly” doing here? I am led tothink that the task of representing must be a bit like getting to work, where youhave to take the right turns in order to follow the most direct route. So what are theinstructions for following the direct route to representation?

When a person says that Y represents X, then that typically signals that theperson endorses some inferences of the form

(†): If Y has property ϕ, then X has property ϕ0,

where ϕ↦ ϕ0 is some particular association of properties (the details of which neednot detain us). Let us call (†) a property transfer rule. For example, if I say that acertain map represents Buenos Aires, then I mean that some facts about BuenosAires can be inferred from facts about the map.

What then is the force of insisting that Y does not merely represent X, but that itrepresents X directly? I suspect that the word “directly” is supposed to signalendorsement of quite liberal use of property transfer rules. But just how liberal?The key question to keep in mind is: Which specification of permitted property-transfer inferences corresponds most closely to the notion of “direct representa-tion” that is favored by realists such as Carroll, Wallace, Saunders, and Timpson?

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When they say that “ψ directly represents reality,” what exactly are they sayingabout the relation between ψ and the world?

Consider first the proposal:

(DR1) Y directly represents X just in case every property of Y is also a property of X.

This proposal is logically consistent, but also absurd. One of the properties that Xhas is being identical to X. Thus, according to DR1, if Y directly represents X thenY ¼ X. Could Wallace and Timpson possible intend this? Does Carroll mean tosay our universe is a subset of R3n � C? If so, then scientific realism is truly aradical point of view. The wave function is an abstract mathematical object. Thus,if the universe is a wave function, then the universe is an abstract mathematicalobject. Perhaps mathematicians will applaud this conclusion, because then puremathematics tells us everything there is to be known about the universe.

I suspect that the realists do not mean their direct representation claim in thesense of DR1. Let us try a more reasonable proposal.

(DR2) Y directly represents X just in case each mathematical property of Y corres-ponds to some physical property of X.

Here we need some precise account of the “mathematical properties” of Y .According to standard set-theoretic foundations of mathematics, the mathematicalproperties of Y are precisely those properties that can be described in the languageof Zermelo-Fraenkel (ZF) set theory. Thus, for example, the mathematical proper-ties of Y would include its size (cardinality). In contrast, arbitrary predicates innatural language do not pick out mathematical properties of Y . For example, “is anabstract object” cannot be articulated in ZF set theory, and so would not count as amathematical property of Y . Thus, DR2 does not say that “anything goes” in termsof the representationally significant properties of Y .

Even so, DR2 is still implausibly profligate in the number of representationallysignificant properties it assigns to the wave function. In particular, for eachdefinable name c in ZF set theory, there is a definable predicate Θc given by

Θc Sð Þ $ c 2 S:

Among these definable set names, we have ∅, ∅f g, and so on. Now, a wavefunction is a function ψ : A ! B, with domain set d0f ¼ A and codomain setd1f ¼ B. Thus, for any definable name c, it makes sense to ask whetherΘc d0ψð Þ, i.e., whether c is contained in the domain of ψ.

Imagine now the following scenario. Two physicists, Jack and Jill, are arguingabout whose wave function is a better representation of the universe. The funnything is, Jack and Jill’s wave functions are both Gaussians, centered on 0, and withthe same standard deviation. If you ask Jack to draw a picture of his wave function,then he draws a Gaussian centered at 0. If you ask Jill to draw a picture of her wave

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function, then she also draws a Gaussian centered at 0. They agree that this pictureis a correct representation of their respective wave functions. They also agree thattheir wave functions are written in the configuration space basis, and that the origin0 represents the same point in the universe. It seems that there is nothing left forthem to disagree about.

And yet, Jack and Jill insist that their wave functions cannot both be correct.According to Jack, the correct wave function ψ has the property that Θ∅ d0ψð Þ, thatis, the empty set is an element of the domain of the wave function. According toJill, the correct wave function ψ0 does not have that property. They both believethat Θ∅ corresponds to a genuine physical property. Jack asserts that this propertyis instantiated, and Jill asserts that it is not.

Jack and Jill would fail their quantum mechanics course. They do not under-stand how the theory works. In using the formalism of quantum theory to representreality, we do not care about these fine-grained set theoretic differences. Iftwo wave functions have the same shape, then we consider them to be the same.If two wave functions can be described via the same equation, then we take them tobe identical. But what is this notion of same shape that we are using here? How canwe tell when two wave functions are the same, at least for the purpose of doingphysics?

At this point, we might want to lay down the ace card of recent philosophy ofscience: the notion of isomorphism. Can’t we just say that two wave functions arerepresentationally equivalent just in case they are isomorphic? In this case, wecould then propose the following criterion for direct representation:

(DR3) Y directly represents X just in case Y and X are isomorphic.

This proposal sounds a lot more plausible than the previous two – especiallybecause the word “isomorphism” is simultaneously precise (within certain fixedcontexts) and flexible (since it means different things in different contexts). Butthat is precisely the problem with DR3: the phrase “Y is isomorphic to X” is nobetter defined than the phrase “Y directly represents X.”

In mathematics, isomorphism is a category-relative concept. If you hand me twomathematical objects and ask, “Are they isomorphic?” then I should reply by asking“Which category do they belong to?” For example, two mathematical objects can beisomorphic qua groups, but nonisomorphic qua topological spaces. Thus, it makes nosense to say that a mathematical object is isomorphic to the world tout court. In orderto make sense, we would first have to specify a relevant type (or category) ofmathematical objects. For example, one might say that the world is isomorphic to atopological space Y , as shorthand for saying that the world has topological structure,and is in this sense isomorphic to Y . But if you giveme a concretemathematical objectA and say that the world is isomorphic to A, then I have no idea what you are saying.

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So, if we want to say that the world is isomorphic to a wave function ψ, then weneed to say what category of mathematical objects we take ψ to belong to. And thatis not going to be easy, for ψ is not a group, or a topological space, or adifferentiable manifold, or any other of the standard types of mathematical struc-ture. There is no category of wave functions; and, there is no nontrivial notion ofisomorphism between wave functions. It will not help to say that ψ and ψ0 areisomorphic wave functions just in case there is a unitary symmetry U such thatUψ ¼ ψ0, for in that case, all wave functions would be isomorphic. The closest wecome to finding a home for ψ is in the category of Hilbert spaces: ψ is an elementof a Hilbert space, which is an object in the category of Hilbert spaces. But that willnot help, because we do not want to say that the world is isomorphic to the Hilbertspace H, but that it is isomorphic to a particular wave function ψ.

There are numerous other problems with analyses of representation in terms ofisomorphism, some of which are discussed in a recent article by Frigg and Nguyen(2016). We mention two further problems here, each of which might be taken todeliver a fatal blow to the account. First, an isomorphism is a function between twomathematical objects, and the world is not a mathematical object. In fact, aspointed out long ago by Reichenbach (1965), the only grip we have on thestructure of the world is by means of our representations.

Second, our account of representational significance should mesh with ouraccount of theoretical equivalence, and many philosophers of science hold viewsof theoretical equivalence according to which equivalent theories need not haveisomorphic models. For example, Halvorson (2012) labels this view as “the modelisomorphism criterion of theoretical equivalence,” and he argues that it must berejected. However, if the model isomorphism criterion of theoretical equivalence isrejected, then we must also reject the claim that representation entails isomorphismbetween the world and one of the theory’s models. We can argue as follows: If twotheories T and T 0 are equivalent, and if T is representationally adequate, then T 0 isalso representationally adequate. But if the models of T are not isomorphic to themodels of T 0, then it cannot be the case that the world is isomorphic to a model ofT and also to a model of T 0. Therefore, to say that T is representationally adequatedoes not entail that the world is isomorphic to one of the models of T .

8.6 Representationally Significant Properties

As we have seen, isomorphism-based analyses of representation have difficultyexplaining how wave functions represent – because there is no obvious candidatenotion of “isomorphism” for wave functions. Perhaps, however, we can attack thisproblem from the other side. Having a notion of isomorphism in place gives us acriterion for identifying representationally significant properties:

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A property ϕ is representationally significant just in case ϕ is invariant underisomorphism.

But of course, we need not already have a notion of isomorphism in place tochoose the representationally significant properties. We can simply say what thoseproperties are.

As we know, it would be disastrous to propose that all mathematical proper-ties of a wave function are representationally significant. For example, theproperty of “having domain that contains the element ∅; ∅f gf g” is a perfectlygood mathematical property that a wave function either possesses or does notpossess. But nobody, to my knowledge, has ever proposed that this mathemat-ical property represents a bona fide physical property. In practice, we simply donot care whether we use a wave function ψ that has that property or a similarwave function ψ0 that lacks that property. Many of these set-theoreticallydefinable properties of a wave function are routinely ignored as “surplusmathematical structure.”

In my experience, physicists cannot usually say explicitly which properties of ψare representationally significant. However, we can determine which properties ofψ they care about by watching what they do. If they treat two wave functions ψ andψ0 as interchangeable, then their behavior suggests that they accord no representa-tional significance to properties that separate these two functions. Here we say thata property Θ separates ψ and ψ0 just in case Θ ψð Þ and ¬Θ ψ0ð Þ.

The art of discriminating between wave functions is not so unlike the fabled artof “chicken sexing.” The skilled chicken-sexer has the ability to judge reliablywhether two chicks are of the same sex. But if you ask what criteria he or she isusing, the chicken-sexer will be at a loss for words. In the same way, the skilledquantum mechanic has the ability to judge whether two wave functions arerepresentationally equivalent. And he or she displays his or her judgment ofrepresentational equivalence by his or her disinterest in the question: Which ofthese two wave functions provides the correct representation of reality?

I am not sure that it would be possible to give a fully explicit account of theequivalence relation of “representational equivalence” for wave functions. None-theless, there are certain sufficient conditions for representational equivalence thatare uncontroversial.

First, two wave functions are representationally equivalent if one is a complexmultiple of the other, i.e., if they lie in the same ray in Hilbert space. Thus, if aproperty Θ of wave functions is not invariant under this relation (of lying in thesame ray) then Θ is not representationally significant. For example, consider theproperty Θ given by

Θ ψð Þ $ ψ 0ð Þ ¼ 1ð Þ:

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Clearly there are two functions ψ and ψ0 such that ψ � ψ0, but Θ ψð Þ and ¬Θ ψ0ð Þ.Therefore, Θ is not a representationally significant property of wave functions.

Second, wave functions are not actually functions at all. In fact, the space ofsquare integrable functions on configuration space is not a Hilbert space. Instead,to define a positive-definite inner product, one has to take equivalence classes offunctions relative to the equivalence relation � of “agreeing except on a set ofmeasure zero.” But now consider the property Θ defined by:

Θ ψð Þ $ ψ 0ð Þj j2 ¼ 1� �

:

Again, there are two functions ψ and ψ0 such that ψ � ψ0, but Θ ψð Þ and ¬Θ ψ0ð Þ.Therefore, Θ is not a representationally significant property of wave functions.

This is not to say that there are no representationally significant properties ofwave functions. For example, consider the property

Θ ψð Þ $ðΔjψ xð Þ j dμ xð Þ ¼ 1:

This propertyΘ can be shown to be invariant under the equivalence relationsmentionedpreviously. Indeed, practitioners of quantum theory know exactly what this property is:It is the property Q 2 Δ½ � of being located in the region Δ. What other invariantproperties are there? Can we give some sort of systematic description of them?

As mentioned before, the Hilbert space formalism is normally taken to representproperties by means of the subspaces of the statespace. Let us think about how thisworks in the case of the space L2 Xð Þ of (equivalence classes of ) wave functions.What does a subspace of L2 Xð Þ look like? Some subspaces correspond to proper-ties of functions. For example, consider the property

Θ ψð Þ � ψ has support in the region Δ:

It is not difficult to see that the set of functions satisfying Θ forms a closedsubspace of L2 Xð Þ. But not every subspace of L2 Xð Þ has such an interpretationin terms of straightforwardly geometric features of functions. For example, letU : L2 Kð Þ ! L2 Xð Þ be the unitary isomorphism between the momentum-spaceand position-space representation of wave functions. Now begin by defining thesame sort of subspace, but relative to the momentum-space representation. That is,let E be the subspace of L2 Kð Þ consisting of functions with support in Δ. Thenatural interpretation of E is: having momentum value in the set Δ. Then U Eð Þ is asubspace of L2 Xð Þ, and hence, represents a quantum-theoretic property Θ. But thisproperty Θ does not manifest itself as a natural property of functions on the originalconfiguration space X. Indeed, it is not clear that it would be possible to express Θwithout making reference to the isomorphism between L2 Kð Þ and L2 Xð Þ.

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We have here a nice concrete example of an issue that philosophers havebeen discussing in the abstract – the issue of abundant versus sparse views ofproperties (see Bricker 1996). The Hilbert space formalism gives a specialversion of the sparse view of properties: Not every subset of L2 Xð Þ corres-ponds to a natural property. One might think initially that this sparse viewmakes life difficult by preventing us from saying certain things. For example,as Wallace (2012) points out, this sparse view entails that “has a definite valueof energy” fails to pick out a property (a consequence which he finds to beunacceptable).

However, there is an obvious problem with trying to take an abundant view ofthe properties of quantum theory, i.e., with taking every subset of L2 Xð Þ to pick outa physical property. The problem is that there are too many such subsets, and theirphysical interpretation is unclear. Nonetheless, the Hilbert space formalism pro-vides a method for identifying those subsets of L2 Xð Þ that represent physicalproperties. In particular, we have the following result:

(SQ) Let H be an abstract Hilbert space of countably infinite dimension. Then eachsubspace of H is of the form U�1 Z 2 Δ½ �, where U : H ! L2 Rð Þ is a unitaryisomorphism, Δ is a Borel subset of R, and Z 2 Δ½ � is the subspace of functionswith support in Δ.

(This result is part of the folklore of functional analysis, and may be reconstructedfrom the results in chapter 9 of the book by Kadison and Ringrose [1991].) Here,we think of L2 Rð Þ as wave functions of some particular dynamical variable Z,which could be position (along some axis), or momentum (along some axis), orenergy, or . . . In this case, U�1 Z 2 Δ½ � is the subspace of wave functions where thevalue of Z lies in Δ. In other words, U�1 Z 2 Δ½ � and Z 2 Δ½ � represent the sameproperty – only, this property’s physical interpretation is more perspicuous in thelatter case.

Thus, there is a nontrivial question about which properties of functions (i.e.,subsets of L2 Xð Þ) represent bona fide, or “natural,” physical properties. Take anarbitrary mathematical predicate of functions, such as

Θ ψð Þ � ψ is a smooth i:e:; infinitely differentiableð Þ function,which seems to be quite natural, at least from a mathematical point of view. Butwhy suppose that Θ represents a natural physical property? What criteria shouldwe use to sort out the genuine predicates from the spurious predicates? Some mightsuggest an operationalist criterion:

(operationalist) A predicate Θ of wave functions represents a natural physical property ifand only if there is a measurement that would verify whether an object’s state ψ hasproperty Θ.

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But that criterion is too imprecise. And, in any case, the operationalist criterion isstricter than quantum theory’s own criterion, which countenances many naturalproperties that cannot be operationally detected.

The language of quantum theory, represented via the Hilbert space formalism,comes with a vocabulary, including a list of predicates.

(QM properties) A predicate Θ of wave functions represents a natural physicalproperty if and only if the set ψ 2 L2 Xð Þ j Θ ψð Þf g is a subspace of L2 Xð Þ.

By this result, the previous criterion can be restated as follows:

(QM properties) A predicate Θ of wave functions represents a natural physicalproperty if and only if there is a dynamical variable Z, and a measurable Δ � R, suchthat Θ ψð Þ if and only if ψ lies in the subspace Z 2 Δ½ �.

These predicates can then be taken as giving quantum theory’s preferred account ofnatural properties. In short, the natural properties are precisely those picked out bysaying that a quantity Z has value in a certain range.

So, we return to the original question: If Θ ψð Þ is the predicate “ψ is asmooth function,” then does Θ pick out a physical property of wave functions?Quantum theory answers this question by saying: Θ represents a physicalproperty only if there is some quantity Z such that that Θ picks out the subspaceZ 2 Δ½ �.When we talk about giving a “physical interpretation” to a subset E of state-

space, the demand is not that E be given an operational interpretation, as, e.g.,corresponding to some measurement operation. Instead, we are simply asking thatthe mathematical object E be describable in words that have some antecedentphysical meaning. It is simply the demand that we understand what the formalismpurports to represent.

8.7 Reading the State Literally

Recall that Wallace and Timpson say that a quantum state realist does two things:(1) he or she believes that the state represents reality directly, and (2) he or shereads the state literally. As we saw, there are various ways of cashing out “Ydirectly represents X.” If you push the notion to the extreme, where Y ¼ X, youwill end up saying stupid things; however, as soon as you start to nuance thisnotion, you start to sound less like a full-blooded realist.

So can we find a firm foothold for realism in the second criterion? Is it thecommitment to a “literal reading of the state” that sets the quantum state realistsapart from their antirealist counterparts? Here we have tapped into a central vein inphilosophical discussions of scientific realism. For example, 40 odd years ago, vanFraassen described scientific realism as the belief that

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The aim of science is to give us a literally true story of what the world is like; and theproper form of acceptance of a theory is to believe that it is true.

(van Fraassen 1976: 623, emphasis added)

The debates of the last 40 years seem not to have brought into question theconnection between realism and literalism. In a recent authoritative account ofscientific realism, Chakravartty reasserts the connection:

Semantically, realism is committed to a literal interpretation of scientific claims aboutthe world.

(Chakravartty 2017: on line, emphasis added)

But something fishy must be going on here. The idea that a scientific theory is a setof claims (i.e., sentences) fell out of favor about 40 years ago. Nowadays, mostphilosophers of science say that a scientific theory consists of a collection ofmodels, plus some claim to the effect that one of these models represents theworld. But if a theory is a collection of models, then how am I supposed to read itliterally? Nor can this problem be brushed away by adopting a different view ofscientific theories. For better or worse, the theories of mathematical physicsinvolve collections of mathematical models, such as Lorentzian manifolds, Hilbertspaces, etc. So how then are we supposed to read these theories literally?

The answer, in short, seems to be: To read a theory literally is to take one of itsmodelsM as a reliable guide to features of the world. But now we are right back towhere we were when considering analyses of “Y directly represents X.” If I am aliteralist about M, then which features of M should I take to be representationallysignificant? The simple answer “all features of M” leads immediately to absurdity.The answer “all mathematical features of M” also leads to a bizarre and untenablepicture. Thus, we are thrown back on a more piecemeal approach, where one has toknow how to interpret the model M, which means being able to distinguish itsrepresentationally significant properties from the insignificant ones.

Indeed, learning how to use a physical theory requires that learning the art of“reading claims off” of a model. Consider, for example, the general theory ofrelativity (GTR), where a model M is a Lorentzian manifold. What might it looklike to readM literally? Well, GTR claims that at each point p 2 M, there is a four-dimensional tangent space Tp. And living on top of Tp there is an infinite tower ofm; nð Þ tensors, for all natural numbers m and n. Are these things I have just saidamong the “scientific claims” of GTR? If I am a realist about GTR, then amI committed to these claims? Should I envision an infinitely extended tangent spaceTp of four dimensions sitting on the tip of my nose, and indeed, a different suchtangent space for each instant of time? Are these tangent spaces “part of thefurniture of the world”? If this is what it means to be a realist about GTR, thenEinstein himself was no realist.

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To make the point more clearly, GTR entails that

For each point p 2 M, there is an open neighborhood O of p, and a coordinate chartϕ : O ! R4.

These coordinate charts are just as much elements of a model of GTR as a wavefunction is an element of a model of QM. Thus, if literalism demands commitmentto the wave function ψ, then it also demands commitment to the coordinate chart ϕ.If quantum state realism is just a “literal reading of QM,” then coordinate chartrealism is just a “literal reading of GTR.”

If you do not think that GTR involves a commitment to an ontology of tangentspaces, coordinate charts, etc., then I can only agree: Not every true statement,made within the language of a theory, is one of the “scientific claims” of thattheory. To say that a model M accurately represents the physical world does notmean that every mathematical thing in M represents a physical thing. Realism,according to Chakravartty, Timpson, Wallace, van Fraassen, et al., requires com-mitment to the scientific claims of a theory, interpreted literally. But you cannotinterpret a mathematical object literally. That simply does not make sense. Thedemand for literal interpretation only makes sense after we have used the formal-ism to express claims in a language that we understand.

Here we have to lay some blame at the door of the semantic view of theories.The semantic view of theories plus realism suggests the idea that one ought tointerpret models literally – an idea that can lead to absurd consequences if notfurther nuanced. A model’s elements need not all play the same representationalrole. For example, suppose that I make a map of Princeton University, on whichI draw several buildings. Suppose that I also draw a picture of a compass in thelower right hand corner of my map – to indicate its orientation. Now, I am arealist about the geography of Princeton, and I believe that my map is a faithfulrepresentation of it. But that does not mean that I believe there is a huge compasslying on the ground just outside of the university. Nor would I say that thecompass on the map is “just a bookkeeping device” or that it “has no representa-tional role.” The compass does have a representational role: It represents a claimabout how my map is related to the actual town of Princeton. And if this compasscan be said to have a representational role, then so can a wave function. (For anilluminating investigation of the notion of “literal interpretation,” see Hirsch2017).

8.8 Spacetime State Realism

The most recent development in the realist ontology program is the proposal toupgrade wave function realism to “spacetime state realism” (see Wallace and

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Timpson 2010; Wallace 2018a). But does this technical maneuver dodge thevarious philosophical problems that confront wave function realism? In order topress the question further, we need to sketch the idea behind spacetime staterealism.

Let us begin with the simplest (and least interesting) case of spacetime staterealism – the case where spacetime consists of a single point. In this case, werepresent a quantum system by means of a C∗-algebra A of observables (Foran account of this formalism, see Ruetsche 2011). The important point is thatA is closed under operations of addition, multiplication, and conjugationA↦A∗. Moreover, there is a preferred multiplicative unit I 2 A, the identityoperator. The prototypical case of a C∗-algebra is the algebra of n� n complexmatrices.

We need a few definitions. An operator A 2 A is said to be self-adjoint just incase A∗ ¼ A, and A is said to be positive just in case A ¼ B∗B for some operatorB 2 A. A function ω : A ! C is said to be a linear functional just in caseω cAþ Bð Þ ¼ cω Að Þ þ ω Bð Þ for all A,B 2 A and c 2 C. A linear functional ω issaid to be positive just in case ω Að Þ 0 for every positive operator A 2 A.A positive linear functional ω is said to be a state just in case ω Ið Þ ¼ 1. We willuse Σ Að Þ to denote the space of states of A.

We can formulate quantum mechanics in the language of C∗-algebras just aswell as we can in the language of Hilbert spaces. Indeed, the self-adjoint operatorsin A represent observables (or more accurately, quantities), and the elements ofΣ Að Þ represent physical states. As a particular case in point, if A is the algebra of2� 2 matrices, then the self-adjoint operators are simply the Hermitian matrices,and the states on A correspond one-to-one with density operators on C2 via theequation

ω Að Þ ¼ Tr WωAð Þ:With these definitions in hand, we can state Wallace and Timpson’s proposal quitesimply:

For a system represented by the algebra A, the properties correspond one-to-one with thestates in Σ Að Þ.This proposal can be made more picturesque and plausible if you think of a “fieldof states,” where each point p in spacetime is assigned a state ωp. And if you feelthat this is just empty mathematics, then it might help to think of the typical case,where ωp is represented concretely by a density operator Wp. Then the fieldp↦Wp of density operators starts to look more like a classical field configuration,where some mathematical object, such as a tensor, is assigned to each point inspace. The only mathematical difference is thatWp is a complex matrix instead of a

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tensor. But as Wallace and Timpson point out, the relative unfamiliarity ofcomplex matrices such as Wp should not rule them out as legitimate values of aphysical field.

To this point, I agree with Wallace and Timpson. What bothers me is not thedifference between tensors and complex matrices. What bothers me is the confla-tion of the various theoretical roles of states, quantities, and properties. The typicaljob of states is to assign values to quantities. So, if we ask states also to serve asvalues of quantities, then the job of states will be to assign states. In order to try tokeep things straight in our heads, we might try to declare some “types.” First, thestandard way of thinking of states is that they are of type Q ! V, where Q is thequantity type, and V is the value type. But now, Wallace and Timpson tell us thatstates are also of type V. In this case, states would be both of type Q ! V and oftype V, resulting in a type confusion.

What’s more, we typically ask a physical theory to provide some sort of “state-to-property” link. For example, the so-called orthodox interpretation of quantumtheory proposes the eigenstate-eigenvalue link:

(EE link) A property E of the system is possessed in state ψ just in case Eψ ¼ ψ.

Wallace and Timpson also propose a state-to-property link. However, their prop-erties are of the form “being in state W ,” and so their proposal reduces to:

(WT link) A system has property W when it is in state W

Or perhaps it would be better to say:

(WT link) A system has the property of being in state W just in case it is in state W .

I suppose this claim is true. But I did not need to learn any physics to draw thatconclusion. This is nothing more than a disquotational theory of truth.

Is it possible that Wallace and Timpson’s proposal only trivializes in the trivialcase – where spacetime consists of a single point? Perhaps their proposal is onlymeant to give an interesting picture in the case where we associate a differentalgebra of observables A Oð Þ to each region O of spacetime. In that case, theirrecipe would yield a much richer structure, something like a co-presheaf of states(see Swanson 2018). But I do not see any reason to think that this additionalmathematical structure can undo the conflation of states and properties that alreadyoccurs at the level of individual algebras.

Finally, even if you can get past these other worries, there is a worry that theWallace-Timpson proposal shows too much. Indeed, there is a case to be made thatany reasonable generalized probability theory can be formulated in the frameworkof C∗-algebras. In that case, it would seem that the Wallace-Timpson proposalyields a realistic physical ontology for any reasonable generalized probabilitytheory. In other words, it is realism on the cheap.

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8.9 The Wave Function as Symbol

We began our discussion with the dilemma: Either the quantum state has onto-logical status, or it does not. We saw that this dilemma cannot be taken seriously,because a state is not a candidate for existence or nonexistence in the physicalsense. Thus the original ontological dilemma was transformed into a representa-tional one: Either the quantum state represents reality, or it does not. But then wediscovered that “represents” can be understood in many different senses – and inthe most extremely realistic sense of “represents,” no sane person would say thatthe quantum state represents the world. Thus, the disagreement between realistsand antirealists – where it is not simply a matter of emotional associations withwords – boils down to different stories about how to use the quantum state torepresent reality.

It is ironic then, that early interpreters of quantum theory – such as Bohr andCarnap – are often assumed to be operationalists about the quantum state. Thatcould not be further from the truth. Both Bohr and Carnap explicitly say that thewave function is not merely a calculational device. Presumably, somebody ran aword-search on Bohr and Carnap’s writings, and having found no hits for “ψrepresents reality directly” or “ψ has ontological status,” they concluded that theseguys must have been antirealists.

There is another possibility that we should take seriously.What if Bohr and Carnapintentionally exercise caution with words like ontological status and direct represen-tation, because those words might lead to a misunderstanding of how quantum theoryworks? Perhaps Bohr and Carnap were groping their way, if ever so haltingly, towarda more articulate account of how the wave function represents reality.

8.9.1 Bohr

Analytic philosophers have been quick to categorize Bohr as an operationalistabout the wave function, citing statements like this one:

the symbolic aspect of Schrödinger’s wave functions appears immediately from the use ofa multidimensional coordinate space, essential for their representation in the case of atomicsystems with several electrons.

(Bohr 1932: 370)

Faye, for example, seems to think that Bohr’s use of “symbolic” is code for“should not be taken literally.”

Thus [for Bohr], the state vector is symbolic. Here “symbolic” means that the state vector’srepresentational function should not be taken literally but be considered a tool for thecalculation of probabilities of observables.

(Faye 2014: on line)

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Faye’s confusion here is understandable. We analytic philosophers of science tendto associate “symbolic” with “nonreferential” or “uninterpreted.” In particular,with regard to a sentence X in a formal calculus, to say that “X is symbolic” meansprecisely that X in uninterpreted, and so lacks a truth value. In other words, whenwe hear “symbol,” we immediately think “does not purport to describe reality.”

However, the considerations of previous sections show that this usage of“symbolic” does not make much sense when X is a mathematical object, such asa wave function. Nor would it make much sense to attribute this usage of“symbolic” to Bohr, who did not use words in exactly the way that analyticphilosophers of science have come to use them. When Bohr uses “symbolic,”I assume that his meaning draws on a his peculiar educational background, whichwas heavy on continental figures such as Kant, Goethe, Hegel, and Helmholtz.I assume that his meaning was also shaped by his interactions with continental-type philosophers such as Ernst Cassirer, and mathematicians such as his brotherHarald. Thus, when Bohr says something is “symbolic,” we should not immedi-ately conclude that he means it in the sense of the uninterpreted predicate calculus.

Indeed, one of Bohr’s students, Christian Møller, asked him explicitly what hemeant by calling the wave function “symbolic.” In a 1928 letter, Bohr replies in somany (!) words:

Regarding the question discussed in your letter about what was meant, when I in my articlein Naturwissenschaften, emphasized so strongly the quantum-theoretical method’ssymbolic character, I am naturally in complete agreement with you that everydescription of natural phenomena must be based on symbols. I merely sought toemphasize the fact, that this circumstance – that in quantum theory, we typically use thesame symbols we use in the classical theory – doesn’t justify our ignoring the largedifference between these theories, and in particular necessitates the greatest caution inthe use of the intuitive concepts [anskuelsformer] to which the classical symbols areconnected. Naturally, one doesn’t easily run this danger with the matrix formulation,where the calculation rules, which diverge so greatly from the previously standardalgebraic ones, hold quantum theory’s special nature before our eyes. Furthermore, touse the word “symbolic” for non-commutative algebra is a way of speaking that goes backlong before quantum theory, and which has entered into standard mathematicalterminology. When one thinks about the wave theory, it is precisely its “visualizability”[anskuelighed] which is simultaneously its strength and its snare, and here by emphasizingthe approach’s [behandlingens] symbolic character, I was trying to bring to mind thedifferences – required by the quantum postulate – from classical theories, which are hardlyever sufficiently heeded.

(Bohr 1928, original in Danish)

As is typical with reading Bohr, one does not feel that the situation has been greatlyclarified. However, one thing is clear: Bohr does not intend to single out thequantum state for operational treatment. If Bohr is an antirealist about the quantum

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state, then he is an antirealist about all of mathematical physics. For Bohr, allmathematical representation is “symbolic,” whether observable or unobservableaspects of reality are being represented. Among the symbolic representations ofphysics, he would include the Fab of Maxwell’s equations, the gab of generalrelativity, as well as functions representing the trajectories of material bodiesthrough spacetime. Bohr’s point might be summed up simply by saying thatmathematical objects are not sentences, and so they cannot “be read literally.”

To understand Bohr’s use of “symbolic,” it might also help to look at aphilosopher whose career ran in parallel with his. In fact, it is well known thatBohr interacted extensively with Ernst Cassirer when the latter was composing hisbook Determinismus und Indeterminismus in der Modernen Physik, first publishedin 1937. Whether there is a more substantial overlap in their usage of “symbolic”will have to await more detailed historical investigations.

Nonetheless, it is clear that there are many common themes in the views of Bohrand Cassirer (see e.g., Pringe 2014). One such common theme is giving carefulthought to the way that mathematical objects can be used to represent the physicalworld. In putting forward his views on this issue, Cassirer is clear that “symbolic”should not be opposed to “representational.” The interesting question is notwhether something is representational, but rather how it represents. In particular,Cassirer believes that the development of mathematics and physics in the ninteenthcentury provides a particularly clear demonstration of the need to expand thenotion of representation beyond a simplistic “similarity of content” account.

Mathematicians and physicists were first to gain a clear awareness of this symbolic characterof their basic implements. . . . In place of the vague demand for a similarity of contentbetween image and thing, we now find expressed a highly complex logical relation, a generalintellectual condition, which the basic concepts of physical knowledge must satisfy.

(Cassirer 1955: 75)

For the former, more narrow, use of symbols, Cassirer uses the word Darstellings-funktion. For the latter, more general, use of symbols, Cassirer uses the wordBedeutungsfunktion. Thus, to relate back to our earlier analysis of “Y representsX,” we might think that Darstellungsfunktion picks out a kind of representationalrelation that licenses many inferences about X from Y , especially inferences havingto do with spatiotemporal properties. The paradigm case, of course, of suchrepresentations are the directly geometric. In contrast, Bedeutungsfunktion picksout a more general kind of representation relation that does not imply geometricsimilarity between X and Y .

Bohr does not avail himself of Cassirer’s classification of symbolic forms.However, he often does speak of things being “unvisualizable” (uanskuelig) –

opening a door to the deep dark recesses of the Kantian tradition. Bohr’s notion of

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representing something to visual intuition doubtless overlaps in important wayswith Cassirer’s notion of Darstellungsfunktion. And if there is any coherence inCassirer’s idea of moving toward Bedeutungsfunktion, then Bohr may be blazingthe same trail. In particular, when Bohr says that subatomic processes cannot bevisualized, he should not be taken as saying that quantum theory is nonrepresenta-tional. Instead, Bohr might be groping his way toward a more nuanced account ofhow mathematics can be used to represent physical reality.

8.9.2 Carnap

We began the chapter with a story about how the early interpreters of quantumtheory were operationalists. That story is often neatly combined with another storythat post-Quinean analytic philosophers love to tell: the story about how silly andstupid the logical positivists were. According to this story, the logical positivistsviewed scientific theories as “mere calculi” for deriving predictions. Thus, thestory concludes, it is no surprise that Bohr et al. were operationalists about thequantum state, given that operationalism had so thoroughly infected the prevailingview of scientific theories.

If you have ever read a serious historical account of the origins of quantumtheory, you know that the first story is mostly propaganda. None of the pioneers ofquantum theory – Bohr, Heisenberg, Dirac, etc. – was a crass operationalist. And ifyou have ever read a serious historical account of twentieth-century philosophy,you also know that the second story is largely Quinean propaganda. In fact, Carnaphimself was a vocal critic of operationalism – long before he felt the pressure ofQuine’s critiques of the positivist program.

Some, especially philosophers, go so far as even to contend that these modern theories,since they are not intuitively understandable, are not at all theories about nature but “mereformalistic constructions”, “mere calculi”. But this is a fundamental misunderstanding ofthe function of a physical theory.

(Carnap 1939: 210)

Notice how Carnap feels the same pressure that Bohr and Cassirer feel – thepressure that the new theories of physics are not “intuitively understandable.”Moreover, like Bohr and Cassirer, he refuses to take the breakdown of intuitiveunderstandability (or anskuelighed, or Darstellbarkeit) to demand a retreat tooperationalism. Instead, Carnap – like Bohr and Cassirer – asks us to think harderabout how our theories purport to represent physical reality.

Like Bohr, Carnap insists that the representational status of the quantum wavefunction is not all that different from the situation of the symbols of classicalmathematical physics.

To Be a Realist about Quantum Theory 157

If we demand from a modern physicist an answer to the question what he means by thesymbol “ψ” of his calculus, and are astonished that he cannot give an answer, we ought torealize that the situation was already the same in classical physics. There the physicistcould not tell us what he meant by the symbol “E” in Maxwell’s equations. . . .Thus thephysicist, although he cannot give us a translation into everyday language, understands thesymbol “ψ” and the laws of quantum mechanics. He possesses the kind of understandingwhich alone is essential in the field of knowledge and science.

(Carnap 1939: 210–211)

Interestingly, the words of Carnap here are echoed – quite unintentionally, I amsure – by Wallace and Timpson.

. . .it’s not as if we really have an intuitive grasp of what an electric or magnetic field is,other than indirectly and by means of instrumental considerations. . . .Thus, it seems thatwe gain a basic understanding of the electromagnetic field by seeing it as a property ofspatial regions, and our further understanding must be mediated by reflecting on its role inthe theory. . . .beyond that there doesn’t seem to be much further to be grasped.

(Wallace and Timpson 2010: 700)

We might just add that the concept of spatial regions does not provide us with atruly Archimedian reference point – for these regions themselves are understood ina mediated way, via their description in physical theory.

At this point, it should be thoroughly unclear how the views of Bohr, Cassirer,and Carnap differ from some of the more moderate and reasonable quantum staterealists. To one such view we now turn.

8.10 The Nomological View

According to the cutting-edge survey of Chen (2018), there are three versions ofwave function realism – the two ψ-field views and a “nomological view,” wherethe wave function represents a law of nature (Goldstein and Zanghì 2013, Esfeld2014, Miller 2014, Callender 2015, Esfeld and Deckert 2017). Thus, if we were toregiment the nomological view, we might say that the wave function plays thetheoretical role of a proposition, or perhaps of a rule for generating propositions.The theoretical role of propositions is, of course, quite different than the theoreticalrole of names or even variables, both of which are used to denote existing things.Thus, only by stretching the word “ontological” beyond the breaking point couldwe say that the nomological view is ontological. No matter what view of laws wetake, a law is not a thing and is not in the domain of quantification of a physicaltheory. Thus, according to the nomological view, the wave function is not a beable.

Why then should the nomological view of the wave function be called a realistview if it does not treat the wave function as corresponding to an existing thing?Presumably, nomologists would say that what makes their view realist is that the

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propositions encoded in ψ are objectively true, i.e., they correspond to reality. Butwhat then are these propositions that are encoded in ψ? Of course, Bohmians havean answer ready at hand: ψ encodes propositions about the trajectories of particles.

Notice that the specific Bohmian answer is not implicit in the very idea that ψencodes true propositions. Even a rank operationalist will say that ψ encodes truepropositions – about the probabilities of measurement outcomes. Only we mightquestion whether these propositions are “objectively true,” because probabilities ofmeasurement outcomes are indexed by measurements, and the latter has yet to beobjectively defined.

So what makes the nomological view realist? Is it simply that ψ encodesobjectively true propositions? Or is it that ψ encodes true propositions aboutparticle trajectories? I would be loath to accept the second answer, because itwould make realism hostage to one idiosyncratic ontological picture, viz., aparticle ontology. Surely one can be a realist and have some sort of gunkyontology, or a field ontology. So, it seems that realist-making feature of thenomological view is merely its commitment to the idea that ψ represents object-ively real features of the world. But now, if that is enough to make a view realist,then Healey’s view is also a realist view. For Healey says that each physicalsituation is correctly represented by at most one quantum state. Healey and thenomologists agree that ψ represents objectively real features of the world.

Nor can we say that the nomological view is more realist than Healey’s becauseit takes ψ to be a direct representation of reality. The representation relationposited by the nomological view is every bit as indirect and nuanced as thatposited by Healey (or by Bohr for that matter). Indeed, the nomological viewincludes an intricate translation scheme from mathematical properties of ψ tovarious meaningful physical statements, some of which are about occurent statesof affairs, and some of which are about how things will change as time progresses.Thus, in terms of how ψ represents, the nomological view is closer to the views ofHealey, Bohr, and Carnap than it is to ψ-field views. The nomologists may behorrified to hear this, for they take great pride in being realists. But recall thatBohm often emphasized that his point of view was not so radically different fromBohr’s. He even offered his point of view as a clarification of Bohr’s. Perhaps thenthe nomological view could be thought of as an attempt to clearly articulate someof the things that Bohr was trying to say about the wave function.

8.11 Conclusion

The primary aim of this chapter was to investigate the meaning of realism aboutquantum theory, and in particular, realism about the quantum state. We found that,for the most part, these phrases are empty of substantive content. They are emotive

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catch phrases that are meant to muster the troops – and perhaps to sell books. Butplease don’t get me wrong. I am not saying that there are no substantive questionsabout how to interpret the quantum state. First of all, dissolving the antirealism/realism distinction does not solve the measurement problem. There is still thethorny issue of why it appears to us that measurements have outcomes. Second,there are genuine disagreements about how to use quantum states – even if thesedisagreements do not correlate directly with a distinction between “real” and“not real.”

First, there is a genuine question of how to think of the relation of quantumstates to physical situations. (For simplicity, I will suppose that a physical situationis picked out by an ordinary language description, for example, by the sorts ofinstructions that one might give to an engineer or to a postdoc in the lab.) At oneextreme, we have objectivists who think that each such situation corresponds to aunique, correct quantum state. At the opposite extreme, we have the QuantumBayesians who propose no correctness standards whatsoever between physicalsituations and quantum states. For these QBists, a quantum state just is a person’spoint of view – it is neither correct nor incorrect, appropriate or inappropriate.Between these two extremes, we have views like Rovelli’s, where each physicalsituation can be described equally by at least two quantum states, depending onone’s choice of a direction of time. Some people also think that Bohr was anonobjectivist about quantum states (see Zinkernagel 2016). However, I find thatview hard to square with Bohr’s repeated pronouncements of the “objectivity ofthe quantum-mechanical description.”

I propose that we stop talking about the ill-defined notion of quantum staterealism, and that we instead start talking about these sorts of questions, e.g.,whether quantum theory comes with objective standards for the ascription of statesto physical situations. First of all, what role do physical situations, described inordinary language, play in this debate? Could we replace “physical situation” withsomething more neutral and description-free, such as “object” or “system”? Theproblem with that suggestion is that the bare notion of an object or a system cannotgive us any sort of standard for comparison. For example, we might say:“According to Healey, for each object X, there is a unique correct quantum state.”But how does Healey individuate objects? If he has different standards for indi-viduating objects than Rovelli has, then their apparently diverging views might infact agree. Thus, the question of appropriate use of quantum states requires atarget, or standard of reference, on which all parties antecedently agree. The notionof a “physical situation” is supposed to offer a plausible standard of reference.

I have already suggested a shift from the ontological question: Do states exist?to the representational question: How do states represent? Now I am suggestingthat this representational question be given a normative reading: What are the rules

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governing the use of quantum states? That, I believe, is the real issue at stake,although it is masked by emotionally charged words such as “ontological status.”

There is a second question, closely related to the first one. Should we applyunitary dynamics without exception? Some people say yes (e.g., Bohm, Everett,Wallace), and others say no (e.g., Ghirardi-Rimini-Weber [GRW], Rovelli,Healey). But even this disagreement is not as clear-cut as it may seem. Even thosewho believe in the universal validity of unitary dynamics allow themselves to use“effective states.” The “true state,” they say, follows unitary dynamics. But forcalculational purposes, there can be great advantages to using the effective state.

I am no verificationist, and so I do not propose that we collapse the distinctionbetween real and effective states. Nonetheless, I am interested here in the rules forusing states, i.e., for deciding whether one ought to use the state that results fromunitary evolution or whether one is permitted to use the state that results fromapplication of the projection postulate. Or, to put it in explicitly representationallanguage: The question is whether the state that results from unitary evolution isthe only one that is “apt” to one’s situation or whether the state resulting from theprojection postulate might also be “apt” to one’s situation. Interestingly, all partiesseem to agree that the state resulting from the projection postulate is “apt” in somesense. Even the most fervent anticollapsers will tell you that the projected state iscorrect for all practical purposes. Then they will remind you that it is not the “real”state. But I would then ask: not the real state of what? We are back again to thequestion of how to identify the target X of our representation via a quantum state.

Acknowledgments

I thank Eddy Chen for guidance about wave function realism, to Catherina Juel forhelp translating Bohr’s letter to Møller, and to Tom Ryckman for sending apreprint of (Ryckman 2017), which got me interested in Cassirer’s view.

References

Albert, D. Z. (1996). “Elementary quantum metaphysics,” pp. 277–284 in J. T. Cushing,A. Fine, and S. Goldstein (eds.), Boston Studies in the Philosophy of Science:Bohmian Mechanics and Quantum Theory: An Appraisal, Vol. 184. Dordrecht:Springer.

Belot, G. (2012). “Quantum states for primitive ontologists,” European Journal forPhilosophy of Science, 2: 67–83.

Bohr, N. (1928). Letter to Christian Møller, 14 June. Archive for the History of QuantumPhysics, 1898–1950, Library of the American Philosophical Society.

Bohr, N. (1932). “Faraday lecture: Chemistry and the quantum theory of atomic consti-tution,” Journal of the Chemical Society, 349–384.

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Bricker, P. (1996). “Properties,” pp. 469–473 in The Encyclopedia of Philosophy, Supple-ment. New York: Simon & Schuster.

Callender, C. (2015). “One world, one beable,” Synthese, 192: 3153–3177.Carnap, R. (1939). “Foundations of logic and mathematics,” pp. 139–213 in O. Neurath,

R. Carnap, and C. Morris (eds.), International Encyclopedia of Unified Science.Chicago: University of Chicago Press.

Carroll, S. (2017). The Big Picture: On the Origins of Life, Meaning, and the UniverseItself. New York: Dutton.

Cassirer, E. (1955). The Philosophy of Symbolic Forms: Language. Vol. 1. New Haven:Yale University Press.

Chakravartty, A. (2017). “Scientific realism,” in E. N. Zalta (ed.), The Stanford Encyclo-pedia of Philosophy (Summer 2017 Edition). https://plato.stanford.edu/archives/sum2017/entries/scientific-realism/

Chen, E. (2017). “Our fundamental physical space: An essay on the metaphysics of thewave function,” Journal of Philosophy, 114: 333–365.

Chen, E. (2018). “Realism about the wave function,” Philosophy Compass, forthcoming.Esfeld, M. (2014). “Quantum Humeanism, or: Physicalism without properties,” The

Philosophical Quarterly, 64: 453–470.Esfeld, M. and Deckert, D.-A. (2017). A Minimalist Ontology of the Natural World. New

York: Routledge.Faye, J. (2014). “Copenhagen interpretation of quantum mechanics,” in E. N. Zalta (ed.),

The Stanford Encyclopedia of Philosophy (Fall 2014 Edition). https://plato.stanford.edu/archives/fall2014/entries/qm-copenhagen/

Forrest, P. (1988). Quantum Metaphysics. Oxford: Blackwell.Frigg, R. and Nguyen, J. (2016). “Scientific representation,” in E. N. Zalta (ed.), The

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Goldstein, S. and Zanghì, N. (2013). “Reality and the role of the wave function in quantumtheory,” pp. 91–109 in A. Ney and D. Z. Albert (eds.), The Wave Function: Essays onthe Metaphysics of Quantum Mechanics. Oxford: Oxford University Press.

Halvorson, H. (2012). “What scientific theories could not be,” Philosophy of Science, 79:183–206.

Healey, R. (2015). “How pragmatism reconciles quantum mechanics with relativity,” 3amInterview. www.3ammagazine.com/3am/how-pragmatism-reconciles-quantum-mechanics-with-relativity-etc/

Healey, R. (2016). “Quantum-Bayesian and pragmatist views of quantum theory,” in E. N.Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), https://plato.stanford.edu/archives/spr2017/entries/quantum-bayesian/

Healey, R. (2017). The Quantum Revolution in Philosophy. Oxford: Oxford UniversityPress.

Hirsch, R. J. (2017). “From Representation to Reality: Essays on the Nature of ScientificExegesis,” PhD thesis, Princeton University. https://catalog.princeton.edu/catalog/10690088

Howard, D. (2004). “Who invented the “Copenhagen interpretation”? A study in myth-ology,” Philosophy of Science, 71: 669–682.

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Miller, E. (2014). “Quantum entanglement, Bohmian mechanics, and Humean superve-nience,” Australasian Journal of Philosophy, 92: 567–583.

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Ney, A. (2012). “The status of our ordinary three dimensions in a quantum universe,”Noûs, 46: 525–560.

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Pringe, H. (2014). “Cassirer and Bohr on intuitive and symbolic knowledge in quantumphysics,” Theoria. Revista de Teoría, Historia y Fundamentos de la Ciencia, 29:417–429.

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A confluence of opposites,” Journal for the History of Analytic Philosophy, 6:194–224.

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Oxford Handbook of the Philosophy of Physics. Oxford: Oxford University Press.Wallace, D. (2018a). “Against wave function realism,” in B. Weslake and S. Dasgupta

(eds.), Current Controversies in the Philosophy of Science. London: Routledge.Wallace, D. (2018b). “Quantum theory as a framework, and its implications for the

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9

Locality and Wave Function Realism

alyssa ney

9.1 Introduction

Wave function realism is a framework for interpreting quantum theories. Applied tononrelativistic versions of quantum mechanics, wave function realism yields ametaphysics according to which the central, fundamental object is the quantumwave function, understood as a field on a high-dimensional space with the structureof a classical configuration space, perhaps supplemented with additional degrees offreedom to capture spin and other variables. Particles and other low-dimensionalobjects are understood by the wave function realist to be ontically derivative objects,constituted ultimately out of wave function stuff. For more sophisticated relativisticquantum theories and quantum field theories, the framework recommends a suitablerelativistic extension of this metaphysics: a field in whatever high-dimensionalspace is capable of capturing the full range of pure quantum states.

The case for such high-dimensional field interpretations varies from one frame-work proponent to another, but a recurrent theme is wave function realism’s abilityto provide ontologies for quantum theories that have some intuitively nice meta-physical features. For example, one may note the fact that quantum entanglementthreatens to force a fundamentally nonseparable metaphysics on the interpreter or,what is to some (Howard 1985, and, he argues, Einstein) worse, a fundamentallynonlocal metaphysics. However, these defects may be seen to drop away in thehigher-dimensional interpretations preferred by the wave function realist. For her,what initially appear to be distinct entities possessing primitive relations andcommunicating instantaneously across distant regions of space are revealed to bemanifestations of a single object, fundamentally possessing only intrinsic featuresand acting locally on a high-dimensional space. This motivation for wave functionrealism is, as I shall explain later, more compelling than that suggested by others,who have argued that one should adopt such a framework simply because it is thesort of thing that is most naturally read off the physics.

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Although some have challenged the wave function realist’s claim to provide aseparable metaphysics for quantum theories (e.g., Myrvold 2015, Lewis 2016),I would say it is the claim that wave function realism provides a local metaphysicsthat is more difficult and less straightforward (I rebut the concerns about separabil-ity in Ney 2019b). And so, this is what I wish to examine in the present chapter.I will introduce and distinguish several senses in which a metaphysics for physicsmay be local, starting with two notions made use of by Bell (1964, 1976). Fromthere, we may evaluate in which sense or senses, if any, the wave function realist’smetaphysics are local; and what, after all, is the virtue of having interpretations forquantum theories that are local in that (those) sense(s). I will focus on thenonrelativistic case. Ney (2019b) examines the extension to relativistic theories.

9.2 Wave Function Realism and its Competitors

It is worth noting at the outset that the interpretational question to which wavefunction realism is intended to provide an answer is to a large extent orthogonal toother interpretational questions, for example, that of what is the most promisingapproach to addressing the measurement problem for quantum theories. Themeasurement problem is the problem of how systems that are indeterminate withrespect to one or another variable may evolve into states of what appear to bedeterminate values of such variables upon measurement. How can indeterminacyappear to evolve into determinacy, given a dynamics that seems to make suchevolution impossible?

There are many approaches available to solving the measurement problem,ranging from collapse theories, to the postulation of hidden variables, to the appealto relative states (or many worlds). In this article I focus on what are termed psi-ontic or objectivist approaches to solving the measurement problem, because myinterest is in what might be the mind-independent metaphysics of quantum theor-ies. This is not to reject the existence or interest of psi-epistemic or subjectivistapproaches, which deny that the role of the quantum state is to represent somemind-independent reality. However, the measurement problem raises the question:Should the quantum dynamics be supplemented, modified, or left alone? Bycontrast, wave function realism and its competitor frameworks are addressed atthe question of what such dynamics describe: What is the ontology of a theory witha dynamics like that? In principle, each rival metaphysical framework may beapplied to any of the dynamical proposals aimed at solving the measurementproblem, although some combinations are more natural than others. I will notaddress all of the possibilities here, though I will provide brief overviews of wavefunction realism and its competitors. (Wave function realism is more natural as aninterpretation of collapse theories like that of Ghirardi, Rimini, and Weber [GRW]

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and many worlds. The competitor primitive ontology approach is more naturallycombined with hidden variable theories like Bohmian mechanics).

As mentioned, wave function realism is a framework for the interpretation ofquantum theories in which the wave function is the central ontological item and isinterpreted as a field on a high-dimensional space that (for the nonrelativistic case)is assumed to have the structure of a classical configuration space. For hiddenvariable theories, this wave function is supplemented with additional ontology,e.g., for Bohmian mechanics, a single particle evolving in a way determined by themotion of the wave function. Wave function realists thus take wave functionrepresentations literally and straightforwardly. As Albert has written

The sorts of physical objects that wave functions are, on this way of thinking, are (plainly)fields – which is to say that they are the sorts of objects whose states one specifies byspecifying the values of some set of numbers at every point in the space where they live,the sorts of objects whose states one specifies (in this case) by specifying the values of twonumbers (one of which is usually referred to as an amplitude, and the other as a phase) atevery point in the universe’s so-called configuration space.

(Albert 1996: 278)

As has been mentioned and will be discussed in more detail later, this simpleinterpretation of the wave function has the advantage of providing a metaphysicsfor quantum theories that is fundamentally separable and local. However, otherinterpreters challenge this reading.

The primitive ontology approach of Dürr, Goldstein, Zanghì, Allori, and Tumulka(Dürr, Goldstein, and Zanghì 1992, Allori et al. 2008, Goldstein and Zanghì 2013)insists that the ontology of quantum theories consists primarily of entities in ordinaryspace or spacetime, for example, particles for Bohmian mechanics and matterdensity fields for collapse theories or many-worlds approaches. On the primitiveontology approach, the wave function is interpreted as real, but not an element of aquantum theory’s primitive ontology: It is not what any physical theory is primar-ily about, not what constitutes the matter in the theory. (See Ney and Phillips 2013for a detailed examination and critique of the notion of a primitive ontology).Instead, the wave function plays some other role, to guide the behavior of thematter, and so it is something more like a law, broadly speaking.

Those adopting the primitive ontology framework (and some of the otherapproaches I describe later in the chapter) complain about the use of the name‘wave function realism’ to apply solely to views according to which the wavefunction is a physical field on a higher-dimensional space, claiming they too arerealists about the wave function, taking it to be a real, mind-independent element ofquantum ontology. In a sense, this complaint is fair, but by now the terminologyhas become so entrenched, I will continue to use it. And anyway, in defense of theterminology, one thing that distinguishes the status of the wave function on the

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wave function realist view from how it is viewed in primitive ontology or other psi-ontic approaches to interpretation is that, for the wave function realist, the wavefunction is real in the classical sense of being res- or thing-like; it is a substance,re-al. This contrasts with its status on other interpretational approaches, in which itoccupies one or another distinct ontological category; rather than being res, itviewed as law, property, or a pattern of relations.

To many, the primitive ontology framework has appeal over wave functionrealism in providing metaphysics for quantum theories that are intuitive in certainrespects. According to most applications of the approach, the fundamental entitiesof the theory inhabit our familiar space or spacetime, and the macroscopic objectswe observe may be built out of these basic constituents (particles or matter fields)in straightforward ways. (For a critique, see Ney and Phillips 2013.) An exceptionis the flash ontology offered as a primitive ontology interpretation of some collapseapproaches; this ontology is surprisingly sparse and unfamiliar. Moreover, theseapproaches hold out the promise of a separable metaphysics. The features ofcomposite objects are determined by the features of their smaller constituentparticles or field values at individual spacetime points. For example, for Bohmianmechanics, one may argue that there are no facts about joint states of the particlesthat fail to be determined by the states of individual particles. Facts about quantumentanglement do fail to be determined by facts about the states of particles takenindividually or together, but on a primitive ontology approach to Bohmian mech-anics, entanglement is a feature of the wave function, not the particles. And so, onecould say that the matter ontology of Bohmian mechanics is perfectly separable.The same may be said for the matter density ontology that the primitive ontologyview attaches to collapse theories.

Nonetheless, such metaphysics are not local. As Bell showed

In a theory in which parameters are added to quantum mechanics to determine the results ofindividual measurements, without changing the statistical predictions, there must be amechanism whereby the setting of one measuring device can influence the reading ofanother instrument, however remote. Moreover, the signal involved must propagateinstantaneously, so that such a theory could not be Lorentz invariant.

(1964/1987: 199)

In situations characterized by the presence of quantum entanglement, measure-ments made on one entity at one location can have immediate influence on thatentity’s entangled partner at a spatially distant region. This is so in situationsevolving according to both collapse and noncollapse dynamics. Even for many-worlds versions of the primitive ontology approach, although locality is oftenclaimed, as Lewis (2016) notes, this locality is only available in the higher-dimensional space that the wave function inhabits. Once the wave function is

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relegated to some other nonprimitive status, as something unfield-like andunmatter-like, this advantage is lost.

A distinct class of approaches to the interpretation of quantum theories seeks toobtain a local metaphysics, but at the cost of rejecting separability. This ischaracteristic of the approaches of Howard and Teller from the eighties. Howard(1985) argues that what seem to be entangled pairs of objects in distant regions ofspace should not be viewed after all as numerically distinct entities. In the extremecase in which every putative entity is entangled with any other, the view becomes aversion of monism. There is only one thing. According to Howard, there is noinstantaneous action between spatially separated entities because there are notmultiple entities after all. He argues that if faced with the choice of preservingseparability or locality, Einstein too would have chosen to reject separability inorder to maintain locality (Howard 1985: 197).

Teller (1986) offers a distinct nonseparable approach, allowing that entangledpairs are fundamentally distinct entities while claiming that entanglement forces usto admit the existence of irreducible relations between pairs that do not reduce toany intrinsic features of the individuals constituting those pairs. He argues (1989)that this allows one to avoid nonlocality, because the view rejects the general claimthat correlations between objects must ever be explained in terms of more funda-mental features of these objects, such as the causal relations between them.

A more recent development of the idea that quantum entanglement should beinterpreted as characterizing a world with fundamental relations not reducible tofeatures of their relata is ontic structural realism, which has been advocated in avariety of forms (Ladyman 1998, Esfeld 2004, Ladyman and Ross 2007, French2014). Such a framework for interpretation provides a metaphysics that is mani-festly nonseparable. However ontic structural realists typically resist the claim ofTeller that the existence of irreducible relations allows one to avoid the conse-quence of nonlocality. (Thanks to Michael Esfeld regarding this point.)

Similarly, another interpretative framework, the spacetime state realism advo-cated by Wallace and Timpson (2010) and Myrvold (2015), aims neither atproviding metaphysics for quantum theories that are either separable or local. Onthis view, even in nonrelativistic quantum mechanics, the wave function charac-terizes highly abstract features of spacetime regions, where the features of com-posite regions do not generally reduce to features of their constituents. Whathappens at one region can instantaneously affect what happens at another. In bothcases, that of the ontic structural realist and of the spacetime state realist, the viewis motivated not by the fact that it provides an intuitive metaphysics with variousattractive and natural features such as separability or locality, but instead by thefact, inter alia, that it stays truer to the way physics represents the world rather thanour expectations about what an ontology for physics should look like.

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9.3 Entanglement and Separability

So far, I have said that wave function realism is a framework aimed at providingmetaphysics for quantum theories that are separable, but it would be good to have astraightforward definition of separability with which to work. We may initiallyconsider the following:

A metaphysics is separable if and only if (i) it includes an ontology of objects or propertiesinstantiated at distinct regions, and (ii) when any objects or properties are instantiated atdistinct regions R1 and R2, all facts about the composite region R1[R2 are determined bythe facts about the objects and properties instantiated at its subregions.

The first clause is needed in order to rule out monistic metaphysics in which thereis only one thing or one spatial location. Separability implies that there are distinctobjects, or at the very least (if one prefers a field metaphysics), distinct field valuesinstantiated at distinct locations in space.

One drawback to this definition is that, as it stands, it requires a separablemetaphysics to be a Humean metaphysics. Since it speaks of all facts beingdetermined by facts about what occurs at individual spatial regions, it makesseparability require that all facts about dispositions, counterfactuals, causation,and laws be determined by what occurs at individual spatial regions. Loewer(1996) defends wave function realism explicitly for its ability to provide aninterpretation of quantum theories compatible with Humean supervenience. Onemight avoid this implication by modifying the second clause of the definition tostate only that the categorical, nondispositional, or non-nomic facts are determinedby the facts about individuals at subregions. We then have the following:

A metaphysics is separable if and only if (i) it includes an ontology of objects or propertiesinstantiated at distinct regions, and (ii) when any objects or properties are instantiated atdistinct regions R1 and R2, all categorical facts about the composite region R1[R2 aredetermined by the facts about the objects and properties instantiated at its subregions.

Some might object that the matters of concern when we discuss entanglementrelations are dispositional – this electron would be measured spin up were its z-spinto be measured. And so we really want a definition of separability that also requiresdispositional features to reduce to localized facts about individual spatial regions.But although these are some of the features of interest, entanglement can appear toforce on us as well, the violation of even this weaker account of separability. For,unless one adopts the Copenhagen-ish view that we can only talk sensibly aboutthe results of measurements or the features of systems when they are in eigenstates,it is the occurrent and categorical spin states of entangled pairs as well, not merelyhow they would behave upon measurement, that appears to be determined onlyjointly, not individually, by objects at distinct spatial regions.

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With this definition of separability in hand, we may see how the wave functionrealist may claim to provide interpretations of quantum theories that recognize thephenomenon of quantum entanglement without committing to fundamental non-separability. To illustrate, consider the EPRB state, in which a pair of atoms isentangled with respect to their z-spin. Suppose our atoms are created in thesinglet state

ψS ¼ 1=√2 jz-upiA jz-downiB � 1=√2 jz-downiA jz-upiBand are then sent in opposite directions toward two Stern-Gerlach magnets, whichwill bend them up or down in accordance with their z-spin toward two respectivemeasurement screens. Consider four locations between the magnets and screenwith the following labels:

R1: where atom A goes at time t should it get deflected upR2: where atom A goes at t should it get deflected downR3: where atom B goes at t should it get deflected upR4: where atom B goes at t should it get deflected down

At time t, the atoms will be an entangled state of position:ψx ¼ 1=√2 jR1iA jR4iB � 1=√2 jR2iA jR3iB

And so, there are facts at t about properties instantiated at the joint regions R1[R4and R2[R3 that are not determined by any facts local to their subregions, e.g.,there is an atom at R1 if and only if there is one at R4. There is an atom at R2 if andonly if there is one at R3. We thus have a violation of separability.

The wave function realist argues that what appears as nonseparability arisesbecause what we are seeing is a three-dimensional manifestation of a morefundamental and higher-dimensional metaphysics that is entirely separable. Theindividual atoms A and B are ultimately constituted out of a field – the quantumwave function. This field is spread not over our familiar three-dimensional space inwhich there are the four locations R1, R2, R3, and R4, but instead over a spacewith the structure of a classical configuration space. This space instead contains(for example) regions we may suggestively label R13, R14, R23, R24. The wavefunction has amplitude at these regions corresponding to the Born Rule probabil-ities for the quantum state. So, in the present case, given the quantum state ψx, thewave function will have nonzero amplitude only at the two locations R14 and R23and it will have an amplitude of ½ at each of these locations. Spin states willcorrespond to additional degrees of freedom. For a system initially appearing tohave N particles then, the dimensionality of the wave function’s space is posited tobe at least 3N.

The wave function realist’s proposed higher-dimensional metaphysics is thusentirely separable. All categorical features are determined by features of the wavefunction instantiated at individual regions in its space. (The wave function has

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phase values in addition to the amplitude values highlighted in this discussion.)That the wave function takes a particular shape across the joint region R14[R23,for example, is entirely determined by its features at the individual regions R14and R23.

So far, what we have been discussing is wave function realism as applied to anonrelativistic quantum mechanics without hidden variables. But as has been men-tioned, wave function realism is only a framework for interpretation: The metaphysicsit entails will vary in details depending on the details of the theory of which it is appliedto be an interpretation. If we are interested in the interpretation of a hidden variablestheory, then in addition to the wave function, there will also exist some entity in thehigh-dimensional space corresponding to these variables. In quantum field theories inwhich particle number fails to be conserved, the dimensionality of the space willinstead be determined by the number of basis states of the quantum field. In summary,the wave function realist’s metaphysics depends on the quantum theory one wants aninterpretation of. It will consist (at least) of the following:

• A background space with the structure of a classical configuration space

• Thewave function, a field on that space, characterizable in terms of an assignment ofamplitude and phase values and evolving according to the dynamics of the theory,e.g., the Schrödinger equation, perhaps supplemented with a collapse mechanism

This metaphysics is postulated to be separable. There are properties instantiated atdistinct spatial regions. But there are no facts about the wave function’s categoricalfeatures at composite regions that are not determined by facts local to theseregions’ respective subregions.

The question I now want to raise is whether this metaphysics is also local.

9.4 Concepts of Locality

The concepts of locality most frequently invoked when purported violations oflocality brought about by quantum entanglement are under discussion are thosehighlighted by Bell. Wiseman (2014) has argued that Bell really had two differentaccounts of what locality may come to in physics. The first is the notion of localityinvoked in his 1964 paper “On the Einstein-Podolsky-Rosen paradox”:

. . . the requirement of locality, or more precisely that the result of a measurement on onesystem be unaffected by operations on a distant system with which it has interacted inthe past.

(Bell 1964: 195)

This is equivalent to what Shimony called “parameter independence.” Applied tothe situation described in the previous section, it is the principle that the

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probabilities for the results of a measurement on atom B are independent of whatwe choose to do to what is at time t the spacelike separated atom A, including whatmeasurements we choose to perform on it.

In his 1976 paper, “The theory of local beables,” however, following Wiseman,we may see Bell invoking a distinct principle that he calls “local causality”:

Let A be localized in a space-time region 1. Let B be a second beable in a second region2 separated from 1 in a spacelike way. . . Now my intuitive notion of local causality is thatevents in 2 should not be ‘causes’ of events in 1, and vice versa.

(Bell 1976: 13)

By “beable,” Bell simply means entity, something that is real. This is a strongerprinciple than the early “locality” principle from 1964. It states not only that theprobabilities for the results of a measurement on one system are independent ofhow we may manipulate another system at a spacelike separation from it, but alsothat these probabilities are independent of the actual measurement results we findwhen we measure that other system. Wiseman argues that it is local causality thatBell took to be the primary locality principle of interest from at least 1976 on. Andit is what he argued must be violated if quantum theory is correct.

Arguably, neither of these interpretations ‘locality’ suffices to explicate thesense in which the metaphysics of the wave function realist is claimed to belocal. For the principles invoked by Bell both concern the existence of causalrelations in spacetime – the second one especially explicitly by invoking factsabout certain events exhibiting spacelike separation. Yet there is no spacetimeinterval defined on the space that the wave function is said to inhabit, nor is thespace of the wave function the space in which light propagates, and so there isno sensible notion of spacelike separation in the wave function realist’s funda-mental metaphysics to let us settle the issue of whether these senses of‘locality’ obtain. To explain the way in which wave function realism may beclaimed to involve a local metaphysics, we must move to a concept of localitythat makes sense in the context of the high-dimensional space of the wavefunction realist.

Sometimes, when Bell discusses his principle of local causality, he states it inbroader terms than we just saw:

What is held sacred is the principle of ‘local causality’ – or ‘no action at a distance’.(Bell 1981: 46)

This principle is generalizable so as to be of use by the wave function realist; andso, we may say:

A metaphysics is local if and only if it contains no instantaneous action across spatialdistances.

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Or perhaps:

A metaphysics is local if and only if it contains no instantaneous and unmediated actionacross spatial distances,

where ‘spatial’ refers to whatever is the spatial background of the metaphysics. Itmay be our familiar three-dimensional space or spacetime. But it may also be thehigh-dimensional spatial background of the wave function realist.

Can we be more precise? In his philosophy of physics textbook, Marc Langeprovides an account which seems aimed at capturing such a notion:

Spatiotemporal locality: For any event E, any finite temporal interval τ > 0, and any finitedistance δ > 0, there is a complete set of causes of E such that for each event C in this set,there is a location at which it occurs that is separated by a distance no greater than δ from alocation at which E occurs, and there is a moment at which C occurs at the former locationthat is separated by an interval no greater than τ from a moment at which E occurs at thelatter location.

(Lange 2002: 15)

But as Myrvold has noted (personal communication), without a metric, it is notclear how the wave function realist can make use of this account. Perhaps no moreprecision can be achieved for the sense of locality claimed by the wave functionrealist than with the intuitive definition of no instantaneous (unmediated) actionacross spatial distances.

To see how the wave function realist’s metaphysics generally satisfies this formof locality, one should be careful about distinguishing the situation for the inter-pretation of different quantum theories, including those with and without collapse.In the case of Everettian quantum theories without collapse, the wave functionsimply evolves unitarily in accordance with the Schrödinger equation or itsrelativistic variant. The wave function spreads out and may interfere with itselfas waves do. But at no point does an event at one point in the space influence anevent somewhere else.

For collapse theories like GRW, the wave function may evolve unitarily, butfrom time to time there is a spontaneous collapse. This involves the entire wavefunction undergoing a hit, which may be represented mathematically by themultiplication of the quantum state by a Gaussian function localized on a particularregion of the space. In this case, it is not correct to say that what happens in oneregion of the wave function’s space acts immediately to influence what happens inanother. Rather, in these models, collapses are not caused by anything about thestate of the wave function at the previous time, but occur spontaneously. One couldsay that there are facts about the wave function at the time prior to collapse thatdetermine how likely it is that the hit is localized at one point rather than another.The probability of the collapse being localized at one point rather than another is

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given by the Born Rule probabilities, which are associated with the amplitudesquared of the wave function at the different points in its space. But there is still noreason to say that the amplitude of the wave function at one distant region R causesa collapse to be localized at another region R’ of the space instantaneously. Even ifthe wave function later becomes more peaked around R’, the collapse is notsomething that takes place at R’, but is rather something that happens across theentire space. So there is not really a localized effect that may be influenced by somedistant cause. The evolution of the wave function through collapse may be jerkyand discontinuous, but it does not result in nonlocal action.

Finally, in the case we are interpreting noncollapse theories with hiddenvariables such as Bohmian mechanics, the wave function behaves identically tohow it does in the nonrelativistic Everettian model. However, in this case, therewill be some additional ontology, such as a particle (the so-called marvelous point)that moves around the wave function’s space in a way described by the theory’sguidance equation

dQk

dt¼ ℏ

mkIm

ψ * ∂kψψ *ψ

Q1; . . . ;QNð Þ

In this case, the behavior of this additional ontology, the particle, is determined bythe state of the wave function in the neighborhood of the place in the high-dimensional space it occupies, and so, there is no threat of nonlocal action. Despitethis fact, Bohmian mechanics is a quantum theory (or solution to the measurementproblem) that combines rather poorly with the interpretational framework of wavefunction realism. After all, the very motivation for adopting Bohmian mechanics(at least as presented in Dürr et al. 1992) depends on an argument that wavefunction realists should not accept, namely that quantum theories are not theoriesabout the behavior of the wave function but rather of something else, matter inthree-dimensional space or spacetime Regarding this point, see Section 5 of thepaper by Alyssa Ney and Ian Phillips (2013). But if one is worried about nonlocalaction, a wave function realist interpretation of Bohmian mechanics could beof help.

9.5 Fundamental and Derivative

So we have seen that wave function realism is a framework most naturallycombined with Everettian and collapse versions of quantum mechanics – theoriesin which the wave function ontology is not supplemented with additional variables.When applied as an interpretation of these theories, wave function realism mayyield a metaphysics that is local in its distinctive, (relatively) fundamental high-dimensional framework. But we may also ask what becomes of the nonlocality that

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appeared in the lower-dimensional spacetime framework. Is there still not an issueof nonlocality there according to the wave function realist?

The answer to this question turns first on whether the wave function realistaccepts the existence of three-dimensional space and spacetime, or of such low-dimensional facts. As has been mentioned, the wave function realist typically does,although he or she will also claim that the low-dimensional facts require noaddition to being beyond the high-dimensional framework he or she posits. Thelow-dimensional facts are derivative, i.e., metaphysically explained (grounded) bythe behavior of the wave function.

One response the wave function realist may give to the question of whetherthere is nonlocal action in the derivative low-dimensional framework is that no, inthat framework, there are correlations between spacelike separated events, but nogenuine causal interaction, because all such correlations have a deeper explanationin terms of the behavior of the wave function. The dynamical explanation for thesecorrelations thus undercuts any causal explanation that may be provided by theexistence of spatially distant events such as measurements on one half of anentangled pair.

I tried out such a line of reasoning in earlier work (Ney 2019a), as do Ismael andSchaffer (2018). However, I now find such a position unsatisfactory. The reason isthat if one wants to argue in this way that there is no immediate causation acrossspatial distances because such causal relations are undercut or screened off by thebehavior of the wave function, then one must similarly do so for all other causalrelations in the low-dimensional framework. For there will always be a wavefunction dynamical explanation available at the more fundamental level. So, unlesswe are to be causal nihilists about what happens in ordinary spacetime, thebehavior of the wave function does not undercut the reality of nonlocal action inspacetime. What the wave function realist can offer is a more fundamental explan-ation of in virtue of what that derivative nonlocal action obtains, one that may givea more satisfying picture of what makes things happen in our world than one thatcontains unexplained nonlocal action. But providing this explanation does notremove low-dimensional facts about nonlocal action.

9.6 Motivating a Local Metaphysics

Assuming the wave function realist can provide a local interpretation of at leastsome versions of quantum theories in at least some sense of ‘local’, we may ask,why should one care?

I will start with some empirical considerations and move toward some that aremore a priori. I have already noted that wave function realism is not successful atsecuring the conceptions of locality used by Bell. And yet, these are the senses of

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locality that some would say are mainly at issue when one worries about theincompatibility of relativity and quantum mechanics. But perhaps there is moreone can say, and the fact that wave function realism provides a metaphysics localin its own space may help alleviate some of the concerns arising about nonlocalityin spacetime.

In his recent book Quantum Ontology, Peter Lewis (2016) states the reason whynonlocal action is in tension with relativity in the following way. Suppose oneallows that there exists at least some instantaneous action at a distance. Then thereis some one time at which one event influences another at a spatial distance from it.For example, something happening right here, right now depends on the simultan-eous mass of a distant star. But according to special relativity, there are no absolutefacts about which spatially distant events are simultaneous with which others. So,in Lewis’s example, there is no fact about the mass of the star right now. Thus, thisaction at a distance is ill-defined according to relativity. Thus, it would seem,according to relativity, there cannot be action at a distance.

However, what looks puzzling, ill-defined, or brute from the perspective of anonfundamental metaphysics may be revealed as expected and explained in termsof a more fundamental metaphysics. To the extent that wave function realismsupports a derivative ontology, it will yield an account of which spacetimeconfigurations exist and are causally related in that derivative ontology. So, atleast Lewis’s concern about the conflict between special relativity and quantummechanics seems avoided if one adopts wave function realism. This is not to saythat other issues, which I would concede are more basic, are not avoided, namely aconflict with Lorentz covariance.

Another important feature of local theories is articulated by Einstein, who, in afamous paper from 1948, argued that local metaphysics seem to be required for thepossibility of physical theories:

For the relative independence of spatially distant things (A and B), this idea ischaracteristic: an external influence on A has no immediate effect on B; this is known asthe ‘principle of local action’, which is applied consistently only in field theory. Thecomplete suspension of this basic principle would make impossible the idea of theexistence of (quasi-)closed systems and, thereby, the establishment of empiricallytestable laws in the sense familiar to us.

(Einstein 1948: 321–322)

The point seems straightforward enough. If what is nearby and observable may beaffected by objects that are spatially distant, then without full knowledge of theoccupants of the total spacetime manifold, how are we to make predictions abouthow the objects we observe will behave? Locality appears required to allow us toformulate testable empirical theories.

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Now this point of Einstein’s is itself contestable. In conversation, Myrvold hasquestioned it, claiming that even in classical physics we are very comfortablewriting down and testing laws knowing full well that there are spatially distantobjects affecting the behavior of local objects. His example is the astrophysicist’sdescription of the motion of Jupiter’s moons. The Sun being 480 million milesaway, Einstein’s reasoning would lead one to believe that the physicist would needto reject its influence, modeling the behavior of the moon solely in terms of nearbyfactors. But this would produce wildly wrong results. This is a clear case in whichthe assumption of spatially distant influences is essential, not an obstacle to theformulation and testing of physical laws. Of course what Einstein rejects is that anexternal influence on A has an immediate effect on B, and one might respond toMyrvold by arguing that relativistic modeling will reject that the Sun’s influence isimmediate. But, astronomical phenomena are modeled quite well by Newtonianphysics, according to which gravitational influence is unmediated and instantan-eous. Einstein seems wrong that physics simply cannot be done when we assumethere are nonlocal influences and build these into our models.

Myrvold is right to object to hyperbole in Einstein’s defense of locality inphysics, but I do not believe this undermines a weaker defense of locality as anassumption guiding the formulation of tractable and testable physical theories. Forin the case of the Sun and Jupiter’s moons, the physics works because we areconsidering the influence of just a few large bodies at a distance away. Thingswould devolve quite quickly if the modeling of Jupiter’s moons needed to take intoaccount immediate and significant influences from many or all distant bodies. Soperhaps this is what Einstein is concerned about, thinking of widespread effectsfrom quantum entanglement that would massively complicate physics, perhapsleading to intractability. And so, for physics “in the sense familiar to us” to work,we discount immediate influence in general and for the most part where this isjustified. Note that with advances in computational modeling, physics can take intoaccount a much larger number of distant bodies successfully. However this doesnot undermine Einstein’s point, as in the age of big data, we are moving away fromphysics in the sense familiar to Einstein circa 1948.

So I propose we can make use of the weak point that we have good inductivereason to believe that physics of the kind that is already familiar to us, whichinvolves modeling systems based on the assumption of (mostly) local influence,is a way of developing inductively successful theories. But can this justificationfor a local metaphysics be used to generate any support for wave functionrealism? I am afraid it cannot – the point is, after all, that the testing of lawsdepends on our ability to manipulate and observe what is happening at aconfined region of space, isolating objects from outside influences. Of coursethe space in which human beings’ manipulations and observations take place is

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not the high-dimensional space of the wave function. Thus, it seems, Einstein’sdefense of locality justifies a local metaphysics in three-dimensional space orspacetime, the framework in which we interact with objects, but not a local wavefunction metaphysics.

Perhaps another case to be made for local interpretations of physical theoriesmay be found in the work of Allori. Allori (2013) defends another view she finds inEinstein, that “the whole of science is nothing more than a refinement of oureveryday thinking.” She elaborates:

The scientific image typically starts close to the manifest image, gradually departing from itif not successful to adequately reproduce the experimental findings. The scientific image isnot necessarily close to the manifest image, because with gradual departure after gradualdeparture we can get pretty far away. . . The point, though, is that the scientist will typicallytend to make minimal and not very radical changes to a previously accepted theoreticalframework.

(Allori 2013: 61)

One might then say that since our prescientific thinking and subsequent physicaltheories postulated local and separable metaphysics, our quantum theories should,if possible, do so as well.

To be clear, Allori is herself not making this point to argue for the localmetaphysics of wave function realism. She is using the point to argue for herpreferred primitive ontology view, because she believes that all previous (i.e.,nonquantum) physical theories also possess a primitive ontology. But one mighthope that her point extends to make a case for a local wave function metaphysicsas well.

Unfortunately, I do not think it does. Because wave function realism also rejectsas fundamental a three-dimensional spatial background, replacing it with anunfamiliar, high-dimensional background, it is not really so plausible to argue thatthis local metaphysics is closer to the manifest image and classical theories thanone that would jettison one or both of separability and locality, but retain the low-dimensional spatial background of our experience. If we agree with Allori thatminimal departures should, where possible, be preferred, the move to higherdimensions is very far from a minimal departure.

Finally, we may move to consider more purely a priori reasons in support of alocal metaphysics. Some of these were brought to bear in the eighteenth century asnatural philosophers struggled with Newton’s characterization of gravitationalforces as acting immediately across spatial distances. Newton himself sometimesclaimed that action at a distance is impossible, for example:

The cause of gravity is what I do not pretend to know and therefore would take more timeto consider of it. . . That gravity should be innate, inherent, and essential to matter, so that

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one body may act upon another at a distance through a vacuum, without the mediation ofanything else, by and through which their action and force may be conveyed from one toanother, is to me so great an absurdity that I believe no man who has in philosophicalmatters a competent faculty of thinking can ever fall into it.

(Newton, letter to Bentley, in Bentley 1838: 202)

To claim nonlocality is absurd is not thereby to offer an argument against it. Nor tomy knowledge did Newton ever offer a clear argument for why action at a distanceis absurd; however, we do find something in the work of Clarke in his correspond-ence with Leibniz:

That one body should attract another without any intermediate means, is not a miracle, buta contradiction: for ‘tis supposing something to act where it is not. But the means by whichtwo bodies attract each other, may be invisible and intangible, and of a different naturefrom mechanism; and yet, acting regularly and constantly, may well be called natural. . .

(Clarke, fourth letter to Leibniz, in Alexander 1956: 53)

Clarke, likeNewton, supposes that gravitymust act locally, even if themeans bywhichit does so may be invisible. And the reason why this must be so is for something to act,it must be located where it acts. Otherwise, it would not be it itself that is so acting, butsomething else, or nothing at all. There is something, I believe, that is sensible aboutthis point and it explains at least one reason why nonlocal action strikes us as deeplyunintuitive and worse, incoherent. And it is, finally, a consideration that may bebrought to be bear in support of wave function realism’s local metaphysics.

9.7 Intuitions

It is my view that the best case the wave function realist has for developing adistinctive local metaphysics comes from such conceptual considerations andintuitions. But one might question whether it is at all desirable to have aninterpretation of quantum theories that conforms to our intuitions. Ladyman andRoss (2007) criticize such interpretational projects, calling them “domesticationsof science.” My project is openly one of the domestication of a large part ofphysics. It is my attitude that quantum theories stand very much in need ofdomestication to the scientific community and greater public (this is not to denythat the project of domestication has already been carried out to a large extent bythe work of those providing clear solutions to the measurement problem).Following out interpretations that are compatible with our intuitions may be usefulfor a number of reasons. I will now mention three benefits that such an interpret-ation may bring. All are unabashedly pragmatic.

First, an interpretation of a physical theory, by providing one with a clearaccount of what the world is like according to the theory, benefits students and

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scientists in allowing them a clearer handle on the theory with which they areworking. Although it is not possible to understand our best scientific theorieswithout having a handle on the mathematics used to state it, a clear metaphysicsto supplement the mathematics can be instrumental in seeing more clearly what thetheory says, allowing one to more easily learn and use it. As an example, thespecial theory of relativity, before it is supplemented with the clear interpretationof a four-dimensional Minkowski spacetime, can seem to lead to paradoxes inmeasurements that are difficult to comprehend – like the paradox of the train andthe tunnel or the twins paradox. These are not genuine paradoxes; there is no suchinconsistency in the theory, but this is much easier to comprehend when one graspsthe theory not purely through the predictions the mathematics produces, butsupplements it with a picture of entities spread out in four-dimensional spacetime,for which facts about elapsed time or spatial distance fail to be absolute. I believesomething similar can come to pass for quantum theories. Once supplemented witha clear metaphysics, what looks paradoxical or surprising becomes clear andnatural and easier to use. And there is no reason why distinct interpretations cannotproduce alternative accounts that are useful in this respect.

Second, an interpretation says things that go beyond what the theory on its ownsays, and in this respect, interpretations can be fruitful in generating new specula-tions or predictions that can then extend the theoretical power of the theory. Shouldone adopt the wave function metaphysics and its attendant higher dimensions, onecan begin to ask more questions about the structure and contents of this higher-dimensional space and learn more facts about it that would simply not be discussedwithout attention to this question of interpretation.

Third, for myself and many other former physics students, the reason we chosephysics as a focus of study was to learn about the fundamental nature of reality.Without an interpretation, physics does not provide this. Under the influence ofCopenhagen, Mermin’s “Shut up and calculate!”, and Feynman’s “I think I cansafely say no one understands quantum mechanics,” students often come toquantum theories puzzled about what they say about the world, but then they aretold not to ask such questions because the theory is impossible to understand. Thisis disappointing, and it drives students out of the field. Not all physics students careabout questions of interpretation and the deep issue of the nature of reality, but forthose that do, it is worth having serious work on interpretation that can give themwhat they are looking for. We need more, not fewer students of physics.

I do not want to leave the reader with the sense that anything I am sayingchallenges the idea that we should not at the same time work on interpretations thatchallenge our thinking. In fact, all of the interpretations of quantum theories thatare available have aspects of unintuitiveness – this is simply unavoidable in theinterpretation of quantum theories. In addition, this is what is so exhilarating about

180 Alyssa Ney

the study of these theories – how they challenge what we previously thought wasobvious. What is being suggested in this last section, however, is that there isnothing problematic about trying to fit these startling aspects of the world into apicture we can understand.

9.8 Conclusion

The wave function realist need not deny that there is a clear sense of locality inwhich our world contains nonlocal influences. This is the sense of local causalitytaken up by Bell from 1976 onwards. The question is whether one should take thisto be a brute fact about our world or should attempt to provide explanations interms of an underlying metaphysics. Wave function realism is such an attempt atexplanation. The virtues of having interpretative options that provide such anexplanation justify the exploration and development of this framework that shouldbe pursued alongside others.

References

Albert, D. Z. (1996). “Elementary quantum metaphysics,” pp. 277–284 in J. T. Cushing,A. Fine, and S. Goldstein (eds.), Boston Studies in the Philosophy of Science:Bohmian Mechanics and Quantum Theory: An Appraisal, Vol. 184. Dordrecht:Springer.

Alexander, H. (ed.) (1956). The Leibniz-Clarke Correspondence. Manchester: ManchesterUniversity Press.

Allori, V. (2013). “Primitive ontology and the structure of fundamental physical theories,”pp. 58–75 in A. Ney and D. Z. Albert (eds.), The Wave Function: Essays on theMetaphysics of Quantum Mechanics. Oxford: Oxford University Press.

Allori, V., Goldstein, S., Tumulka, R., and Zanghì, N. (2008). “On the common structureof Bohmian mechanics and the Ghirardi-Rimini-Weber theory,” The British Journalfor the Philosophy of Science, 59: 353–389.

Bell, J. (1964). “On the Einstein-Podolsky-Rosen paradox,” Physics, 1: 195–200.Bell, J. (1976). “A theory of local beables,” Epistemological Letters, 9: 11–24.Bell, J. (1981). “Bertlmann’s socks and the nature of reality,” Journal de Physique

Colloques, 42: 41–62.Bentley, R. (1838). The Works of Richard Bentley, A. Dyce (ed.). London: Francis

Macpherson.Dürr, D., Goldstein, S., and Zanghì, N. (1992). “Quantum equilibrium and the origin of

absolute uncertainty,” Journal of Statistical Physics, 67: 1–75.Einstein, A. (1948). “Quanten-mechanik und Wirklichkeit,” Dialectica, 2: 320–324.Esfeld, M. (2004). “Quantum entanglement and a metaphysics of relations,” Studies in

History and Philosophy of Modern Physics, 35: 601–617.French, S. (2014). The Structure of the World. Oxford: Oxford University Press.Goldstein, S. and Zanghì, N. (2013). “Reality and the role of the wave function in quantum

theory,” pp. 91–109 in A. Ney and D. Z. Albert (eds.), The Wave Function: Essays onthe Metaphysics of Quantum Mechanics. Oxford: Oxford University Press.

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Howard, D. (1985). “Einstein on locality and separability,” Studies in History and Phil-osophy of Science, 16: 171–201.

Ismael, J. and Schaffer, J. (2018). “Quantum holism: Nonseparability as common ground,”Synthese, https://link.springer.com/article/10.1007/s11229-016-1201-2.

Ladyman, J. (1998). “What is structural realism?,” Studies in History and Philosophy ofScience, 29: 409–424.

Ladyman, J. and Ross, D. (2007). Every Thing Must Go. Oxford: Oxford University Press.Lange, M. (2002). An Introduction to the Philosophy of Physics. Oxford: Blackwell.Lewis, P. (2016). Quantum Ontology. New York: Oxford University Press.Loewer, B. (1996). “Humean supervenience,” Philosophical Topics, 24: 101–127.Myrvold, W. (2015). “What is a wave function?,” Synthese, 192: 3247–3274.Ney, A. (2019a). “Separability, locality, and higher dimensions in quantum mechanics,”

forthcoming in B. Weslake and S. Dasgupta (eds.), Current Controversies in Phil-osophy of Science. London: Routledge.

Ney, A. (2019b). “Wave function realism in a relativistic setting,” forthcoming in D. Glick,G. Darby, and A. Marmodoro (eds.). The Foundation of Reality: Fundamental, Space,and Time. Oxford: Oxford University Press.

Ney, A. and Phillips, K. (2013). “Does an adequate physical theory demand a primitiveontology?,” Philosophy of Science, 80: 454–474.

Teller, P. (1986). “Relational holism and quantum mechanics,” British Journal for thePhilosophy of Science, 37: 71–81.

Teller, P. (1989). “Relativity, relational holism, and the Bell inequalities,” pp. 208–223 inJ. T. Cushing and E. McMullin (eds.), Philosophical Consequences of QuantumTheory: Reflections on Bell’s Theorem. Notre Dame: University of Notre Dame Press.

Wallace, D. and Timpson, C. (2010). “Quantum mechanics on spacetime I: Spacetime staterealism,” British Journal for the Philosophy of Science, 61: 697–727.

Wiseman, H. (2014). “Two Bell’s theorems of John Bell,” Journal of Physics A, 47:424001.

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Part III

Individuality, Distinguishability, and Locality

10

Making Sense of Nonindividuals in Quantum Mechanics

jonas r. b. arenhart, otavio bueno, and decio krause

“It is only a slight exaggeration to say that good physics has at timesbeen spoiled by poor philosophy.”

(Heisenberg 1998: 211)

10.1 Motivation

As the epigraph by Heisenberg suggests, physics and philosophy may both benefitfrom a constructive exchange in which one may enlighten the other. Physics canilluminate philosophy, and philosophy can illuminate physics. Of course, somemay think that philosophy has nothing to contribute to physics (see Weinberg1992), and although we shall not provide a detailed defense of why we takephilosophy to be relevant for science in general, we want to defend the relevanceof ontology, as a field of metaphysics, to physics and to what physics is about. Westress, in this work, through a case study, the way in which ontology, as aphilosophical field, can engage with physics, particularly in clearing the groundfor the understanding of the nature of physical reality.

Ontology is concerned with what exists and with what kinds of things exist.Although this description may sound abstract and far from the concerns of physics,the relation between ontology and physics is a close one. Of course, we are notclaiming that physics cannot be successful without ontology. If that were the case,ontology would be required for physics, and it is not. However, physicists workwith ontological problems all the time. For instance, when it is claimed that, ingeneral relativity, space and time are no longer independent and that a new kind ofentity is required, spacetime, this is a physical move with significant ontologicalconsequences. It directly affects how the furniture of the world looks.

Physicists need not be concerned with ontological problems raised by physics,just as one need not be familiar with the Peano axioms in order to be able to usearithmetical operations. Nevertheless, ontology is part of the enterprise, shared by

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most physicists, of obtaining information about how the world works and what it ismade of. What kinds of things are there? Particles, fields, space, time? What arethey like? Answering questions like these is part of the articulation of an under-standing of physical reality. As a result, the furniture of the world is involved insuch understanding. Ignoring those questions and their importance may preventone from getting closer to the most fundamental problems.

In this chapter, we will not focus on such general questions, but rather on a veryspecific case study: Assuming that quantum theories deal with “particles” of somekind (point particles in orthodox nonrelativistic quantum mechanics, field excita-tions in quantum field theories), what kind of entity can such particles be? Onepossible answer, the one we shall examine here, is that they are not the usual kindof object found in daily life – individuals. Rather, we follow a suggestion by ErwinSchrödinger (among others, as will become clear later), according to whichquantum mechanics poses a revolutionary kind of entity – nonindividuals. Whilephysics, as a scientific field, is not concerned with whether entities posited by aspecific physical theory are individuals or not, answering this question is part of thequest for a better understanding of physical reality. Here lies, in large measure, therelevance of ontology.

10.2 Introduction

There is little doubt that quantum entities are difficult to categorize. Quantummechanics introduces so many oddities that it is easier to state what quantum entitiesare not than to affirm what they are. (We use ‘entity’ here as a term that is neutralregarding whether the things that are referred to have well-defined identity condi-tions or not). According to some of the first creators of quantum theory, quantumentities are nonindividuals. This view is now known as the Received View onquantum non-individuality (henceforth, for the sake of brevity, “Received View”;see French and Krause 2006: chapter 3, for further historical details on this view).

In a section aptly called “A particle is not an individual,” Erwin Schrödinger(1998) advanced one of the formulations of the Received View. One passage isworth quoting in full:

This essay deals with the elementary particle, more particularly with a certain feature thatthis concept has acquired – or rather lost – in quantum mechanics. I mean this: that theelementary particle is not an individual; it cannot be identified, it lacks “sameness” . . . Intechnical language it is covered by saying that the particles “obey” a new fangled statistics,either Bose-Einstein or Fermi-Dirac statistics. The implication, far from obvious, is that theunsuspected epithet “this” is not quite properly applicable to, say, an electron, except withcaution, in a restricted sense, and sometimes not at all.

(Schrödinger 1998: 197)

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Several significant points are made in this passage. It is noted that quantumparticles (i) are not individuals, (ii) cannot be identified, (iii) lack “sameness,”and (iv) cannot be referred to by the use of “this,” at least not typically. Of course,it is not clear, by considering this quotation alone, what Schrödinger’s conceptionof identification, individuality, and sameness ultimately is, nor is it specified whatthe proper relations among these concepts are. But a central feature of his viewbecomes salient in another important passage. He notes:

I beg to emphasize this and I beg you to believe it: it is not a question of our being able toascertain the identity in some instances and not being able to do so in others. It is beyonddoubt that the question of “sameness”, of identity, really and truly has no meaning.

(Schrödinger 1996: 121–122)

Here, it is emphasized that the very question of the identity of quantum entities, thequestion of their “sameness,” has no meaning. As a result, the difference betweenthese entities, provided their sameness is meaningless, has no meaning either. Onestill needs to examine, of course, what exactly is the relation between the lack ofsameness (or identity) of quantum entities, on the one hand, and their lack ofindividuality, on the other. It seems that Schrödinger takes them all to be conceptu-ally the same: to “lose” one’s individuality just is to lose one’s identity. On hisview, the question of the identity of quantum particles in general makes no sense.The proper understanding of the relations between these concepts, and the kind ofview that results from them in the context of quantum particles, is the topic of thischapter.

These issues were also central to another contributor to the development ofquantum theory. In a classical passage, in which the issues of identity and individu-ality were prominent, Hermann Weyl points out:

. . . the possibility that one of the identical twins Mike and Ike is in the quantum state E1

and the other in the quantum state E2 does not include two differentiable cases which arepermuted on permuting Mike and Ike; it is impossible for either of these individuals toretain his identity so that one of them will always be able to say ‘I’m Mike’ and the other‘I’m Ike’. Even in principle one cannot demand an alibi of an electron!

(Weyl 1950: 241)

The questions of discernibility and of an “alibi” of a quantum particle are clearlyposed. Once quantum particles, such as electrons, are in an entangled state, itcannot be determined which particle is in which state. In other words, it cannot besettled which particle is which. There is nothing – no property, no special ingredi-ent – that could act as an alibi to discern electrons. In this respect, it is theirindiscernibility rather than their identity that should take center stage. Differentlyfrom what Schrödinger suggests, perhaps identity need not lose its meaning,provided that indiscernible things can still be numerically distinct (or identical).

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As will becomes clear, to articulate this proposal it is required that identity andindiscernibility be distinguished. In classical logic and standard mathematics,identity is formulated in terms of indiscernibility. So, in order to keep one andchange the other, one needs to resist this identification and clearly separate the twonotions. (We will return to this later).

As these quotations illustrate, when it comes to the investigation of the nature ofquantum entities, various possibilities are open. One can examine the commonal-ities between the conceptions underlying Schrödinger’s and Weyl’s approaches orpursue their differences. A major feature that is common to both is that they seemto suggest that something is lost by quantum entities: something that marks adifference between quantum entities and classical entities.

In this chapter, we address the articulation of the Received View and theconception of nonindividuality that it attributes to quantum entities. As we discussin Section 10.3, the conception can be formulated in distinct ways, some moreradical, others more conservative, at least with regard to the role of the concept ofidentity as used in quantum theories. The main issue turns on the behavior ofidentity and its relation with individuality. Central to the Received View is theclaim that identity makes no sense, a claim that, as just noted, Schrödinger seemsto have favored. We discuss, in Section 10.4, how to make metaphysical sense ofthat idea. The bare claim that identity makes no sense should be accompanied byan account of how this view entails that particles are not individuals. In Section10.5, we discuss the formal consequences of the idea, and apply the ReceivedView to suggest a revision of classical logic. In Section 10.6, we draw someconsequences of this case study to the significance of research in the foundations ofphysics.

10.3 The Received View

Common to the claims of both Schrödinger and Weyl quoted earlier is an import-ant point: What is responsible for the strange metaphysical behavior of quantumparticles is the statistics they obey. Behind this trait one finds encapsulated the so-called permutation symmetry (PS). According to PS, quantum states should besymmetric (or antisymmetric) with regard to the permutation of labels of particles.As a result, if we are to represent a system composed by two particles x1 and x2, sothat one of the particles is in a region A and the other is in a region B, it cannot bedetermined which particle is in which region. (A qualification is in order: Assum-ing that the underlying “space” is Newtonian and thus, mathematically, its top-ology is Hausdorff, it follows that two separate points can always be discerned bydisjoint open balls centered on the points [see Krause 2019]. Attention to theinterface between the mathematical framework and the physical setup is

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important). In this case, nonsymmetric wave functions, ΨA x1ð ÞΨB x2ð Þ orΨA x2ð ÞΨB x1ð Þ, are unable to describe the situation alone; what is needed is asuperposition of both of them:

ΨAB ¼ ΨA x1ð ÞΨB x2ð Þ � ΨA x2ð ÞΨB x1ð Þ, except for a normalization factor

(10.1)

Thus, the permutation of A with B results either in the same state ΨAB in the case ofbosons or in the state �ΨAB in the fermions’ case (see French and Krause 2006:chapter 4, for a discussion of the physical aspects of this situation and an examin-ation of how far the metaphysics can go). More importantly, the square of theresulting wave function, which gives us the relevant probabilities, is preserved(since ΨABj j2 ¼ �ΨABj j2). Hence, if A stands for an arbitrary observable and P is apermutation operator, then

ΨABh jA ΨABj i ¼ PΨABh jA PΨABj i: (10.2)

As it turns out, this is as far as one can go based on the quotes given previously.Any additional step will break the shared agreement, given that different metaphysicalconclusions will be drawn in light of the same physical fact.

In both of his claims stated earlier, Schrödinger seems to identify “sameness”and identity, so that the fact that one cannot attribute sameness to the particles alsomeans that one cannot attribute identity to them. Individuality is lost as a result ofthe lack of sameness. Given that it makes no sense to state that one particle is thesame as the other and given that it is not possible to refer to a particle as “this” one,particles are no longer individuals.

Of course, the issue is more complex than these considerations suggest. If it werepossible to determine that there is one particle and then another, it would certainlymake sense to state that they are different. But this is not quite what Schrödingerclaims. At this point, an additional ingredient should be added to make clear whatSchrödinger’s conception of individuality ultimately is. With regard to the typicalprinciple of individuality of the metaphysicians, which accounts for what an entity isin contrast to others, Schrödinger advances a particular spacetime principle ofindividuation, one which accounts for the individuality of an item in terms of itsspatiotemporal position (see French and Krause 2006: chapter 1, for details).

In discussing the individuality of familiar objects, Schrödinger (1998: 204)claims that science has taken for granted the permanence of pieces of matter, andthis is what accounts for the identity and individuality of objects. This is mani-fested in one’s confidence when the identity of familiar objects becomes an issue:

When a familiar object reenters our ken, it is usually recognized as a continuation ofprevious appearances, as being the same thing. The relative permanence of individualpieces of matter is the most momentous feature of both everyday life and scientific

Making Sense of Nonindividuals in Quantum Mechanics 189

experience. If a familiar article, say an earthenware jug, disappears from your room, youare quite sure somebody must have taken it away. If after a time it reappears, you maydoubt whether it really is the same one – breakable objects in such circumstances are oftennot. You may not be able to decide the issue, but you will have no doubt that the doubtfulsameness has an indisputable meaning – that there is an unambiguous answer toyour query.

(Schrödinger 1998: 204)

Compare the view articulated in this passage with the one Schrödinger advancedearlier when he claimed that the notion of identity makes no sense for quantumentities (see the quotation from Schrödinger 1996: 121–122, in the previoussection). While ordinary objects typically are supposed to have well-definedidentity conditions, which allows one to answer questions about their identity overtime (even if, in some cases, one may be unable to decide the issue), for quantumobjects such questions do not even make sense. As a result, there is simply no factof the matter regarding the individuality (as well as the identity or sameness) ofquantum particles. In fact, in the case of quantum particles, situations involvingdistinct observations of an object through time generate problems that prevent theindividuality of the items in question from making sense. As Schrödinger notes

Even if you observe a similar particle a very short time later at a spot very near to the first,and even if you have every reason to assume a causal connection between the first and thesecond observation, there is no true, unambiguous meaning in the assertion that it is thesame particle you have observed in the two cases. The circumstances may be such that theyrender it highly convenient and desirable to express oneself so, but it is only anabbreviation of speech; for there are other cases where the “sameness” becomes entirelymeaningless . . .

(Schrödinger 1996: 121)

Schrödinger highlights the need for identity in order to claim that an entity isobserved in distinct places at distinct times. It is then just one step to add thatwithout the possibility that a particle observed at one instant of time t1 is the sameas a particle observed at a later time t2, individuality is lost. Given that, on thisview, it makes no sense to state that those particles are the same (or different),identity loses its meaning. As we have already noted, identity, individuality, andsameness are taken as conceptually the same by Schrödinger.

Based on these considerations, a straightforward version of the Received Viewemerges. Quantum particles are not individuals, given that they have no well-determined trajectories in spacetime, and it is not possible to identify distinctdetections of an entity as being detections of the same entity (we will return tothis view in the next section and will provide additional details there).

However, this is not the only way to articulate the Received View via spacetimecontinuity. Another form is to keep the restriction that quantum entities fail to have

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well-defined spatiotemporal trajectories (as is the case in most versions of thetheory), and thus insist that these entities are nonindividuals in this sense, but notconnect this lack of individuality with a lack of identity. That is, one could keep aform of spacetime nonindividuality, but separate individuality from the logicalrelation of identity (see the suggestion in French and Krause 2006: 153 andArenhart 2017). Perhaps Schrödinger could be interpreted as suggesting this view:Identity does not apply to quantum particles, but this is no violation of the Principleof Identity of Indiscernibles (PII), according to which, some quality alwaysdiscerns numerically distinct items. Since, on this view, PII is supposed to applyonly to individuals, which quantum particles are not, such particles do not providea violation of the principle. Along these lines, another version of the ReceivedView could be advanced, along with a distinct conception of nonindividual.

However another option, which is less revisionary regarding the role of identitythan Schrödinger’s, can still be pursued. One can take literally the claim thatquantum particles are nonindividuals but resist to follow Schrödinger in makingthe further claim that identity, as a logical relation, loses its meaning. In fact, ifWeyl’s “alibi” is taken to refer to a principle of individuality, then it makes sense toclaim that, even though no alibi is available for quantum particles, this lack of alibineeds not be connected with the lack of meaning of identity (hence, identity wouldbe metaphysically deflated, as recommended by Bueno 2014). In order to do that, itis enough that one resists binding so tightly an item’s individuality with its identity.Let us explore this option further, given that pursuing this route provides anadditional (alternative) version of the Received View. (See also Arenhart 2017for a discussion of alternative formulations of the Received View, which do notinvolve abandoning identity).

One can take the alibi Weyl refers to as involving a property that distinguishes aparticle bearing it from any other particle. It is always possible to differentiate aclassical particle (that is, a particle described by classical mechanics) from anotherparticle by at least one property. Of course, it cannot be a state-independent property(given that particles of the same kind share such properties), but at least theirspatiotemporal location distinguishes them. On this view, no two classical particlesoccupy the same location at the same time, due to a principle of impenetrability. Thismeans that numerically distinct particles have their difference grounded in a prop-erty that accounts for their numerical diversity and their individuality.

This trait leads to the validity of PII in classical mechanics. According to thisprinciple, as noted previously, numerically distinct particles are always discernibleby some quality. Entities are individuals due to the fact that once two of them arepresent, there is always some property that accounts for their numerical difference:This is their alibi. In light of PII, an alibi is always available in classical mechanics.As a result, classical particles are indeed individuals.

Making Sense of Nonindividuals in Quantum Mechanics 191

In contrast, quantum particles have no alibi – nothing that accounts for theirindividuality. Not even spatiotemporal location can be employed to this effect. Dueto the permutation symmetry, quantum particles are indiscernible by their proper-ties, including both state-dependent and state-independent ones. Hence, the versionof PII presented earlier, according to which there is always some property thataccounts for the numerical diversity of particles, fails in quantum mechanics. Theresult is clear: As Weyl noted, there is no alibi for quantum entities (see French andKrause 2006: chapter 4, for further discussion).

It could be argued that, if properties are unable to account for the numericaldifference of quantum particles, perhaps some relations could do that, such as therelation “to have spin opposite to” in a given spatial direction. But this proposal isstill unable to account for the particles’ individuality. After all, if x has spinopposite to y, y also has spin opposite to x (the relation is symmetric). While noparticle has spin opposite to itself (the relation is irreflexive), there is no quantummechanical fact of the matter to determine which of x or y has spin up in a givendirection, and which has spin down in that same direction. Thus, those relations,called weakly discerning relations, in principle can account for the numericaldiversity of the particles (although whether they do account for that is stilldebatable; see French and Krause 2006: chapter 4). Despite that, they are unableto provide an alibi for the particles in question, because weakly discerning relationsare unable to individuate such particles. Accounting for the particles’ numericaldiversity (if at all) is the closest one can get in quantum mechanics to discernibility(see Muller and Saunders 2008, and the discussion in Lowe 2016).

However, if weakly discerning relations are implemented in a mathematicalcontext whose underlying set theory is ZFC (Zermelo-Fraenkel set theory with theaxiom of choice), as is the case of Muller and Saunders (2008), all entities becomefully discernible and identifiable in virtue of the resources of set theory alone (wewill return to this point and provide the argument later). Thus, there is a tensionbetween the motivation for the introduction of weakly discerning relations and theadopted set-theoretic framework.

In principle, if the option of maintaining that identity holds for quantumparticles can be fully worked out, one could claim that they are different oridentical, without thereby implying that they are individuals. What is required, aswe have been suggesting, is that their individuality be grounded in some kind ofalibi (in Weyl’s sense) that is not formulated in terms of identity.

There are additional possibilities to articulate alibis (that is, principles ofindividuality) without requiring the removal of identity (see Arenhart 2017). It isenough that the content of identity be deflated from the metaphysical content thatwould be required if identity also played the role of a principle of individuality.(For a defense that identity should be deflated, see again Bueno 2014). As will

192 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause

become clear later (when a formal approach to identity is discussed), identity canbe thought of as something very minimal and without much metaphysical content,just in terms of two features: reflexivity (that is, every object is identical to itself )and substitutivity (if x is identical to y, then if x is F, so is y). One could add somemetaphysical content to identity, so that it can be used as a principle ofindividuality. But that changes identity by making it more substantive than itneeds to be. Schrödinger, of course, does not seem to follow this path since heappears to keep identity and individuality very closely connected. As a result,insisting on nonindividuality requires abandoning identity, at least for quantumentities. In what follows, we investigate the prospects for a Schrödingerianapproach to nonindividuality.

10.4 Making Sense of Losing Identity

If one is to pursue the option that seems to be suggested by Schrödinger – namely,that individuality and identity go together, and that one cannot have nonindividualswithout abandoning also the relation of identity – it is crucial to explain in detailwhy identity should go hand in hand with individuality regarding quantumparticles.

One of the possible ways of doing that consists in exploring the relation betweenidentity and individuality through the notion of haecceity, as it is done, forinstance, by French and Krause (2006). Basically, a haecceity is a nonqualitativeproperty uniquely instantiated by an object, something like an individual essencepossessed by a single individual. Each individual has its own essence, which, ofcourse, accounts for its individuality (see further discussion in Lowe 2003).

Being nonqualitative, a haecceity does not count as a quality able to discern amongtwo individuals. That is, two individuals may share every qualitative property, but stillnot be the same individual, due to the fact that they have distinct haecceities. As aresult, PII (restricted to qualitative properties) may fail and, despite that, individualityis still saved in light of a haecceity. In this sense, having a haecceity iswhat French andKrause (2006: chapter 1) call a “Transcendental Individuality” principle: that whichconfers individuality over and above an item’s qualities.

This point has an important formal counterpart. If the underlying mathematicalframework in which quantum theories are formulated is that of ZFC, one shouldconsider, in particular, the resulting models – the structures in which quantummechanics holds. Within these structures, it is possible that certain entities cannotbe discerned based on the resources of quantum theories alone. However, outsidesuch structures, it is possible to discern the entities in question, and this can bethought of as a formal expression of transcendental individuality (clearly, therelevant structures cannot be rigid).

Making Sense of Nonindividuals in Quantum Mechanics 193

The problem, however, is not to account for an item’s individuality, but ratherfor its nonindividuality. How can haecceity achieve that? The answer seems to be:through the notion of identity. As French and Krause put it

. . . the idea is apparently simple: regarded in haecceistic terms, “TranscendentalIndividuality” can be understood as the identity of an object with itself; that is, 'a ¼ a'.We shall then defend the claim that the notion of non-individuality can be captured in thequantum context by formal systems in which self-identity is not always well-defined, sothat the reflexive law of identity, namely, 8x x ¼ xð Þ, is not valid in general.

(French and Krause 2006: 13–14)

That is, a haecceity may be formally represented by self-identity. Plato’s individu-ality, if it were attributed by a haecceity, would consist in his bearing the propertyof being identical with Plato. This, of course, is a nonqualitative property, and it isable to connect identity (as a logical concept) and individuality (as a metaphysicalconcept). This quote also provides the basic idea to make sense of nonindividualitywithin a framework that takes into account the Schrödingerian claims that quantumentities are not individuals and that identity makes no sense for them. In order toaccommodate metaphysically the idea that identity has no meaning for quantumparticles, it is enough that the reflexive law of identity fails or does not hold ingeneral. Thus, not everything is self-identical. In light of this connection betweenself-identity and haecceity, those entities for which the law fails are nonindivi-duals: They lack a haecceity, which is the individuation principle.

French and Krause acknowledge explicitly that the connection between identityand individuality is particularly tight:

We are supposing a strong relationship between individuality and identity . . . for we havecharacterized ‘non-individuals’ as those entities for which the relation of self-identitya ¼ a does not make sense.

(French and Krause 2006: 248)

This is only one of the possible ways to accommodate metaphysically the combin-ation of nonindividuality and the loss of identity. This proposal allows one to makea good case for the failure of identity, given that the relation between individualityand identity is very clearly established in this approach. However, in addition toburdening identity with the role of attributing individuality, there is anotherdisadvantage of adopting this approach to nonindividuality: It takes us very farfrom the Schrödingerian ideas with which we started. Of course, it allows us tomake sense of the claim that identity and difference do not apply to quantumentities. But the lack of haecceity arguably was not what Schrödinger had in mindin his discussion of identity and identification of quantum particles. Rather, asdiscussed previously, he seems to favor an account of individuality framed expli-citly in terms of spatiotemporal trajectories.

194 Jonas R. B. Arenhart, Otávio Bueno, and Décio Krause

The following two conditions seem to articulate better the conception of anentity being individuated by its spatiotemporal position, in the sense that an entitysatisfying these two conditions should be counted as an individual (see also Bueno2014 and Bueno 2019):

(A) Identity conditions: an individual has (clearly determined) identity conditions.

(B) Persistence conditions: an individual persists over time (despite changes).

Note that these minimal conditions are satisfied by what are typically consideredindividuals (such as, chairs, cherries, or chariots). In particular, as Schrödingeremphasizes, an earthenware jug would be an individual according to this approach,and so are classical particles, given that their well-determined trajectories groundboth their persistence and their identity conditions (see French and Krause 2006:chapter 2). Quantum particles in Bohmian mechanics are also individualsaccording to this characterization: They have trajectories attributed by hiddenvariables.

It should be noted that there are at least two ways of satisfying persistencecondition (B):

(B.1) Essential traits: as long as certain essential traits (or necessary properties) ofan individual are preserved, the individual remains in existence.

(B.2) Closest continuers: given an individual i that satisfies condition (A), at eachmoment of time the closest continuer individual to i (the one that shares mostproperties with i) is taken to be i (Nozick 1981: chapter 1).

Of course, a haecceity could be an essential trait, and in this way, haecceities couldbe used to account for the permanence of an individual. Given that we have alreadysuggested avoiding a theory of haecceities to account for individuality and to framean approach to nonindividuality, we favor the less metaphysically committingoption (B.2). The idea is that an individual persists through a sequence of closestcontinuers, which, taken together, account for the permanence of an individualover time despite the changes it undergoes.

Given this theory of individuality, formulated by the conjunction of conditions(A) and (B.2), for something to be a nonindividual, three options emerge: condi-tion (A) can be violated; condition (B.2) can be undermined, or both conditions canfail. Quantum entities, as the discussion of Schrödinger’s view indicates, violateboth conditions. This is a Humean point: There appears to be no causal connectionthat would allow one to determine that similar objects detected in differentmoments in time are, in fact, the same. In the quantum case, consider somequantum entities that have no continuous trajectory. One cannot look for aquantum mechanical justification to connect two observations of two such entities

Making Sense of Nonindividuals in Quantum Mechanics 195

through a single trajectory. Nothing in the theory allows us to do that (unless one isa Bohmian). As a result, as we have seen, Schrödinger claimed that identity makesno sense for those entities, given that there is no fact of the matter to determinewhether the two observations correspond to the same entity or not. The question ofthe identity of the observed entities ends up being entirely ungrounded.

This accounts for both the nonindividuality of the particles and the fact thatidentity does not apply to them. This metaphysical picture is closer to whatSchrödinger had in mind, it seems, and it is less inflated than the one first suggestedby French and Krause (2006), which proceeds through the concept of haecceity.However, both approaches require a corresponding rejection of the overall validityof identity. One of the ways to accommodate such a limitation of identity isthrough logics that restrict identity, the so-called nonreflexive logics. We turn tothem now.

10.5 The Formal Approach to Identity

It is important to be clear about what identity is, particularly when it is stated thatquantum objects lack identity. Throughout this chapter, we have been using theterm “identity” in the sense of what is typically called standard identity (or simply“identity,” for short) as conceptualized by classical logic and standardmathematics.

But there is a pretheoretical conception of identity (let us call it a numericalidentity, for lack of a better word). This conception states that every object isidentical just to itself and to nothing else. According to this informal view, when itis coupled with the tight connection between identity and individuality just dis-cussed, it follows from the fact that all objects have identity that they are individ-uals. So, under this interpretation, the informal notion of identity is closely relatedto that of individuality.

The informal view of identity discussed earlier is generally thought to be encapsu-lated in classical logic or in other systems of logic that share the basic features ofidentity. Let us focus on classical logic, for the sake of brevity. Classical logic takesidentity as a binary relation between objects of the domain. Identity statements areusually written as a ¼ b (or as a 6¼ b, depending on the case), in order to express thatthe objects denoted by a and by b are the same, they are identical (or are not the same,they are different). This intuitively means that there are no two objects, but just one,which can be named (described) by such expressions. A typical case is the famousidentity statement ‘Hesperus = Phosphorus’ (Frege 1960). True statements such asthis make reference to the identity of the objects of the domain, and ‘Hesperus’ and‘Phosphorus’ denote the same object. One should consider both the syntactic char-acterization of the notion of identity (given by the binary predicate ‘=’) and the

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semantic characterization in which the identity of the objects of the domain ofinterpretation is at issue (we denote the domain by D).

Let us consider the semantics first. The identity of D is taken to be the setID ¼ x; xh i : x 2 Df g. This clearly presupposes that the identity of the elements ofD is well determined, and the metamathematical framework is consistent with thisfact. If one assumes a standard semantics, that is, a semantics built in a standard settheory such as ZFC, then this assumption is met, given that the identity of all sets ispresupposed (we will return to this point later). ID is a set. According to Cantor, itis thus “a collection of definite and distinct objects of our intuition or of ourthought” (Cantor 1915/1955: 85). This informal, circular idea of a set (of course, itis not a definition), which accounts for sets in terms of collections, is couched interms of numerical identity.

The problem concerns the syntactic side. Is it possible to axiomatize a first-orderlogic having a primitive binary predicate for identity having the set ID as its soleinterpretation? That is, is it possible to provide a definition (or a group ofpostulates) such that the identity predicate has just ID as a model? The answer,we argue, is negative. Let us see why.

To begin with, it is important not to confuse numerical identity with the notionof identity in standard logic and mathematics. Arguably, it is primarily the latterthat can be rigorously dealt with. Suppose that the language L under considerationis first-order. Two cases emerge. First, L contains just a finite set of primitivepredicates. In this case, we can “define” identity by exhausting all predicates. Anexample suffices: Suppose that the predicates are two monadic predicates P and Qand a binary predicate R. Thus, a ¼ b can be “defined” as follows:

a ¼ b $ Pa $ Pbð Þ∧ Qa $ Qbð Þ∧8x Rxa $ Rxbð Þ∧ Rax $ Rbxð Þð Þ: (10.3)

The problem with this “definition” is that clearly there can be additional predicatesnot belonging to the language that could distinguish a and b, not to mention thepossibility of there being some kind of haecceity that achieves that (as we noted inthe beginning of the previous section when we made the point about the “formalcounterpart” regarding haecceity). In fact, Eq. (10.3) stands only for theindiscernibility of a and b regarding the predicates of L.

Second, usually first-order languages introduce identity as a primitive binarypredicate ‘=’. In this case, the standard formulation makes use of two postulates,namely

(R) Reflexivity: 8x x ¼ xð Þ(S) Substitutivity: x ¼ y ! αx ! α y=x½ �ð Þ, where x and y are individual variables,αx is a formula having x free, and α[y/x] results from the substitution of y for x insome free occurrences of x, in which x and y are distinct variables.

Making Sense of Nonindividuals in Quantum Mechanics 197

(Note the use of identity in the very formulation of the substitutivity rule: Thevariables x and y need to be distinct, that is, not identical). From these postulates, itfollows that identity is symmetric and transitive. Thus, it is an equivalence relationas well as a congruence relation due to the presence of substitutivity. Logicians saythat identity is the finest congruence over the domain in the sense that if ffi isanother congruence, then a ¼ b entails a ffi b, for all a and b.

However things are not so easy. Postulates (R) and (S) cannot guarantee that theinterpretation of the predicate ‘=’ is the set ID. In fact, it can be shown that acongruence, other than identity, can be defined over the domain that also modelsthe predicate of identity (da Costa and Bueno 2009, Krause and Arenhart 2018). Inother words, from the point of view of L, it cannot be known whether one isworking with a structure where ‘=’ is interpreted as the identity of the domain D,namely, the set ID, or in terms of another structure that has the defined congruenceas the interpretation of syntactic identity. These structures are elementaryequivalent.

Leaving first-order languages behind, higher-order languages should then beconsidered. It suffices to consider L as a second-order language (the generalizationto other higher-order languages is immediate). In this case, identity can be(allegedly) “defined” in terms of indiscernibility (indistinguishability) by what iscalled Leibniz law, namely:

x ¼ y if, and only if, 8F Fx $ Fyð Þ, (10.4)

where x and y are variables for individuals, and F is a variable for properties ofindividuals. The right side of the biconditional expresses the indiscernibility of xand y, and it states that the objects that stand for x and y have the same properties(hence they also share all relations).

The problem now is with the semantics. Suppose that the domain is the none-mpty set D ¼ 1; 2; 3; 4; 5f g and that our second-order language has three monadicpredicate constants – P, Q, R – and two individual constants – a and b. Considerthe following interpretation: 1 is assigned to a and 2 to b. Furthermore, theextensions of the predicates are interpreted as the following sets: A ¼ 1; 2; 3f g,B ¼ 1; 2; 4f g, and C ¼ 1; 2; 5f g. Thus, since 1 and 2 belong to all sets, it followsthat a and b have all properties in common. In other words, the right side ofEq. (10.4) holds, despite the fact that 1 6¼ 2.

The only way of guaranteeing that Eq. (10.4) will have its full intuitive meaningis to add all subsets of D to the semantics, that is, to consider what Church callsprincipal interpretations (Church 1956: 307). But then, as is well known, com-pleteness is lost.

As these considerations make clear, identity is not a simple concept when onetries to provide a rigorous account of the intuitive idea. But from a logical point of

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view, this is what classical logic presents us with. Based on this theory of identity,which is called classical theory of identity (CTI), we can consider strongersystems, such as various set theories.

As is well known, there are several nonequivalent set theories with distinctproperties and which yield significantly different, and even incompatible, the-orems. For instance, ZFC includes the axiom of choice; Quine-Rosser’s NF(New Foundations) system does not: It is incompatible with this axiom (Forster2014). In ZFC, if consistent, there is no Russell set, namely R ¼ x : x =2 xf g, but insome paraconsistent set theories, this set is legitimate (da Costa, Krause, andBueno 2007). It can be proved, in ZFC, that there are sets that are not Lebesguemeasurable, but in “Solovay set theory” all sets are Lebesgue measurable (Mait-land Wright 1973). What is remarkable is that all these set theories invoke the sametheory of identity (CTI). Thus, our considerations apply to all of them.

It is undeniable that set theory is the most widely used basis for standardmathematics, that is, the part of mathematics that can be developed in theoriessuch as ZFC. This is also the mathematics that underlies quantum theories. In fact,it is unclear what kind of quantum mechanics could be developed in a system suchas NF, given its incompatibility with the axiom of choice (AC). After all, AC isnecessary for the usual mathematical formulation of quantum mechanics, so that itcan be guaranteed, for instance, that the relevant Hilbert spaces have a basis (ofcourse, quantum mechanics can be developed in many different ways that need notrely on von Neumann’s approach; see Styer et al. 2002).

It is a remarkable fact, we noted, that in all of these set-theoretic frameworks, allobjects are individuals, in the sense that all of them have identity. In other words,given any objects (that is, any sets; the case of Urelemente will be mentionedsoon), there is always a way to distinguish them, if not effectively, at least inprinciple. The proof is immediate. Given a certain object a, which is either a set ora Urelement, the postulates of a set theory enable us to form the set {a}, thesingleton of a (as is well known, there are pure set theories, containing only sets,and impure set theories, systems that also include atoms – the Urelemente in theoriginal Cantor’s terminology. These atoms are not sets but can be elements ofsets). Define the “property” Ida xð Þ≕ x 2 af g. The only object that has such aproperty is a itself, so a has at least one property distinguishing it from any otherobject. Leibniz law applies and, thus, there cannot be indistinguishable but non-identical objects.

Indiscernible entities can be accommodated in a set theory via equivalencerelations. The elements of an equivalent class can be taken as representing thesame object, but this is clearly a mathematical trick and does not work as part of aphilosophically well-motivated proposal. A trick similar to this is used in orthodoxquantum mechanics when symmetric and antisymmetric wave functions are

Making Sense of Nonindividuals in Quantum Mechanics 199

chosen to stand for certain quantum systems: Functions are selected that do notalter the probabilities when particle labels are exchanged. (This trick was called“Weyl’s strategy” because it was used by Hermann Weyl; see French and Krause2006: section 6.5.1).

As a result, within standard mathematics, there are no absolutely indiscernibleobjects as quantum objects are said to be in certain situations. Thus, if weuse standard mathematics in our preferred formulation of quantum mechanics(the same point applies to quantum field theories), from the simple fact that thereare two quantum objects, it results from the mathematics alone that the objects aredifferent (they are not identical), and by Leibniz law, there is at least one propertythat one of them has and the other does not. However, if the objects in question areindiscernible, such as two bosons in the same state, which property would that be?The assumption of the existence of such a property amounts to the introduction ofhidden variables – even in those formulations of quantum mechanics that do notaccept them. But the fact that there is such a property follows from Leibniz law(which, as noted, is part of the package formed by classical mathematics, whichincludes a corresponding logic, and the standard theory of identity). Thus, in anysituation, given two quantum objects, there is a difference between them. Such adifference cannot be given by a substratum (a haecceity), because the existence ofsuch a substratum is ruled out in quantum theories (see Teller 1998). The differ-ence can be expressed in terms of a bundle theory of properties, which leads to theconclusion that there is a property that only one of the quantum objects in questionhave, but not the other. The problem is that, according to quantum theories,assuming their usual interpretations, this is not a viable possibility. Otherwise,quantum objects would be discernible. In the end, what is needed is a frameworkthat does not preclude the possibility of indiscernible but potentially distinctsystems of entities – a framework that makes room for nonindividuals.

An appropriate, philosophically well-motivated, strategy would then be to leavestandard set theories behind and adopt a set theory in which identity is not taken tohold in general, namely, a quasi-set theory. This is a mathematical frameworkwhich can be used as a metamathematics for quantum theories (see French andKrause 2006, Domenech, Holik, and Krause 2008, Krause and Arenhart 2016). Inthis theory, collections (called quasi-sets) can be formed by absolutely indiscern-ible elements without thereby becoming identical. As a result, Leibniz law isviolated for some objects (although it remains valid for another kind of objects,called classical). These collections of indiscernible entities can have a cardinal,called its quasi-cardinal, even if they do not have an ordinal. The theory provides aframework to examine collections of objects without ordering them, withoutidentifying or individuating them. And differently from classical set theories, thetheory offers a framework in which nonindividuals can be formulated and

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thoroughly studied without the incoherence found in the use of classical settheories for the formulation of the foundations of quantum mechanics (for details,see French and Krause 2006, Krause and Arenhart 2016). We conclude this chapterby noting the significance of foundational studies of physics, of which quasi-settheories provide a clear case.

10.6 Conclusion

In this chapter, the metaphysical underpinnings of the idea that quantum entitiesare nonindividuals have been examined. Schrödinger’s claim that identity does notmake sense for quantum entities was interpreted, and the connections between thisclaim and some issues related to continuous trajectories in quantum theory wereinvestigated. The resulting metaphysics of nonindividuals assumes a tight connec-tion between identity and individuality, so that if individuality has to go, so doesidentity. Given that identity is a logical relation, which is part of classical math-ematics, and since almost every system of logic has a version with identity, it isimportant to provide an account of what it is like for a system of logic to have noidentity. A few more words on the importance of developing such formalisms andtheir relation with physics are, thus, in order.

First, consider some reasons to look for alternative mathematics (and logic) forquantum mechanics. Leaving aside historical considerations (an up-to-date analy-sis, which also considers some historical facts, can be found in Maudlin 2018), themotivation has to do with the foundations of physics. One could argue that physicsworks fine with standard (Leibnizian) mathematics (and logic), as it can be seen byconsidering any book on quantum mechanics. In particular, the argument goes,questions about the foundations of physics could be regarded as “mere philosoph-ical” problems that, on their own, contribute nothing to the clear understanding ofphysics. That this view is untenable becomes clear by considering some ofthe papers in Cao (1999) and the significant insights that a careful reflection onthe foundations of physics provides (for the sake of brevity, we will not revisit thevarious arguments here).

It could be argued that something similar happens with current physical theor-ies. Current physics works fine with two incompatible theories, namely, thestandard model of particle physics and general relativity. The former providesthe best way developed so far to account for the physics of the small, while thelatter offers the best physics of the big, as it were. One or the other is applieddepending on the subject matter under study. However, these two theories arelogically incompatible with one another, for gravity has not been quantized.Should the situation be left at that, with everyone being encouraged to accept thatphysics has reached its final limit, and no unification is ultimately necessary?

Making Sense of Nonindividuals in Quantum Mechanics 201

Of course not. Novelties have always emerged when foundational issues havebeen pursued. In mathematics, this is undeniably the case, as the development oflogic and set theory clearly illustrates. There is no reason to think that quantummechanics would be any different. Indeed, the relevance of string theories, loopgravity, and any other attempt to find a more fundamental theory, in particular, thequantization of gravity, cannot be appreciated without acknowledging the signifi-cance of foundational research. In fact, without the latter, it would be difficult tomake sense of why physicists systematically pursue such enterprises.

To look for more suitable mathematical bases for a coherent metaphysicalconception of quantum objects as nonindividuals is reasonable and even necessary(to prevent inconsistencies). Arguably, no one seems to know, and perhaps no onewill ever know, what quantum entities ultimately are. All one has are one’stheories. Even the concept of particle changes from theory to theory (see Falken-burg 2007: chapter 6). Foundational research provides some perspective andinsight to pursue the search for understanding that is integral to the attempt ofmaking sense of these issues as well as their significance.

This brings the second topic to be addressed here: ontology. Physicists, ingeneral, have a broad and intuitive idea of what ontology is, but some of themdo not find it relevant to their work in physics. Ontology, it was noted, is tradition-ally occupied with what there is (in the world) – with existential questions, such as:Are there winged horses? Are there electrons? Are there transcendental numbers?Metaphysics is more general and includes ontology as a proper part. For instance,Democritus’ claim that the world is composed by atoms (indivisibles) is a meta-physical view. It concerns the basic structures of the world. Even in classical logicone finds metaphysical assumptions. For example, classical propositional logicassumes a metaphysical, semantic principle to the effect that the truth of a complexformula depends on the truth of its component formulas, usually referred to asFrege’s Principle of Compositionality (Szabó 2017).

Physics is not different, and it also has its share of metaphysical claims. One ofthem, crucial to the entire discussion examined in this work, is that quantumobjects – whether they are particles in orthodox quantum mechanics or fieldexcitations in quantum field theories – are ultimately nonindividuals. This claim,of course, does not force us to assume that nonindividuals exist. Situations can bepresented in a conditional form: If there are things like quantum entities, then theycan be interpreted as nonindividuals. This process of interpretation, itself anintegral part of foundational research, provides a possible approach to the under-standing of the nature of such entities.

Just to be clear, we are not asserting that quantum objects are nonindividuals. Itis unclear how this claim could be established. Rather, the goal is to develop theview as coherently as possible and indicate how it helps one to understand

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quantum entities. Bohmian mechanics (BM) works with a “classical” metaphysicsinvolving “classical” individuals, each one having their own identity. But Boh-mians should also be careful and acknowledge that their hypothesis is just ahypothesis, a metaphysical view among several others. The physics works finewith BM, at least at the level of its nonrelativistic counterpart (we make nocommitment regarding relativistic BM). Clearly, Bohmian mechanics providesan additional example of the significance of foundational research.

To conclude, placed in classical metaphysics and in standard underlying math-ematics (based on ZFC), quantum objects cannot be completely indiscernible. Theresulting theory can provide predictions of quantum phenomena, such as in thetwo-slit experiment (see Holland 2010), but these are just predictions that are ableto make the physics work. The problem, however, is the logic that is being invokedis inconsistent with the indiscernibility of the phenomena in question. Quantumstatistics, Gibbs paradox, and many other quantum phenomena presuppose abso-lute indiscernibility. As Wiczek and Devine have said, “in the microworld, weneed uniformity of a strong kind: complete indistinguishability” (Wilczek andDevine 1987: 135). So, from a logical point of view, predictions are not enough:One needs a proper foundational mathematical framework. The commitment thatclassical logic and standard mathematics have to Leibniz law questions theiradequacy to accommodate, in a proper way, truly indistinguishable things.A different path is then called for.

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Bueno, O. (2014). “Why identity is fundamental,” American Philosophical Quarterly, 51:325–332.

Bueno, O. (2019). “Weyl, identity, indistinguishability, realism,” in A. Cordero (ed.),Philosophers Look at Quantum Mechanics. Dordrecht: Springer.

Cantor, G. (1915/1955). Contributions to the Founding of the Theory of TransfiniteNumbers. New York: Dover.

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Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton UniversityPress.

da Costa, N. C. A. and Bueno, O. (2009). “Non-reflexive logics,” Revista Brasileira deFilosofia, 232: 181–196.

da Costa, N. C. A., Krause, D., and Bueno, O. (2007). “Paraconsistent logics and para-consistency,” pp. 791–911 in D. Jacquette (ed.), Philosophy of Logic. Amsterdam:North-Holland.

Domenech, G., Holik, F., and Krause, D. (2008). “Q-Spaces and the foundations ofquantum mechanics,” Foundations of Physics, 38: 969–994.

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Falkenburg, B. (2007). Particle Metaphysics: A Critical Account of Subatomic Reality.Dordrecht: Springer.

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Frege, G. (1960). “On sense and reference,” pp. 56–78 in P. Geach and M. Black (eds.),Translations from the Philosophical Writings of Gottlob Frege. Oxford: BasilBlackwell.

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Krause, D. (2019). “Does Newtonian space provide identity for quantum systems?,”Foundations of Science, https://link.springer.com/article/10.1007/s10699-018-9561-3

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Krause, D. and Arenhart, J. R. B. (2018). “Quantum non-individuality: Backgroundconcepts and possibilities,” pp. 281–306 in S. Wuppuluri and F. A. Doria. (eds.),The Map and the Territory: Exploring the Foundations of Science, Thought andReality. Dordrecht: Springer.

Lowe, E. J. (2003). “Individuation,” pp. 75–95 in M. J. Loux and D. W. Zimmerman(eds.), The Oxford Handbook of Metaphysics. Oxford: Oxford University Press.

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Modern Physics. London: Penguin Books.

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11

From Quantum to Classical Physics:The Role of Distinguishability

ruth kastner

11.1 Introduction

This chapter consists of three main sections. In Section 11.2, I review NickHuggett’s finding regarding the nonrelevance of permutable labeling for theclassical/quantum divide (Huggett 1999). In Section 11.3, I review the derivationsof the classical and quantum statistics and argue that a form of separability is a keyfeature of the quantum-to-classical transition. In Section 11.4, I consider thequestion: What allows separability to serve as a form of distinguishability in theclassical limit? First, let us review some basic considerations regarding issues ofindividuality and distinguishability.

Steven French (2015) has noted that the concept of individuality is primarily ametaphysical issue, while that of distinguishability is primarily an epistemologicalissue. Nevertheless, distinguishability does have bearing on ontological questionssuch as:

What is an individual?Are there any true individuals?Does Leibniz’ Principle of Identity of Indiscernibles apply to nature?

However, I will not enter here into the metaphysical debate concerning questionssuch as: What is an individual? and Are quantum systems individuals? Rather,I will focus on the issue of distinguishability regarding the quantum-classicaldivide and attempt to identify some ontological features that may underlie theform of distinguishability obtaining in that context.

11.2 Huggett’s Finding on Haecceitism and Classical Objects

Davis Lewis (1986) introduced the term haecceitism, which denotes a form ofstrong individuality: An individual’s identity is taken as a primitive “this-ness”

205

which transcends all its qualitative features. This concept can be identified with theterm ‘transcendent individuality’ (TI) as discussed in French and Redhead (1988).Anti-haecceitism consists in saying that an individual’s identity is constituted by itsqualitative features and nothing more. Although the precise definition of haecceit-ism is still a matter under discussion, for our purposes we can think of it as thecapacity of an entity to carry a label or “name,” where that label is not contingenton any of its qualitative features. Thus, what makes a person named Fred “Fred theindividual” is his primitive this-ness, not the color of his eyes or hair or his height,weight, etc.

Now let us consider this notion as applied to some typically classical objects,such as a pair of coins that are assumed to be completely identical and can be either“heads” or “tails.” Give them name-labels, say ‘Fred’ and ‘Joe’ � their assumedhaecceitism is represented by their name-labels. In this context, haecceitismimplies that if we consider the case in which Fred and Joe are in different states(one of them being “heads” and the other “tails”), then interchanging Fred and Joeconstitutes two different possible configurations. If we include the cases in whichFred and Joe are both “heads” or both “tails,” we have four possible states of thecoins, as seen in Figure 11.1.

In contrast, for a pair of hypothetical “boson coins,” the usual story is that thereis no such thing as Fred or Joe � no nameable identity that transcends thequalitative properties of the quantum coins. So the possible configurations are justthree in number (Figure 11.2).

It should be noted that French and Redhead (1988) dissent from this usualidentification of individuality with the capacity to carry a label such that permuta-tion of the labels establishes a different state of the total system. They argue that aform of individuality can still be retained for quantum systems if one argues thatcertain states are not accessible to the total system. For purposes of this discussion,

Figure 11.1 States of two classical coins.

206 Ruth Kastner

we work with Huggett’s formulation, but note that his interpretation of themetaphysical bearing of the labels is not obligatory.

Huggett (1999) notes that there are two ways of describing the state-space of acomposite system such as the set of two coins. We can either use a phase space(Γ-space) description, which specifies which component system is in which state,or we can use a distribution space (Z-space) description, which just specifies howmany systems are in each state. The Γ-space description assumes that eachcomponent can be meaningfully labeled and/or distinguished from the others, soit supports haecceitism in that respect. In contrast, since Z-space specifies only theoccupancy number of each state, without identifying any particular system withany particular state, it does not support haecceitism in the same way. Since it istypically supposed that the key distinction between classical and quantum objectsis the ability of the former to carry a label, one would think that the two kinds ofdescriptions � phase space and distribution space � would lead to different kindsof statistics; i.e., classical and quantum statistics, respectively.

However, Huggett shows that if we assume that classical objects are impene-trable, i.e., that no more than one such object can never occupy a given spacetimepoint, then it turns out that the Γ- and Z-space descriptions give the same empiricalpredictions. Thus, we cannot use any experimental data to decide between them.This means that there is no empirical support for the idea that classical andquantum objects differ fundamentally in their metaphysical nature as individuals.

The basic argument goes like this: In terms of the coin analogy, we have to pretendthat there are no other qualitative differences between the coins and forbid the twocoins, Fred and Joe (they can keep their labels), from occupying the same state. Ofcourse, real coins would not fulfill this criterion. For themore realistic case of classicalgas molecules, the operative condition is that no two molecules can ever occupy thesame individual phase space state, since they can never be at the same spacetime point.

Figure 11.2 States of two hypothetical “boson” coins.

From Quantum to Classical Physics: The Role of Distinguishability 207

In the case of the idealized coins, if we forbid them from occupying the samestate, there are now only two available composite Γ-states for Fred and Joe � theones in which they are in different “heads” or “tails” states (Figure 11.3).

Additionally, since both of these correspond to the distribution “one coin in eachstate,” the frequency of this distribution is 2/2 = 1. Meanwhile, the frequency ofthis distribution in terms of the Z-space representation, which ignores the phasespace configurations, is just 1/1 = 1. We see that, for the idealized classical coins,the frequency of occurrence of the distribution is exactly the same in eitherrepresentation, so there is no empirical difference between the two spaces � theyboth predict the same probabilities. Huggett shows that this holds in general, for anarbitrary number of systems and states (i.e., in which the frequency of a givendistribution may differ from unity, in contrast to the trivial example shown earlier).

Thus, it turns out that there is no empirical support for the Γ-space descriptionover the Z-space description for classical systems, if they are correctly character-ized as impenetrable, and thus no empirical support for haecceitism as applying toclassical objects—if one identifies that as a criterion for haecceitism, as doesHuggett. Based on the dissent of French and Redhead (1988), from the criteriondescribed earlier for transcendental individuality, these authors could, of course,still argue that both quantum and classical systems possess metaphysical individu-ality. What is off the table, in view of Huggett’s argument, is the idea there is anyempirical support for a fundamental difference between quantum and classicalsystems regarding their status as individuals. While somewhat bewildering for ourintuitions about the difference between classical and quantum objects, we actuallyneed this result. Why? Because, in keeping with the correspondence principle, theclassical (Maxwell-Boltzmann) distribution must (and does) emerge as a limit from

Figure 11.3 States of two classical coins if we forbid them from occupying thesame state.

208 Ruth Kastner

quantum statistics (either Bose-Einstein or Fermi-Dirac). That is, the quantumdistributions transition smoothly into the classical distribution. This calls intoquestion the idea that classical objects have any sort of “digital” or on/offdistinguishability or individuality feature or features that differ from quantumobjects.

Thus, the challenge facing us is that the transition from the quantum domain to theclassical domain seems continuous, not discrete and essential (as in a change ofintrinsic character or essence). This is another puzzling feature of the micro/macrodivide. In the next section, we review the derivations of each kind of distribution andconsider what clues we might find therein to better understand the ontology under-lying the transition between the quantum and classical statistics.

11.3 Classical versus Quantum Statistics

Let us begin by simply listing the three major distributions: the classical Maxwell-Boltzmann, Bose-Einstein for bosons, and Fermi-Dirac for fermions, respectively:

�n MBð Þi ¼ N

e�βεiPje�βεj

(11.1a)

�n BEð Þi ¼ 1

eβ εi�μð Þ � 1(11.1b)

�n FDð Þi ¼ 1

eβ εi�μð Þ þ 1(11.1c)

In the quantum distributions Eq. (11.1b) and Eq. (11.1c), the chemical potential μ(related to the number of degrees of freedom N) necessarily enters for systems witha fixed N. This will turn out to be significant, as we shall see.

Now, recall that classical distributions can only be wavelike or particle-like. Theparticle-like classical distribution (applying to systems such as ideal gases) is justthe Maxwell-Boltzmann distribution Eq. (11.1a). Meanwhile, the classical wavedistribution is what was applied to blackbody radiation prior to the advent ofquantum theory, resulting in the Rayleigh-Jeans distribution and the “ultravioletcatastrophe”:

P εð ÞRJ ¼2ε2kT

hcð Þ2 (11.2)

In view of “wave-particle duality,” it is well known that the quantum distri-butions interpolate between these two extremes, as follows. Consider the

From Quantum to Classical Physics: The Role of Distinguishability 209

correct quantum distribution for electromagnetic blackbody radiation (the“Planck Distribution”):

P εð Þ ¼ 2ε3

hcð Þ21

eε=kT � 1(11.3)

For energies ε small compared to kT, this becomes

P ε � kTð Þ ¼ 2ε3

hcð Þ21

1þ ε=kT þ � � �ð Þ � 1� 2ε3

hcð Þ2kT

ε¼ 2ε2kT

hcð Þ2 , (11.4)

i.e., it yields the Rayleigh-Jeans law Eq. (11.2).On the other hand, for ε large compared to kT, the exponential in the denomin-

ator of Eq. (11.3) swamps the unity term and we get

P ε � kTð Þ / 1

eε=kT(11.5)

which is the Maxwell-Boltzmann distribution, reflecting particle-like (or at leastdiscrete) behavior on the part of the radiation. (To better reveal the basic form ofthe distribution in this limit, we neglect the factors corresponding to the density ofstates for blackbody radiation.)

Thus, we see that the quantum statistics interpolate between wavelike andparticle-like behavior. This is a key aspect of quantum systems as opposed toclassical systems; the latter can be unambiguously categorized as either waves orparticles. In contrast, the quantum statistics must cover both situations in the samedistribution � indicating that they describe entities that are (somehow) both waveand particle.

If we look at the assumptions that go into deriving the various distributions, wecan get some additional clues as to the key differences between the classical andthe quantum situation. For example, the classical Maxwell-Boltzmann distributionfor an ideal gas (e.g., a system of Nmolecules) is obtained from a partition functionassumed to be the direct product of the N individual molecular partition functions.This cannot be done for the quantum statistics for fixed N. We will now considerthose issues in detail.

First, let us recall the general procedure for obtaining a distribution describ-ing the mean number of systems �nr in a given energy state εr. This procedureholds regardless of the type of system considered (whether quantum orclassical).

• The number of degrees of freedom having energy εr is denoted nr.

• Thus, the possible energy states ER of the whole gas (having N particles) are:

ER ¼ n1ε1 þ n2ε2 þ � � � ¼X

rnrεr (11.6)

210 Ruth Kastner

and

N ¼X

rnr (11.7)

At this point, it may already be noted that Eq. (11.6) represents a distribution overthe possible energy states, and in that sense is the “Z-space” representation. Since this isa general derivation (leading to both classical and quantum statistics), it is clear that theZ-space representation is applicable for both cases, reinforcing Huggett’s observation.

Now, for the case in which the total system of N degrees of freedom is taken ascapable of exchanging energy with its environment at temperature T (the “canon-ical ensemble”), the probability that the total system is in the state R is given by:

PR ¼ Ce�βER (11.8)

where β ¼ 1=kT .The constant of proportionality C is 1/Z, where Z is the total system partition

function:

Z ¼X

je�βEj (11.9)

So the probability that the gas is in state R is:

PR ¼ e�βER

Z(11.10)

From this we can find the average number of degrees of freedom in energy state εr:

�nr ¼XR

nrPR ¼PRnr e�βER

Z¼ � 1

β1Z

XR

∂∂εr

e�βPr

nrεr ¼ � 1β1Z

∂Z∂εr

(11.11)

So that in compact form,

�nr ¼ � 1β∂ ln Z∂εr

(11.12)

Again, this is a general result for any partition function Z.Now, for a single degree of freedom with possible energy states εi, the partition

function ζ (i.e., the weighted sum over the possible energy states) is given by:

ζ ¼X

je�βεj (11.13)

So, analogously with Eq. (11.10), the probability that a single system is in state εi is

P εið Þ ¼ e�βεiPje�βεj

¼ e�βεi

ζ(11.14)

From Quantum to Classical Physics: The Role of Distinguishability 211

We make note of this because for a classical gas of N degrees of freedom, one findsthe average number simply by taking Z(N) for the entire gas as the product of theindividual partition functions:

Z Nð Þ ¼ ζN (11.15)

So that, using Eq. (11.12), the distribution for �ni becomes:

�ni ¼ � 1β∂ ln Z Nð Þ

∂εi¼ � 1

βN∂ ln ζ∂εi

¼ Ne�βεiPje�βεj

(11.16)

which is just the Maxwell-Boltzmann distribution Eq. (11.1a).However, one cannot use the expression Eq. (11.15) for quantum systems

that have a constrained number of degrees of freedom N � and this is ofcrucial significance. Instead, one must incorporate the restriction to N by wayof the chemical potential μ, which acts as a Lagrange multiplier. This dictatesthat we are working with the “grand canonical ensemble,” which allows N tovary. The corresponding grand canonical partition function Z

^

is obtained asfollows:

Z^¼

XR

exp �βER½ � exp ½βμN�

¼X

n1, n2, n3, ...exp �β n1ε1 þ n2ε2 þ n3ε3 þ . . .ð Þ½ � exp ½βμN� ¼

¼X

n1, n2, n3, ...exp �β n1ε1 þ n2ε2 þ n3ε3 þ . . .ð Þ½ � exp βμ n1 þ n2 þ n3 þ . . .ð Þ½ � ¼

¼X

n1, n2, n3, ...e�βn1 ε1�μð Þe�βn2 ε2�μð Þe�βn3 ε3�μð Þ . . .

¼X∞n1¼0

e�βn1 ε1�μð Þ ! X∞

n2¼0

e�βn2 ε2�μð Þ ! X∞

n3¼0

e�βn3 ε3�μð Þ !

. . . (11.17)

This is just a product of infinite sums of the formP∞

n¼0 xn ¼ 1= 1� xð Þ, xj j < 1;

and given that μ < εr, we therefore have

Z^¼ 1

1� e�β ε1�μð Þ

� �1

1� e�β ε2�μð Þ

� �1

1� e�β ε3�μð Þ

� �. . . (11.18)

Taking logs of both sides, we get the more useful form:

ln Z^¼ �

X∞r¼0

ln 1� e�β εr�μð Þ� �

(11.19)

212 Ruth Kastner

Then we can use Eq. (11.12) to get the distribution for average occupationnumber �ns:

�ns ¼ � 1β∂ ln Z

^

∂εs¼ � 1

βN

∂∂εi

�X∞r¼0

ln 1� e�β εr�μð Þ� �" #

¼ e�β εs�μð Þ

1� e�β εs�μð Þ ¼1

eβ εs�μð Þ � 1(11.20)

which is the Bose-Einstein distribution.The first thing to notice here (besides the fact that we could not use Eq. (11.15)

to obtain this quantum distribution) is that the total number of degrees of freedom,N, seems to have “disappeared.” It got “dissolved” into infinite sums over all thepossible values of the ns εsð Þ. Thus, ironically, N has to become a variable in orderto be able to “fix” N for a gas of quantum systems. We recover N as the sum overthe average occupation numbers �ns:

N � �N ¼Xs

�ns (11.21)

The situation is similar for fermions, except that they obey the Pauli Exclusionprinciple which limits state occupancy to zero or one. Without going through thederivation here, we note that, given the restriction described earlier on occupancy,the inability to express the partition function Z(N) as a direct product of Nindividual degrees of freedom yields for the mean occupancy number:

�n FDð Þs ¼ 1

eβ εs�μð Þ þ 1(11.22)

which is the Fermi-Dirac distribution.Thus, for both bosons and fermions, the chemical potential μ is involved in a

crucial, nonseparable way. Its relation to N is fixed by (11.21), i.e.,

N ¼ �N ¼Xs

1

eβ εs�μð Þ 1(11.23)

Does the chemical potential play any role in the classical case? Yes, but onlytrivially, as a normalizing factor. In the “dilute” (low-occupancy) limit yielding theclassical case, the exponential factor involving μ approaches the particle number Ndivided by the single-particle partition function, i.e.,

eβμ ! N

ζ¼ NP

je�βεj

(11.24)

From Quantum to Classical Physics: The Role of Distinguishability 213

So that the Maxwell-Boltzmann distribution can be expressed in terms of μ as:

�n MBð Þi ¼ N

ζe�βεi ¼ eβμe�βεi ¼ 1

eβ εi�μð Þ (11.25)

In this form, it is easy to see that the classical case emerges from the quantumdistributions when εi � μ for all i.

In summary, we make the following observations based on the derivations of therespective distributions. For a classical system comprising N degrees of freedom,we can simply assume that N is fixed, and use the “canonical ensemble” to obtainthe distribution. For that purpose, we can express the total canonical partitionfunction Z(N) as simply the product of the individual partition functions ζ i for thecomponent degrees of freedom.

However, for a quantum system with fixed N (and nonvanishing mass), wecannot use the canonical ensemble; we must use the “grand canonical ensemble”(i.e., partition function Z

^

) – representing a system in contact with both an energyand particle reservoir. That is, we must in-principle allow N to vary. The physicalcontent of this procedure is as follows: The chemical potential μ is a Lagrangemultiplier, representing a constraint force that is present in the quantum case, evenif there is no contact with an external particle reservoir. Thus the natural physicalinterpretation is that the quantum degrees of freedom are imposing a constraintforce on one another that is not present in the classical case.

Based on the previous discussion, what can we conclude about the classical/quantum divide? We cannot treat a collection of N quantum objects as elements ofseparable probability spaces, because in that case we do not obtain the quantum-mechanical statistics. Nonseparability of the spaces confirms that we are dealingwith quantum coherence, with all its attendant features such as entanglement andthe requirement for symmetrization rendering labels superfluous (the latter usuallyand reasonably understood as reflecting indistinguishability). Moreover, themutual constraint of quantum systems expressed by the chemical potential μ (evenat T = 0) reflects a peculiarly quantum sort of physical correlation or interaction notpresent in the classical case, probably expressing the so-called ‘exchange forces’associated with symmetrization (of course, this is a misnomer; there is no real“force” operating here in the usual physical sense).

Thus, our finding is that there is no empirical support for any ‘digital’ on/off formof metaphysical individuality at the classical/quantum border. Since the classicalstatistics are straightforwardly obtainable as a limit from the quantum statistics, andrepresentable in terms of Z-space, we can confirm Huggett’s result that the capacityof classical systems to carry labels that simply permute to form new system states(i.e., new Γ-space configurations) is not reflected in the statistics. Yet clearly, theclassical limit brings with it some sort of new capacity for permutable labeling of the

214 Ruth Kastner

component systems, in that the collective partition function can be obtained fromindividual partition functions ζ i in-principle capable of carrying the permutablelabel i. This indicates that in the classical limit, the component systems acquire aform of distinguishability. In the next section, we investigate the nature of thisemergence, in the classical limit, of the capacity to carry a label.

11.4 Whence Quasi-Classical Distinguishability in the Dilute Limit?

The dilute limit, yielding classicality, is known to be obtained in the “smallwavelength limit,” through the use of the so-called thermal wavelength λth. Theusual ways of obtaining the λth condition can be criticized for conflating classicaland quantum quantities. For example, one typical method for deriving λth is bytreating it as a kind of quantum-mechanical position uncertainty Δx, correspondingto the root-mean-square uncertainty ΔpRMS of the momentum of the componentdegrees of freedom (at a given temperature T). That is, one starts with the expression:

ΔpRMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2h i � ph i2

q(11.26)

The average momentum ph i appearing in Eq. (11.26) is assumed to be zerobecause of “random motion” � thus, it is an average over N independent degreesof freedom, not an expectation value for any quantum state. ΔpRMS is then taken asequal to the square root of the average squared momentum p2

� �, which is obtained

from the equipartition theorem:

p2� � ¼ 3mkT (11.27)

So, from Eq. (11.26) and Eq. (11.27), the quantity ΔpRMS is taken to be:

ΔpRMS ¼ffiffiffiffiffiffiffiffiffiffiffi3mkT

p(11.28)

and this (despite the fact that it is not a real momentum uncertainty but rather aroot-mean-square error) is plugged into the uncertainty relation to obtain a corres-ponding thermal wavelength λth:

λth ¼ hffiffiffiffiffiffiffiffiffiffiffi3mkT

p (11.29)

Clearly, in this context, λth is assumed to be a kind of position uncertainty. It is thendemanded that this be much smaller than the average interparticle spacing d, where:

d ¼ V

N

� �1=3

(11.30)

From Quantum to Classical Physics: The Role of Distinguishability 215

the idea being that this condition makes the gas ‘dilute’ (i.e., no particles everoccupying same position x; and many positions unoccupied, see Figure 11.4.) Sothe thermal wavelength condition for classicality becomes:

hffiffiffiffiffiffiffiffiffiffiffi3mkT

p � V

N

� �1=3

(11.31)

However, as alluded to earlier, the preceding derivation applies a classical, notquantum, uncertainty to momentum in two distinct ways:

• Averaging over N systems assumed to be in random motion to get a vanishingvalue for ph i

• Using the classical Equipartition Theorem to get a value for p2� �

The resulting momentum uncertainty is only an epistemic average over manydifferent phase space points, not an intrinsic quantum uncertainty arising from asingle state having an intrinsic “coarse graining,” represented by a finite-sizedelement of phase space. Therefore, it is arguably not the correct quantity forapplicability of the uncertainty relation between position and momentum. More-over, the derivation of the Equipartition Theorem, which is used to obtain thatroot-mean-square “momentum uncertainty,” presupposes classical Maxwell-Boltzmann statistics!

In addition, the derivation conflates a wavelength with a position uncertainty,which is problematic: If a system has a well-defined wavelength λ, then it hasinfinite position uncertainty. Thus, λth is really being interpreted as the spread of a“wave packet,” despite the fact that the states available to the systems making up

Figure 11.4 One way of picturing the thermal wavelength condition for theclassical limit of a quantum distribution.

216 Ruth Kastner

the gas are not necessarily described by wave packets (i.e., in the quantum limit,they are plane waves).

Of course, it is well known that classical behavior emerges in the small-wavelength limit, so rather than try to pretend that a wavelength is a positionuncertainty, one can simply work with the de Broglie wavelength of the averagemomentum (still using the Equipartition Theorem) to obtain the same conditionEq. (11.31). But in this case, one cannot explain the classical behavior in this limitby saying that the gas is dilute, as pictured in Figure 11.4, because it retains anonlocal character arising from the presumed exact wavelengths of its degrees offreedom. So the question is: Why does condition Eq. (11.31) seem to work so wellas a criterion for the classical limit?

It turns out that if we reexpress Eq. (11.31) in terms of thermal energy kT, wefind the condition (neglecting numerical constants of order unity):

kT � h2

m

N

V

� �2=3

(11.32)

But the quantity on the right hand side is then recognized as the Fermi energy,which is the chemical potential μ for fermions at T = 0:

EF ¼ h2

m

N

V

� �2=3

¼ μ T ¼ 0ð Þ (11.33)

And in fact, this condition kT � EF is also the well-known condition for theclassical limit of the Fermi-Dirac distribution. So what happens in this limit thatcould justify a classical description? Once again, the chemical potential μ Tð Þ hasmuch to teach us.

Specifically, μ is crucially related to the Helmholtz free energy F, defined as

F ¼ U � TS (11.34)

The chemical potential μ is the change in F when adding a degree of freedom (at agiven T and V), i.e.,

μ ¼ ΔFΔN

� �T ,V

(11.35)

Now, μ is of large magnitude and negative in the classical limit of large T, smallN/V, and small λth, as can be seen from the well-known relation (see, e.g., Kelly2002 for details):

μ ���!clas

�kT lnV

Nλ3th

!(11.36)

From Quantum to Classical Physics: The Role of Distinguishability 217

If we can understand the physical significance of the negativity (and largemagnitude) of μ in the classical limit as expressed by Eq. (11.36), we may hopeto gain insight into the ontology of the quantum/classical transition. A largenegative value of μ ¼ ΔF=ΔN means that F decreases significantly when a degreeof freedom is added to the system. Since F ¼ U � TS, increasing the entropy S isthe only way to decrease F. This indicates that the addition of a degree of freedomin the classical limit increases the entropy far more than it increases the internalenergy U. The higher the temperature T, the larger this decrease, and the morenegative μ becomes. Thus, a large negative μ corresponds to a large entropyincrease of the whole system whenever a degree of freedom is added, accompaniedby a negligible increase in internal energy. The same basic situation applies tobosons, although in that case μ can never be positive, and with increasing T, itbecomes more and more negative while more of the higher energy states εs becomepopulated (and the system becomes more dilute in terms of state occupancy).

We can therefore summarize as follows, guided by the clues provided by thebehavior of the chemical potential. In the classical (dilute) limit, adding a degree offreedom results in a statistically independent increase in the overall state-space,increasing the entropy with comparatively small increase in U. In contrast, in thequantum domain, we have two cases: (i) for fermions, adding a degree of freedomincreases U more than it increases TS; and (ii) for bosons, the entropy term (�TS)is always larger in magnitude than U, but the increase in U is nonnegligiblecompared to the increase in the magnitude of TS. The physical origin of therelatively small increase in TS when adding a degree of freedom in the quantumlimit is the following: The new degree of freedom has to find an energy levelcontingent on the preexisting energy level structure, which reduces the availabilityof states that would have been available in the classical case. Thus the entropyincrease (which is a measure of the increase in the number of available states) ismuch smaller than what obtains in the classical case. Once again, the quantumdegrees of freedom “know about each other” and evidently have some form ofinteraction (quantified by the chemical potential), even at T = 0 when there are nothermal interactions at all. They are not independent and separable.

11.5 Conclusions

By examining the derivations of the quantum and classical distributions, we havefound that separability of the individual probability spaces fails in the quantumdomain. In contrast, in the classical limit, the probability spaces of the componentdegrees of freedom are fully separable. In addition, by examining the role of thechemical potential μ, we find a clear manifestation of the highly nonclassicalconstraints that quantum degrees of freedom impose on one another via the so-

218 Ruth Kastner

called exchange forces corresponding to the need for symmetrization. We alsoconfirm, via the behavior of μ, that in the classical limit, adding a degree offreedom gives rise to new energy states for the whole system of N degrees offreedom, independently of the state occupancies of the preexisting N � 1 degreesof freedom � increasing entropy with minimal increase in internal energy.

Thus, classical systems (those obeying Maxwell-Boltzmann statistics) have aform of separability and independence not applying to quantum systems. Thisseparability amounts to distinguishability, since one could in-principle apply labelsto the N individual state-spaces making up the collective state-space (as in aΓ-space representation).

However, it is notable that impenetrability does not come into the picture in anyfundamental way, because we need only consider energy states (not position) inorder to obtain Maxwell-Boltzmann statistics. Nevertheless, what about our intu-ition (also reflected in the usual derivation of the thermal wavelength criterion asillustrated in Figure 11.4) that classical objects do not overlap in spacetime and arefundamentally independent from one another? Einstein addressed this classicalnotion of separability in terms of his “being thus” concept:

An essential aspect of this arrangement of things in physics is that they lay claim, at a certaintime, to an existence independent of one another, provided that these objects ‘are situated indifferent parts of space.’ Unless one makes this kind of assumption about the independenceof the existence (the ‘being-thus’) of objects which are far apart from one another in space�which stems in the first place from everyday thinking � physical thinking in the familiarsense would not be possible. It is also hard to see any way of formulating and testing the lawsof physics unless one makes a clear distinction of this kind.

(Einstein 1948/1971: 170)

Of course, when he made this statement, Einstein was resisting the quantumnonlocality and/or nonseparability that was evident in the context of the famousEPR experiment (Einstein, Podolsky, Rosen 1935). It has since become clear that itis indeed possible, and necessary, to formulate and test the laws of physics withoutrelying on this sort of classical picture at all levels.

We can trace the emergence of Einstein’s “being thus” in the classical limit bynoting that the latter obtains for high thermal energies kT Eq. (11.32). What can bededuced from that depends on one’s interpretation of the quantum formalism. In aunitary-only account, high thermal energies enable decoherence arguments toproceed (Joos and Zeh 1985), although that account has been criticized on thebasis of entanglement relativity and circularity (e.g., Fields 2010, Dugić andJeknić-Dugić 2012, Kastner 2014, 2016a).

Another interpretive approach is to take the projection postulate of vonNeumann as a real physical process (i.e., a “collapse” interpretation). Onesuch approach is actually a different theory from quantum mechanics: The

From Quantum to Classical Physics: The Role of Distinguishability 219

Ghirardi-Rimini-Weber (GRW; Ghirardi, Rimini, and Weber 1985) mechanismrequires an ad hoc modification to the Schrödinger evolution. In contrast, acollapse interpretation that does not change the basic quantum theory is the(Relativistic) Transactional Interpretation (RTI). The original TI, as proposed inCramer (1986), was limited to the nonrelativistic domain and took emitters andabsorbers as primitive. The extension of TI to the relativistic domain (RTI) by thepresent author has allowed a quantitative definition of emitters and absorbers fromunderlying principles (Kastner 2012: chapter 6, Kastner 2016a), and full refutationof the consistency challenge raised by Maudlin (1996) (the refutation is presentedin Kastner 2016b). The RTI takes the advanced states as playing a physical role inmeasurement by breaking the linearity of the evolution and giving rise to the vonNeumann “measurement transition” (Kastner 2012: chapter 3, 2016a). High ther-mal energies kT give rise to frequent inelastic scatterings among the degrees offreedom of the gas and thermal photons. According to RTI, inelastic scatteringscorrespond to collapses, which serve to localize the component degrees of freedom� giving them effective separate and distinct spacetime trajectories, conferringindependence, and thus restoring Einstein’s notion of being thus. Another advan-tage of this approach is to provide a physical grounding for the Second Law ofThermodynamics at the micro-level (see Kastner 2017).

It should also be noted that, under the RTI model with a real nonunitarytransition defining “measurement,” interference truly does disappear uponmeasurement, in contrast to the usual assumption of unitary-only evolution.This resolves the issue alluded to (for example) in French and Redhead (1988),who say:

But, of course, ontologically speaking, ‘interference’ is never strictly absent. That, after all,is what constitutes the ‘problem of measurement’ in QM, so the involvement of everyelectron with the state of every other electron in the universe, although negligible forpractical purposes, remains an ontological commitment of QM, under the interpretationwhere the particles are treated as individuals.

(French and Redhead 1988: 245)

In contrast, under RTI, the previously discussed form of global interference doesvanish upon the nonunitary transition in which a transaction is actualized. Sincetransactions are very frequent in the conditions defining the classical limit, this canbe seen as directly supporting the independence, or being thus, of systems in theclassical limit.

Whichever interpretation one adopts, in the domain of high thermal energies kT,one gets at least effective determinacy of position over time, and thus a uniquespacetime trajectory for each degree of freedom. Such a trajectory confers the capacityfor a unique label and therefore supports distinguishability of the degree of freedom towhich it corresponds. This does not amount to a haecceitistic label, because it is

220 Ruth Kastner

conferred based on qualitative features of the degrees of freedom (i.e., their trajector-ies). Nevertheless, as noted by French and Redhead, one may still regard all systems(classical and quantum) as haecceitistic under a suitable interpretation of individuality.This chapter takes no position on that metaphysical issue.

Acknowledgments

The author is grateful for valuable correspondence from Jeffrey Bub and StevenFrench.

References

Cramer J. G. (1986). “The transactional interpretation of quantum mechanics,”’ Reviews ofModern Physics, 58: 647–688.

Dugić, M. and Jeknić-Dugić, J. (2012). “Parallel decoherence in composite quantumsystems,” Pramana, 79: 199–209.

Einstein, A. (1948/1971). “Quantum mechanics and reality,” pp. 168–173 in M. Born(trans.). The Born-Einstein Letters. London: Walker and Co.

Einstein, A., Podolsky, B., and Rosen, N. (1935). “Can quantum mechanical description ofreality be considered complete?”, Physical Review, 47: 777–780.

Fields, C. (2010). “Quantum Darwinism requires an extra-theoretical assumption ofencoding redundancy,” International Journal of Theoretical Physics, 49: 2523–2527.

French, S. (2015). “Identity and individuality in quantum theory,” in E. N. Zalta (ed.), TheStanford Encyclopedia of Philosophy (Fall 2015 Edition), https://plato.stanford.edu/archives/fall2015/entries/qt-idind/

French, S. and Redhead, M. (1988). “Quantum physics and the identity of indiscernibles,”The British Journal for the Philosophy of Science, 39: 233–246.

Ghirardi, G. C., Rimini, A., and Weber, T. (1986). “Unified dynamics for microscopic andmacroscopic systems,” Physical Review D, 34: 470–491.

Huggett, N. (1999). “Atomic metaphysics,” The Journal of Philosophy, 96: 5–24.Joos, J. and Zeh, D. H. (1985). “The emergence of classical properties through interaction

with the environment,” Zeitschrift für Physik B, 59: 223–243.Kastner, R. E. (2012). The Transactional Interpretation of Quantum Mechanics: The

Reality of Possibility. Cambridge: Cambridge University Press.Kastner, R. E. (2014). “Einselection of pointer observables: The new H-theorem?”, Studies

in History and Philosophy of Modern Physics, 48: 56–58.Kastner, R. E. (2016a). “The Transactional Interpretation and its evolution into the 21st

century: An overview,” Philosophy Compass, 11: 923–932. Preprint version: https://arxiv.org/abs/1608.00660.

Kastner, R. E. (2016b). “The Relativistic Transactional Interpretation: Immune to theMaudlin challenge,” https://arxiv.org/abs/1610.04609

Kastner, R. E. (2017). “On quantum non-unitarity as a basis for the second law ofthermodynamics,” Entropy, 19: 106. Preprint version: https://arxiv.org/abs/1612.08734.

Kelly, J. (2002). “Semiclassical statistical mechanics,” (lecture notes), www.physics.umd.edu/courses/Phys603/kelly/Notes/Semiclassical.pdf

Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell.Maudlin, T. (1996). Quantum Nonlocality and Relativity. Oxford: Blackwell.

From Quantum to Classical Physics: The Role of Distinguishability 221

12

Individuality and the Account of Nonlocality:The Case for the Particle Ontology in Quantum Physics

michael esfeld

12.1 The Measurement Problem

Quantum mechanics is a highly successful theory as far as the prediction, confirm-ation, and application of measurement outcome statistics is concerned. The centraltool for these predictions is the Born Rule, according to which, in brief, the squaredmodulus of the wave function of a quantum system indicates the probability to finda particle at a certain location if a measurement is made. Consequently, themeasurement outcomes show a ψj j2 distribution. Furthermore, all the informationthat we obtain in experiments is knowledge about positions, as Bell (2004: 166)stressed: the measurement outcomes are recorded in macroscopic positions, suchas the positions of dots on a screen, pointer positions, etc. and finally brainconfigurations, and they provide information about where the investigated objectsare. This insight holds whatever observables one defines and measures in terms ofoperators. Thus, for instance, all the information that the outcome of a spinmeasurement of an electron by means of a Stern-Gerlach experiment provides isinformation about the wave packet in which the electron is located, etc.

However, an algorithm to calculate measurement outcome statistics is not aphysical theory. Physics is about nature, physis in ancient Greek. Consequently, aphysical theory has to (i) spell out an ontology of what there is in nature accordingto the theory, (ii) provide a dynamics for the elements of the ontology and (iii)deduce measurement outcome statistics from the ontology and dynamics bytreating measurement interactions within the ontology and dynamics; in order todo so, the ontology and dynamics have to be linked with an appropriate probabilitymeasure. Thus, the question is: What is the law that describes the individualprocesses that occur in nature (dynamics) and what are the entities that make upthese individual processes (ontology)?

Consider as an illustration the double-slit experiment in quantum mechanics:One can do this experiment with individual particles, say prepare a source that

222

emits one particle every morning at 8 a.m. so that one gets an outcome recorded inthe form of exactly one dot on a screen every morning. The question then is whathappens between the emission of the particle from the source and the measurementrecord on the screen. Is there a particle that goes through one of the two slits?A wave that goes through both slits and that contracts afterwards to be recorded asa dot on a screen? Or something else? The constraint on the ontology and dynamicsthat are to answer this question is that they have to account for the characteristicdistribution of the dots on the screen that shows up if one runs this experimentmany times. In other words, the ontology and the dynamics have to explain themeasurement outcome distribution.

It is not possible to infer the law that describes the individual processes thatoccur in nature from the rule to calculate the measurement outcome statistics. Thecharacteristic pattern of the measurement outcome distribution in the double-slitexperiment by no means reveals what happens between the source and the screen.Trying to make such inferences runs into the famous measurement problem ofquantum physics. The measurement outcomes show a ψj j2 distribution, the law forthe evolution of the wave function ψ is the Schrödinger equation, but the Schrö-dinger evolution does, in general, not lead to measurement outcomes. Moreprecisely, the by now standard formulation of the measurement problem is theone of Maudlin (1995: 7):

1.A The wave-function of a system is complete, i.e., the wave-function specifies(directly or indirectly) all of the physical properties of a system.

2.A The wave-function always evolves in accord with a linear dynamical equation(e.g., the Schrödinger equation).

3.A Measurements of, e.g., the spin of an electron always (or at least usually) havedeterminate outcomes, i.e., at the end of the measurement the measuring device iseither in a state which indicates spin up (and not down) or spin down (and not up).

Any two of these propositions are consistent with one another, but the conjunctionof all three of them is inconsistent. This can be easily illustrated by Schrödinger’scat paradox (Schrödinger 1935: 812): If the entire system is completely describedby the wave function, and if the wave function always evolves according to theSchrödinger equation, then, due to the linearity of this wave equation, superpos-itions and entangled states will, in general, be preserved. Consequently, a meas-urement of the cat will, in general, not have a determinate outcome: At the end ofthe measurement, the cat will not be in the state of either being alive or being dead.

Hence, the measurement problem is not just a philosophical problem of theinterpretation of a given formalism. It concerns the very formulation of a consistentquantum theory. Even if one takes (1.A) and (2.A) to define the core formalism of

Individuality and the Account of Nonlocality 223

quantum mechanics and abandons (3.A), one has to put forward a formulation ofquantum physics that establishes a link with at least the appearance of determinatemeasurement outcomes. If one retains (3.A), one has to develop a formulation of aquantum theory that goes beyond a theory in which only a wave function and alinear dynamical equation for the evolution of the wave function figure. Accord-ingly, the solution space for the formulation of a consistent quantum theory can bedivided into many-worlds theories, rejecting (3.A); collapse theories, rejecting(2.A); and additional variable theories, rejecting (1.A).

However, research in the last decade has made clear that we do not face threeequally distinct possibilities to solve the measurement problem, but just two: Themain dividing line is between endorsing (3.A) and rejecting it. If one endorses(3.A), the consequence is not that one has to abandon either (1.A) or (2.A), but thatone has to amend both (1.A) and (2.A). Determinate measurement outcomes asdescribed in (3.A) are outcomes occurring in ordinary physical space, that is, inthree-dimensional space or four-dimensional spacetime. Hence, endorsing (3.A)entails being committed to the existence of a determinate configuration of matter inphysical space that constitutes measurement outcomes (such as a live cat or anapparatus configuration that indicates spin up, etc.). If one does so, one cannot stopat amending (2.A). The central issue then is not whether or not a collapse term forthe wave function has to be added to the Schrödinger equation, because even withthe addition of such a term, this equation still is an equation for the evolution of thewave function, by contrast to an equation for the evolution of a configuration ofmatter in physical space. Consequently, over and above the Schrödinger equation –however amended – a law or rule is called for that establishes an explicit linkbetween the wave function and the configuration of matter in physical space. Bythe same token, (1.A) has to be changed in such a way that reference is made to theconfiguration of matter in physical space and not just the quantum state as encodedin the wave function (see Allori et al. 2008). This fact underlines the point madeearlier: We need both a dynamics, filling in proposition (2.A), and an ontology,filling in proposition (1.A), that specifies the entities to which the dynamicsrefers and whose evolution it describes. The way in which these two interplaythen has to account for the measurement outcomes and their distribution, filling inproposition (3.A).

This point can be further illustrated by a second formulation of the measurementproblem that Maudlin (1995: 11) provides:

1.B The wave function of a system is complete, i.e., the wave function specifies(directly or indirectly) all of the physical properties of a system.

2.B The wave function always evolves in accord with a deterministic dynamicalequation (e.g., the Schrödinger equation).

224 Michael Esfeld

3.B Measurement situations which are described by identical initial wave functionssometimes have different outcomes, and the probability of each possible outcome isgiven (at least approximately) by the Born Rule.

Again, any two of these propositions are consistent with one another, but theconjunction of all three of them is inconsistent. Again, the issue is what the law isand what the physical entities are to which the law refers such that, if one takes(3.B) for granted, configurations of matter in physical space that constitute definitemeasurement outcomes are accounted for.

All mathematical formulations of nonrelativistic quantummechanics work with aformalism in terms of a definite, finite number of point particles and a wave functionthat is attributed to these particles, with the basic law for the evolution of the wavefunction being the Schrödinger equation. The wave function is defined on configur-ation space, thereby taking for granted that the particles have a position in three-dimensional space: For N particles, the configuration space has 3N dimensions sothat each point of configuration space represents a possible configuration of the Nparticles in three-dimensional space. This fact speaks also in quantum physicsagainst regarding configuration space as the physical space, because its dimensiondepends on a definite number of particles admitted in three-dimensional space.

Even if one pursues an ontology of configuration space being the physical spacein quantum physics, one can add further stuff than the wave function to configur-ation space, such as, e.g., the position of a world-particle in configuration space, ortake the wave function to collapse occasionally in configuration space in order tosolve the measurement problem (see Albert 2015: chapters 6–8). However, theproblem remains how to connect what there is in configuration space and itsevolution with our experience of three-dimensional physical objects, their relativepositions, and motions. That experience is the main reason to retain (3.A and 3.B).

This chapter is situated in the framework that envisages abandoning (3.A and 3.B)only as a last resort, that is, only in case it turned out that the consequences of the optionsthat endorse (3.A and 3.B) were even more unpalatable than the ones of rejecting (3.Aand 3.B). It seeks tomake a contribution to assessing the options that are available in thisframework, namely, the option that starts from amending the Schrödinger dynamics byadmitting a dynamics of the collapse of the wave function – (not 2.A and 2.B); see nextsection – and the option that starts from admitting particles, although the informationabout their positions is not contained in the wave function – (not 1.A and 1.B); Section12.3. The chapter closes with a few remarks on quantum field theory (Section 12.4).

12.2 The Collapse Solution and Its Ontology

Against this background, let us take (3.A and 3.B) for granted. On the one hand,the fact that all formulations of nonrelativistic quantum mechanics work with a

Individuality and the Account of Nonlocality 225

formalism in terms of a definite, finite number of point particles and a wavefunction that is attributed to these particles suggests to propose a particle ontologyfor quantum physics. Following this suggestion, the basic ontology – that is, theobjects in nature to which the formalism refers – is the same in classical andquantum mechanics. By contrast, the dynamics that is postulated for these objectsis radically different: There is no wave function in classical mechanics. This viewis supported by the fact that, as mentioned in the previous section, all recordedmeasurement outcomes consist in definite positions of macroscopic, discreteobjects that provide information about where microscopic, discrete objects (i.e.,particles) are located. Furthermore, the measurement instruments are composed ofparticles that, hence, are located where these instruments are.

On the other hand, the dynamics as given by the wave function evolvingaccording to the Schrödinger equation does not describe the evolution of particlepositions: It does not provide for determinate trajectories of individual particles.Even if one starts with an initial condition of precise information about particlepositions, the Schrödinger equation will, in general, describe these particles asevolving into a superposition of different trajectories. Consequently, the Schrö-dinger evolution does not establish an intertemporal identity of these objects: Itfails to distinguish them. Moreover, the Heisenberg uncertainty relations put a limiton the epistemic accessibility of particle positions: Operators for position andmomentum cannot both be measured with arbitrary accuracy on a quantum system.These facts motivate going for another quantum ontology than the one of particles.In brief, if there are no precise particle positions, it makes no sense to maintain aparticle ontology. An object that does not have a precise position is not a particle,but something else. The foremost candidate for that something else is a wave, sincethe Schrödinger equation is a wave equation.

There is a proposal for a quantum ontology that takes the wave equation todescribe a wave in three-dimensional physical space, namely, a continuous matterdensity field (see Ghirardi, Grassi, and Benatti 1995). Determinate measurementoutcomes – as well as the formation of discrete objects in general – are accountedfor in terms of a spontaneous contraction of the matter density field at certainpoints or regions of space. This spontaneous contraction is represented in terms ofthe collapse of the wave function. Consequently, Ghirardi et al. (1995) use amodified Schrödinger dynamics that breaks the linearity and the determinism ofthe Schrödinger equation by including the collapse of the wave function undercertain circumstances – rejection of (2.A and 2.B).

On the quantum dynamics proposed by Ghirardi, Rimini, and Weber (GRW; seeGhirardi, Rimini, and Weber 1986), the wave function undergoes spontaneousjumps in configuration space at random times, distributed according to the Poissondistribution with rate N λ, with N being the particle number and λ being the mean

226 Michael Esfeld

collapse rate. Between two successive jumps, the wave function Ψt evolvesaccording to the usual Schrödinger equation. At the time of a jump thekth component of the wave function Ψt undergoes an instantaneous collapseaccording to

Ψt x1; . . . ; xk; . . . ; xNð Þ↦Lxxk

� �1=2Ψt x1; . . . ; xk; . . . ; xNð Þ

Lxxk

� �1=2Ψt

��������

(12.1)

where the localization operator Lxxk is given as a multiplication operator of the form

Lxxk≔1

2πσ2ð Þ3=2e�

12σ2

xk�xð Þ2 (12.2)

and x, the center of the collapse, is a random position distributed according to the

probability density p xð Þ ¼ Lxxk

� �1=2Ψt

��������2

. This modified Schrödinger evolution

captures in a mathematically precise way what the collapse postulate in thetextbooks introduces by a fiat, namely the collapse of the wave function so thatit can represent localized objects in physical space, including in particular meas-urement outcomes. GRW thereby introduces two additional parameters, the meanrate λ as well as the width σ of the localization operator. An accepted value of λ isof the order of 1015s�1. This value implies that the spontaneous localizationprocess for a single particle occurs only at astronomical time scales of the orderof 1015s, while for a macroscopic system of N~1023 particles, the collapse happensso fast that possible superpositions are resolved long before they would be experi-mentally observable. Moreover, the value of σ can be regarded as localizationwidth; an accepted value is of the order of 10�7m.

A further law then is needed to link this modified Schrödinger equation with awave ontology in the guise of an ontology of a continuous matter density fieldmt xð Þ in space:

mt xð Þ ¼XNk¼1

mk

ðdx1 . . . dxN δ

3 x� xkð Þ Ψt x1; . . . ; xNð Þj j2 (12.3)

This field mt xð Þ is to be understood as the density of matter in three-dimensionalphysical space at time t (see Allori et al. 2008: section 3.1). The thus defined theoryof a GRW collapse dynamics describing the evolution of a matter density field inphysical space is known as GRWm.

Consequently, although GRWm is formulated in terms of particle numbers,there are no particles in the ontology. More generally speaking, there is no plurality

Individuality and the Account of Nonlocality 227

of fundamental physical systems. There is just one object in the universe, namely amatter density field that stretches out throughout space and that has varyingdegrees of density at different points of space, with these degrees changing intime. Hence, there are no individual systems in nature according to this theory sothat the issue of identity and distinguishability of individual quantum systems doesnot arise. That notwithstanding, this theory accounts for measurement outcomesthat appear as individual particle outcomes in terms of a spontaneous contraction ofthe matter density field at certain locations. Position thus is distinguished in theform of degrees of matter density at points of space. It is the only fundamentalphysical property, as Allori et al. (2014) point out:

Moreover, the matter that we postulate in GRWm and whose density is given by the mfunction does not ipso facto have any such properties as mass or charge; it can only assumevarious levels of density.

(Allori et al. 2014: 331–332)

This ontology is not committed to a dualism of an absolute space and time onthe one hand and a matter field that fills that space and that develops in time on theother hand. It can also be conceived as a relationalism about space and time,namely, a field relationalism (see Rovelli 1997 for a field relationalism in thecontext of general relativity theory): There is just the matter density field as anautonomous entity (substance), with an internal differentiation into various degreesof density, and change of these degrees. The geometry of a three-dimensional,Euclidean space then is a means to represent that differentiation, and time is ameans to represent that change. In any case, on this view, matter is a continuousstuff, known as gunk, and it is a primitive stuff or bare substratum that, moreover,admits different degrees of density as a primitive matter of fact. There is nothingthat accounts for the difference in degrees of density of matter in different regionsof space as expressed by the m function in the formalism.

Making the collapse postulate of textbook quantum mechanics precise byamending the Schrödinger equation with the two new parameters λ and σ, indicat-ing the mean rate of spontaneous collapse and the width of the localizationoperator, paradoxically has the consequence that the GRW formalism cannotreproduce the predictions that textbook quantum mechanics achieves by applyingthe Born Rule in all situations. Rather than being a decisive drawback, this,however, opens up the way for testing collapse theories like GRW against theoriesthat solve the measurement problem without the collapse postulate (see Curceanuet al. 2016 for such experiments).

On a more fundamental level, however, it is in dispute whether the ontology of acontinuous matter density field that develops according to the GRW equation issufficient to solve the measurement problem. The reason is the so-called problem

228 Michael Esfeld

of the tails of the wave function. This problem arises from the fact that the GRWformalism mathematically implements the collapse postulate by multiplying thewave function with a Gaussian, such that the collapsed wave function, althoughbeing sharply peaked in a small region of configuration space, does not actuallyvanish outside that region; it has tails spreading to infinity. On this basis, one canobject that GRWm does not achieve its aim, namely to describe measurementoutcomes in the form of macrophysical objects having a definite position (see e.g.,Maudlin 2010: 135–138). However, one can also make a case for the view that thismathematical fact does not prevent GRWm from accounting for definite measure-ment outcomes in physical space (see e.g., Wallace 2014, Egg and Esfeld 2015:section 3).

The main drawback of GRWm, arguably, is its account of quantum nonlocality,which occurs when the wave function collapses all over space. Consider a simpleexample, namely the thought experiment of one particle in a box that Einsteinpresented at the Solvay conference in 1927 (the following presentation is based onde Broglie’s version of the thought experiment in de Broglie 1964: 28–29, and onNorsen 2005). The box is split in two halves that are sent in opposite directions,say from Brussels to Paris and Tokyo. When the half-box arriving in Tokyo isopened and found to be empty, there is on all accounts of quantum mechanics thatacknowledge that measurements have outcomes a fact that the particle is in thehalf-box in Paris.

On GRWm, the particle is a matter density field that stretches over the wholebox and that is split in two halves of equal density when the box is split, thesematter densities travelling in opposite directions. Upon interaction with a meas-urement device, one of these matter densities (the one in Tokyo in the examplegiven earlier) vanishes, while the matter density in the other half-box (the one inParis) increases so that the whole matter is concentrated in one of the half-boxes.One might be tempted to say that some matter travels from Tokyo to Paris;however, because it is impossible to assign any finite velocity to this travel, theuse of the term ‘travel’ is inappropriate. For lack of a better expression, let us saythat some matter is delocated from Tokyo to Paris (this expression was proposedby Matthias Egg; see Egg and Esfeld 2014: 193). For even if the spontaneouslocalization of the wave function is conceived as a continuous process, as inGhirardi, Pearle, and Rimini (1990), the time it takes for the matter density todisappear in one place and to reappear in another place does not depend on thedistance between the two places. This delocation of matter, which is not a travelwith any finite velocity, is a quite mysterious process that the GRWm ontologyasks us to countenance.

Apart from the matter density ontology, there is another ontology availablefor the GRW collapse formalism. This ontology goes back to Bell (2004:

Individuality and the Account of Nonlocality 229

chapter 22, originally published 1987): Whenever there is a spontaneouslocalization of the wave function in configuration space, that development ofthe wave function in configuration space represents an event occurring at apoint in physical space. These point-events are known as flashes; the term‘flash’ was coined by Tumulka (2006: 826). According to the GRW flash theory(GRWf), the flashes are all there is in physical space. Macroscopic objects are,in the terms of Bell (2004: 205), galaxies of such flashes. Consequently, thetemporal development of the wave function in configuration space does notrepresent the distribution of matter in physical space. It represents the objectiveprobabilities for the occurrence of further flashes, given an initial configurationof flashes. Hence, space is not filled with persisting objects such as particles orfields. There only is a sparse distribution of single events. These events areindividual and distinguishable, because they are absolutely discernible by theirposition in space; but there is no intertemporal identity of anything, becausethese events are ephemeral.

GRWf is committed to absolute space and time (or spacetime) as the substancewithin which the flashes occur. There can also be times at which there are noflashes at all. The flashes, again, are bare particulars. There is no informativeanswer to the question of what distinguishes an empty spacetime point from a pointat which a flash occurs: it is a primitive matter of fact that there are flashes at somepoints of spacetime. The flashes are only characterized by their spacetime location.They come into existence at some points of spacetime out of nothing and theydisappear into nothing.

The most serious drawback of GRWf is that this theory covers only thespontaneous appearance and disappearance of flashes, but offers no account ofinteractions, given that there are no persisting objects at all. The idea that motivatesthe GRW collapse dynamics is that a macroscopic object such as a measurementdevice consists of a great number of particles so that the entanglement of the wavefunction of the apparatus with the one of the measured quantum objects will beimmediately reduced due to the spontaneous localization of the wave function ofthe apparatus. However, even if one supposes that a macroscopic object such as ameasurement apparatus can be conceived as a galaxy of flashes (however see thereservations of Maudlin 2011: 257–258), there is on GRWf nothing with which theapparatus could interact: There is no particle that enters it, no matter density, and ingeneral, no field that gets in touch with it either (even if one conceives the wavefunction as a field, it is a field in configuration space and not a field in physicalspace). There is only one flash (standing for what is usually supposed to be aquantum object) in its past light cone, but there is nothing left of that flash withwhich the apparatus could interact. In brief, there simply are no objects that couldinteract in GRWf.

230 Michael Esfeld

12.3 The Particle Solution and Its Dynamics

One may regard the flashes as particles that are deprived of their trajectories, sothat there are only disconnected particle localizations left, being represented by thecollapses of the wave function in GRWf. Nonetheless, of course, particles withouttrajectories and only disconnected point-like localization events is no longer aparticle ontology. Let us therefore now consider a particle ontology for quantummechanics. The quantum theory in that sense is the one going back to de Broglie(1928) and Bohm (1952). Its dominant contemporary version is known as Boh-mian mechanics (BM; see Dürr, Goldstein, and Zanghì 2013). BM is based on thefollowing four axioms:

1. Particle configuration: There always is a configuration of N permanent pointparticles in the universe that are characterized only by their positions x1, . . . , xNin three-dimensional, physical space at any time t.

2. Guiding equation: A wave function is attributed to the particle configuration,being the central dynamical parameter for its evolution. The wave function hasthe task to determine a velocity field along which the particles move, given theirpositions. It accomplishes this task by figuring in the law of motion of theparticles, which is known as the guiding equation:

dxkdt

¼ ℏmk

Imrkψψ

x1; . . . ; xNð Þ (12.4)

This equation yields the evolution of the kth particle at a time t as depending on,via the wave function, the position of all the other particles at that time.

3. Schrödinger equation: The wave function always evolves according to theusual Schrödinger equation.

4. Typicality measure: On the basis of the universal wave function Ψ, a typicalitymeasure can be defined in terms of the Ψj j2-density. Given that typicalitymeasure, it can then be shown that for almost all initial conditions, the distribu-tion of particle configurations in an ensemble of subsystems of the universe thatadmit of a wave function ψ of their own (known as effective wave function) is aψj j2-distribution. A universe in which this distribution of the particles insubconfigurations obtains is considered to be in quantum equilibrium.

Assuming that the actual universe is a typical Bohmian universe in that it is inquantum equilibrium, one can hence deduce the Born Rule for the calculation ofmeasurement outcome statistics on subsystems of the universe in BM (instead ofsimply stipulating that rule). In a nutshell, the axiom of Ψj j2 providing a typicalitymeasure with Ψ being the universal wave function justifies applying the ψj j2-rulefor the calculation of the probabilities of measurement outcomes on particular

Individuality and the Account of Nonlocality 231

systems within the universe, with ψ being the effective wave function of theparticular systems in question (see Dürr et al. 2013: chapter 2).

Axiom (1) defines the ontology of the theory. The universe is a configuration ofpoint particles that, consequently, always have a precise position relative to oneanother: They stand in determinate distance relations to each other. Indeed, BMdoes not require the commitment to an absolute space in which the particles areembedded and an absolute time in which their configuration evolves. The geom-etry and the time with its metric can be conceived as a mere means to represent theparticle configuration and its change, that is, the change in the relative distances ofthe particles (see Esfeld and Deckert 2017: chapter 3.2 for the philosophicalargument, and Vassallo and Ip 2016 for a relationalist formulation of BM).Consequently, the particles are individuals that are absolutely discernible by theirposition in a configuration. They have an identity in time that is provided by thecontinuous trajectory that their motion, i.e., the change in their relative positions,traces out. In the framework of relationalism about space and time, one can employthe distance to individuate the particles, so that the particles in BM are not bareparticulars: The distance relations make them the objects that they are and accountfor their numerical plurality (see Esfeld and Deckert 2017: chapter 2.1).

This ontology contains the core of the solution to the measurement problem thatBM provides: There are always point particles with definite positions, and theseparticles compose the macroscopic objects. Hence, Schrödinger’s cat is alwayseither alive or dead, a radioactive atom is always either decayed or not decayed, anelectron in the double-slit experiment with both slides open always goes eitherthrough the upper or the lower slit, etc. That is to say: There are no superpositionsof anything in nature. Superpositions concern only the wave function and itsdynamics according to the Schrödinger equation – Axiom (3) – but not the matter –the objects – that exist in the world, although, of course, the superpositions in thewave function and its dynamics can be relevant for the explanation of trajectoriesof the matter in physical space.

Axiom (2) then provides the particle dynamics. As it is evident from the guidingequation, the evolution of the position of any particle depends, strictly speaking, onthe position of all the other particles in the universe via the wave function. This isthe manner in which BM implements quantum nonlocality: It is correlated particlemotion, with the correlation being established by the wave function and beingindependent of the distance of the particles. However, it is only correlated motion.By contrast to the collapse dynamics with the wave function representing anontology of a wave in physical space in the guise of a matter density field(GRWm), there is never a delocation of matter in physical space. There are alwaysonly particles, moving on continuous trajectories and thus with a finite velocity inphysical space (in the relativistic setting, a velocity that is not greater than the

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velocity of light), with their motions being correlated with one another as aprimitive matter of fact. The particle ontology and this dynamics provide aconsiderable advantage of the Bohmian account of measurement results over thecollapse one: The particles are always there in space, instead of spontaneouslylocalizing upon the collapse of the wave function – either from nowhere, as theflashes do in GRWf, or being delocated all over space, as the matter density does inGRWm.

The ontology of BM – known as the primitive ontology, that is, the referent ofthe formalism – is given by the particles and their relative distances as well as thechange of these distances (i.e., the particle motion). The wave function is definedby its dynamical role of providing the velocity field along which the particlesmove. The wave function is set out on configuration space. It can be conceived as awave or a field; but, then, it is a wave or field on configuration space by contrast toan entity in physical space over and above the particles: The wave function doesnot have values at the points of physical space. It is therefore misleading toconsider BM as an ontology of a dualism of particles and a wave and to imaginethe wave function as a pilot wave that guides or pilots the particles in physicalspace. These are metaphorical ways of speaking, since the wave function cannot beor represent a wave in physical space.

The wave function is nomological in the sense that it is introduced and definedthrough its dynamical role for the particle motion. Consequent upon its beingnomological in that sense, all the stances in the metaphysics of laws of nature areapplicable to the wave function. In particular, in recent years, a BohmianHumeanism has been developed according to which the universal wave functionis fixed by or supervenes on the history of the particle positions, being a variablethat figures in the Humean best system (see Miller 2014, Esfeld 2014, Callender2015, Bhogal and Perry 2017). This stance is applicable to all the theories thatintroduce the wave function through its dynamical role for the evolution of theconfiguration of matter in physical space, including GRWm and GRWf (seeDowker and Herbauts 2005 for a precise physical model based on GRWf).

This Humean stance makes clear that one can be a scientific realist withoutsubscribing to an ontological commitment to the wave function and without fallinginto instrumentalism about thewave function: The role of thewave function is, in thefirst place, a dynamical one through its position in the law of motion – Axiom (2) –with its role for the calculation of measurement outcome statistics deriving from thatnomological role – Axiom (4). But the laws, including the dynamical parametersthat figure in them, can be the axioms of the system that achieves the best balancebetween simplicity and information in representing the particle motion, as onHumeanism, instead of being entities that exist in the physical world over and abovethe configuration of matter (see Esfeld and Deckert 2017: chapter 2.3).

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The particle positions are an additional parameter in BM in the sense that thewave function and its evolution according to the Schrödinger equation do notcontain the information about the exact particle positions and their evolution. Allthere is to the Bohmian particles are their relative positions, that is, the distancesamong them. Although the parameter of particle mass figures in the guidingequation – Axiom (4) – mass cannot be considered as an intrinsic property of theparticles in BM. It is not situated where the particles are, but rather in thesuperposed wave packets. The same holds for the charge (see e.g., Brown,Dewdney, and Horton 1995, Brown, Elby, and Weingard 1996, and referencestherein; see also, most recently, Pylkkänen, Hiley, and Pättiniemi 2015 and Esfeldet al. 2017).

The particle positions are the only additional parameter. Theorems like the oneof Kochen and Specker (1967) prove that it is not possible to take the operators orobservables of quantum mechanics to have definite values independently ofmeasurement contexts, on pain of violating the predictions of quantum mechanicsfor measurement outcome statistics. Nonetheless, these theorems leave the possi-bility open to admit one additional parameter of the quantum objects that has adefinite value, without the precise information about that value figuring in the wavefunction. The natural choice for the additional parameter then is position, since allmeasurement outcomes consist in macroscopically recorded positions of discreteobjects. In making this choice, one therefore lays the ground for solving themeasurement problem. By contrast, pursuing a strategy that accords definite valuesto different parameters over time – as done in so-called modal interpretations apartfrom BM – falls victim to the measurement problem, as has been proven byMaudlin (1995: 13–14).

It is often maintained that, apart from position, all the other operators orobservables are treated as contextual properties in BM, in the sense that theyacquire a definite value, signifying that they are realized as properties of quantumsystems, only in the context of measurements. But this is wrong-headed. Theoperators or observables defined on a Hilbert space are never properties ofanything. Suggesting that measurements somehow bring into existence propertiesthat are contextual, in the sense that they do not exist independently of measure-ment situations, is a confused manner of talking. The operators or observablesdefined on a Hilbert space are instruments that provide information about how thequantum objects behave in certain situations, and that finally comes down toinformation about how their positions develop. Thus, there is no property of spinthat quantum objects possess, and there is no contextual property of a definitevalue of spin in a certain direction that quantum objects acquire in the context ofmeasurement situations. What these measurements do, essentially, is to provideinformation about particle positions. For instance, the measurement result “spin up”

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of the measurement of an electron provides the information that the particle issituated in the upwards wave packet, and not in the downwards one, etc. (see Bell2004: chapter 4, originally published 1971, and Norsen 2014). A similar remarkapplies to the matter density field in GRWm and the flashes in GRWf; again, theyonly have position, and the operators or observables provide information abouttheir position (i.e., the position of flashes or the matter density at points of space).

Consequently, although if one formulates the relationship of the operators orobservables defined on a Hilbert space in first order logic, the result is that thisrelationship does not fulfill the laws of classical logic; but there is no reason toassume that the world does not conform to the laws of classical logic, on pain ofconfusing operators with properties of objects in the world. BM in particular showswhy there is no problem with classical logic in quantum mechanics. More preciselyand more generally speaking, when it comes to the ontology of what there is in theworld and the dynamics of these entities, one solves the measurement problem bysticking to classical logic, and one would remain trapped by this problem if onewere to abandon classical logic in the formulation of an ontology of quantumphysics.

The upshot of these considerations is this one: In classical physics, one can beliberal about the properties of objects. That is to say, one can take the parametersthat figure in the dynamical equations of classical physical theories to designateproperties of the objects that these theories admit, such as the particles. Quantumphysics teaches us that such a liberal ontological attitude is misplaced: Theparameters that one employs to describe the dynamics of the physical objects –

including, in particular, their behavior in measurement situations – cannot withoutfurther reflection be attributed as properties to the objects. This holds even for theclassical parameters of mass and charge, which are, in quantum physics, situated atthe level of the wave function and, hence, cannot be considered as intrinsicproperties of the objects, although their value remains constant.

Quantum physics thereby teaches us that it is mandatory to draw the followingdistinction: On the one hand, there is the basic or primitive ontology of the theory,namely the referents of the formalism, which are supposed to be simply there innature, such as particles, which are then only characterized by what is necessaryand minimally sufficient for something to be a particle, namely relative positions(the same holds for the flashes in GRWf and the positions of matter density valuesin GRWm). On the other hand, there is the dynamical structure of the theory. Alland only the parameters that are introduced in terms of the functional or causal rolethat they exercise for the evolution of the elements of the basic or primitiveontology belong to the dynamical structure. They are nomological in the sensethat they are there to perform a certain role for the evolution of the referents of thetheory through their place in the laws of that evolution (but, of course, they are not

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themselves laws – not even the universal wave function in BM is a law, because thetheory admits of models with different universal wave functions). It then depends onthe stance that one takes with respect to laws whether or not one accords thedynamical structure a place in the ontology over and above the basic or primitiveontology or takes it merely to be a means of representation (as on Humeanism).

Not only the operators in quantum mechanics, but also the classical parametersof mass and charge belong to the dynamical structure. Mass and charge are alsointroduced in classical mechanics through their functional role for the particlemotion (see Mach 1919: 241 on mass in Newtonian mechanics). Consequently,also in classical mechanics, the distinction is available between, on the one hand,the primitive ontology of the theory in the guise of particle positions and particlemotion and, on the other hand, the dynamical structure in terms of mass andcharge, forces and fields, and energy and potentials introduced through their causalrole for particle motion. Thus, also in classical mechanics, one can suspend orrefuse an ontological commitment to the dynamical structure, being committedonly to the particle positions and their change (see Hall 2009: section 5.2 andEsfeld and Deckert 2017: chapter 2.3). In short, quantum physics simply makesevident that it is mandatory to draw a distinction that was already there from thevery beginning.

Although the particle positions make up the ontology in BM, there are limitsto their accessibility. These limits are given in Axiom (4), which implies thatBM cannot make more precise predictions about measurement outcomes onsubsystems of the universe than those generated by using the Born Rule. Thelink between the dynamical laws of BM and the Ψj j2-density on the level of theuniversal wave function as typicality measure is at least as tight as the linkbetween the dynamical laws of classical mechanics in the Hamiltonian formu-lation and the Lebesgue measure (see Goldstein and Struyve 2007). Indeed, inBM, the quantum probabilities have the same status as the probabilities inclassical statistical mechanics: They enter into the theory as the answer to thequestion of what evolution of a given system we can typically expect insituations in which the evolution of the system is highly sensitive to slightvariations of its initial conditions, and we do not know the exact initial condi-tions. In such situations, the deterministic laws of motion cannot be employedto generate deterministic predictions. Nonetheless, the probabilities are object-ive: They capture patterns in the evolution of the objects in the universe thatshow up when one considers many situations of the same type, such as manycoin tosses in classical physics or running the double-slit experiment with manyparticles in quantum physics. To put it in a nutshell, the Bohmian universe islike a classical universe in which not the motion of the planets, but the coin tossis the standard situation.

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However, one may wonder whether, if one lived in a classical universe of cointosses, one would endorse the Hamiltonian laws as providing the dynamics of thatuniverse; Hamiltonian mechanics could then well be a minority position like BMtoday. The reason for endorsing Hamiltonian mechanics in that classical casewould be the same as the one for endorsing BM in the quantum case: We neednot only statistical predictions, but also a dynamics that describes the individualprocesses occurring in nature, on pain of falling into a measurement problem in theguise of the inability to account for the occurrence of determinate measurementoutcomes. That is to say: In a classical universe of coin tosses, the Hamiltonianlaws would be useless for predictions, as the calculation of Bohmian particletrajectories is useless for predictions. But both are indispensable for physics as atheory of nature. This is not only philosophical ontology, it is the business ofphysics to provide dynamical laws that apply to the individual processes in nature.

Nonetheless, in contrast to the relationship between classical statisticalmechanics and classical mechanics, there is a principled limit to the accessibilityof initial conditions of physical systems in the quantum case. That limit becomesevident, for instance, in the famous Heisenberg uncertainty relations. It is trivialthat measurement is an interaction, so that the measurement changes the measuredsystem, and can thus not simply reveal the position and velocity that it hadindependently of the measurement interaction. Some limit to the accessibility ofphysical systems may follow from the triviality that any measurement is aninteraction. Thus, it is well known in classical physics that no observer withinthe universe could obtain the data that Laplace’s demon would need for itspredictions. However, the quantum case is not simply an illustration of thattriviality, since there is a precise principled limit of the epistemic accessibility ofquantum systems (as illustrated, for instance, by the Heisenberg uncertaintyrelations). In BM, this principled limit follows from applying the theory as definedearlier to measurement interactions (see Dürr et al. 2013: chapter 2).

In the last resort, of course, it is the particle motion in the world that makespossible stable particle correlations such that one particle configuration (say ameasurement device or a brain) records the position and traces the motion of otherparticles and particle configurations, and it is also the particle motion in the worldthat puts a limit on such correlations. The laws of BM, including the typicalitymeasure and the assumption that the actual universe is a typical Bohmian universe,bring out these facts about the actual particle motion. Again, the wave function inconfiguration space represents that particle motion; it is not the wave function thatputs the limit on the epistemic accessibility of the particle positions, although weunderstand that limit by representing the particle motion through the wave func-tion. Instead of taking this limit to be a drawback and hoping for a physical theorylike classical mechanics in which there are paradigmatic cases of deterministic

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laws of motion enabling deterministic predictions (e.g., the motion of the planets),it is fortunate and by no means trivial that there are such stable particle correlationsin the universe at all so that we can represent actual particle positions and motionsand make reliable predictions.

As these considerations make clear, posing a limit to the accessibility of theobjects in physical space is by no means a feature that is peculiar to BM. Such alimit applies not only to the particles in BM, but, for instance, also to the flashdistribution in GRWf and the matter density field in GRWm (see Cowan andTumulka 2016). In any case of a quantum ontology of objects in physical space, ifthese objects were fully accessible, we could employ the thus gained informationto exploit quantum nonlocality for superluminal signaling. This limited accessibil-ity of the particle configuration, the flash distribution or the matter density fieldconfirms that if one endorses proposition (3.A and 3.B) of the measurementproblem, i.e., determinate measurement outcomes whose statistical distributionsare given by the Born Rule, the situation is not that one has to reject eitherproposition (1.A and 1.B) or proposition (2.A and 2.B). One has in this case tomodify both proposition (1.A and 1.B) and proposition (2.A and 2.B). If one startsfrom admitting particle positions that are not revealed by the wave function –

rejection of proposition (1. A and 1.B) – one can retain the Schrödinger dynamicsfor the wave function – proposition (2.A and 2.B) – but then this is not thecomplete dynamics: The central dynamical law is the law of the evolution of theadditional variables, namely, the guiding equation that tells us how the particlepositions evolve in physical space. If one starts from amending the Schrödingerequation by collapse parameters – rejection of proposition (2.A and 2.B) – one canretain the wave function and its dynamics as describing the evolution of the objectsin physical space – flashes, matter density field; proposition (1.A and 1.B) – butthat distribution then nevertheless is “hidden” in the sense that it follows from thetheory that it is not fully accessible.

At least three conclusions can be drawn from this situation:

1. Any theory that admits definite measurement outcomes distinguishes position –be it particle positions, positions of flashes, or values of matter density at pointsof space. All the other observables are accounted for on this basis.

2. If one is not prepared to accept a principled limit to the epistemic accessibilityof the objects in physical space, one remains trapped by the measurementproblem, because one then does not have a dynamics at one’s disposal thataccounts for determinate measurement outcomes.

3. The solution space to the measurement problem reduces to this one: Either oneabandons determinate measurement outcomes in physical space, in which caseone can retain the propositions of the wave function being complete and its

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evolving always according to the Schrödinger equation; one then has to comeup with an Everett-style account of why it appears to us as if there weredeterminate measurement outcomes in physical space. Or, one retains determin-ate measurement outcomes in physical space, and the account of these meas-urement outcomes then commits one to endorsing a distribution of objects inphysical space whose evolution cannot be given by the Schrödinger equationand is not fully accessible.

BM can be seen as the answer to the following question: What is the minimaldeviation from classical mechanics that is necessary in order to obtain quantummechanics? BM shows that the physical ontology can remain the same – pointparticles moving on continuous trajectories – and that the status of probabilities canremain unchanged. What has to change is the dynamics, that is, a wave functionparameter has to be introduced with the wave function binding the evolution of theparticle positions together independently of their distance in physical space.

That notwithstanding, this conceptualization of quantum nonlocality is likeNewtonian gravitation, in that there is never any matter instantaneously delocatedin space. As in Newtonian gravitation, the distribution of the particle positions,velocities, and masses all over space at any time t fixes the acceleration of theparticles at that t, so in BM the distribution of the particle positions and theuniversal wave function at any time t fix the velocity of the particles at that t. Ofcourse, Newtonian gravitation concerns all particles indiscriminately and dependson the square of their distance, whereas quantum nonlocality is de facto highlyselective, i.e., concerns de facto only specific particles, and is independent of theirdistance. Nonetheless, in both cases, nonlocality means that there are correlationsin the particle motion without these correlations being mediated by the instantan-eous transport of anything all over space. There is no reason to change more.Doing so only leads to unpalatable consequences beyond the quantum nonlocalitywith which one has to come to terms anyway.

12.4 Permanent Particles in Quantum Field Theory

The same conclusions apply to quantum field theory (QFT). Let us briefly point outwhy (for details see Esfeld and Deckert 2017: chapter 4.2). The measurementproblem hits QFT in the same way as quantum mechanics (see Barrett 2014).Again, we have a highly successful formalism to calculate measurement outcomestatistics at our disposal. However, the measurement problem as formulated byMaudlin (1995) arises as soon as it comes to accommodating measurementoutcomes in physical space, for this formalism does as such not include a dynamicsthat describes the individual processes in nature that lead to determinate

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measurement outcomes. In particular, only the formalism to calculate measurementoutcome statistics is Lorentz-invariant (i.e., it is irrelevant for it in which temporalorder space-like separated measurement outcomes occur). But we do not have arelativistic, Lorentz-invariant dynamics of the individual processes at our disposalthat lead to determinate measurement outcomes in physical space (despite whatmay look like claims to the contrary in the context of GRWf and GRWm; seeTumulka 2006 and Bedingham et al. 2014 for these claims; see Barrett 2014 andEsfeld and Gisin 2014 for pointing out their limits).

Despite its name, QFT is not an ontology of fields. The fields in the formalismare operator valued fields, by contrast to fields that have determinate values at thepoints of physical space. In the standard model of particle physics, fields are thereto model the interactions (i.e., the electromagnetic, the weak, and the stronginteraction, without gravitation). If one endorses an ontology of fields in physicalspace, then the problem is, like in GRWm for quantum mechanics, to formulate acredible dynamics of the contraction of fields so that they can constitute determin-ate measurement outcomes and, in general, discrete macroscopic objects.

However, a particle ontology also faces obstacles in QFT, namely, new obs-tacles that do not arise in quantum mechanics: In QFT, measurement records notonly fail to keep track of particle trajectories but, moreover, they fail to keep trackof a fixed number of particles. Also in QFT, as in any other area of physics, theexperimental evidence recorded by the measurement devices is particle evidence.However this evidence includes what appears to be particle creation and annihila-tion events, so that there seems to be no fixed number of particles that persist.

Nonetheless, these experimental facts as such do not entitle any inferences forontology. More precisely, as it is a non sequitur to take particle trajectories to beruled out in quantum mechanics due to the Heisenberg uncertainty relations, so it isa non sequitur to take permanent particles moving on definite trajectories accordingto a deterministic law to be ruled out in QFT due to the statistics of particle creationand annihilation phenomena. In both cases, the experimental evidence leaves openwhether the particle trajectories do not exist or are simply not accessible inmeasurements, and whether the particles come into being and are annihilated, orit is simply not possible to keep track of them in the experiments. These issueshave to be settled by the theory. The standard for assessing the theory is thesolution to the measurement problem.

It is possible to pursue a Bohmian solution to the measurement problem in QFTalong the same lines as in quantum mechanics. As BM has no ambition to improveon the statistical predictions of measurement outcomes in quantum mechanics, butdeduces these predictions from the axiom that the universe is in quantum equilib-rium, so there is no ambition that a Bohmian solution to the measurement problemin QFT can resolve the mathematical difficulties that QFT currently faces. That is

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to say: Bohmian QFT has to rely on cut-offs, as does the standard model of particlephysics when it comes to dynamical laws of interactions (by contrast to scatteringtheory). Given appropriate cut-offs, one can formulate a Bohmian theory for QFTin the same way as for quantum mechanics: On what is known as Bohmian Diracsea QFT, the ontology is one of a very large, but finite and fixed number ofpermanent point particles that move on continuous trajectories as given by adeterministic dynamical law (guiding equation) by means of the universal wavefunction.

More precisely, one can define a ground state for these particles that is a state ofequilibrium. This state is one of a homogeneous particle motion, in the sense thatthe particle interactions cancel each other out. Consequently, the particle motion isnot accessible. This state corresponds to what is known as the vacuum state.However, on this view, it is not at all a vacuum, but a sea full of particles (knownas the Dirac sea) in which the particles are not accessible. What is accessible, andwhat is effectively modeled by the Fock space formalism of calculating measure-ment outcome statistics, are the excitations of this ground state that show up inwhat appears to be particle creation and annihilation events. Again, by defining atypicality measure on the level of the universal wave function, one can derive thepredictions of measurement outcome statistics in the guise of, in this case, statisticsof excitation events from the ground state. Thus, again, the quantum probabilitiesare due to a – principled – limit to the accessibility of the particle motion (see Colinand Struyve 2007, Esfeld and Deckert 2017: chapter 4.2).

When pursuing a solution to the measurement problem in terms of an ontologyof particles, it is worthwhile to go down all the Bohmian way also in QFT. What isknown as Bell-type Bohmian QFT (Bell 2004: chapter 19, and further elaboratedon in Dürr et al. 2005) goes only half the way down: On this theory, the particlescome into and go out of existence with statistical jumps between sectors ofdifferent particle numbers in the dynamics. However, this proposal amounts toelevating what is known as quasi-particles that are dependent on the contingentchoice of a reference frame to the status of particles in the ontology. Furthermore, itis committed to absolute space as the substance in which the particles come intoand go out of existence. The Bell-type (quasi-) particles are much like the GRWfflashes, apart from the fact that they can persist for a limited time, instead of beingephemeral. By contrast, on the Bohmian Dirac sea theory, the particles arepermanent so that they can be conceived of as being individuated by the distancesamong them in any given configuration and as having an identity in changethrough the continuous trajectories that their motion trace out. Consequently, thereis no need for a commitment to a surplus structure in the guise of absolute spaceand time in the ontology; there is no need for a medium in which the particles exist(viz. come into and go out of existence). Probabilities then come in through linking

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the deterministic dynamics with a typicality measure. Filling negative energy stateswith particles is no problem on this ontology, because the only property of theparticles is their position; energy is not a property of anything, but a variable in theformalism to track particle motion.

In sum, on the Bohmian Dirac sea ontology, the account of measurementoutcomes is of the same type as in Bohmian quantum mechanics, with thedifference that there are many more particles in the sea than one would expect inan ontology of particle positions that are only given by the distances among theparticles (corresponding to empty space in the representation in terms of a space inwhich the particles are embedded). The account again has two stages: The ontol-ogy of particles – in this case, the excitations of particles against the background ofthe particle motion in the Dirac sea – accounts for the presence of the measuredquantum objects as well as the one of the macroscopic systems; the latter areconstituted by these particle excitations, with the particle dynamics that yieldsthese excitations explaining their stability. The measurement outcome statisticsthen are accounted for in terms of the limited accessibility of the quantum particlesby means of defining a typicality measure from which one then deduces theformalism to calculate these statistics.

In any case, the objects that one poses in an ontology of quantum physics aretheoretical entities. They are admitted to explain the phenomena as given by themeasurement outcome statistics. That is why the solution to the measurementproblem is the standard for assessing these proposals. In any case, if one admitsquantum objects in physical space beyond the wave function, there is a limit totheir accessibility; the wave function has, in this case, an exclusively dynamicalstatus, namely, yielding the dynamics for these objects. The Bohmian solution tothe measurement problem provides the least deviation from the ontology ofclassical mechanics that is necessary to accommodate quantum physics, both inthe case of quantum mechanics and in the case of quantum field theory. There is nocogent reason to go beyond that minimum.

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Ghirardi, G. C., Pearle, P., and Rimini, A. (1990). “Markov processes in Hilbert space andcontinuous spontaneous localization of systems of identical particles,” PhysicalReview A, 42: 78–89.

Ghirardi, G. C., Rimini, A., and Weber, T. (1986). “Unified dynamics for microscopic andmacroscopic systems,” Physical Review D, 34: 470–491.

Goldstein, S. and Struyve, W. (2007). “On the uniqueness of quantum equilibrium inBohmian mechanics,” Journal of Statistical Physics, 128: 1197–1209.

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Kochen, S. and Specker, E. (1967). “The problem of hidden variables in quantummechanics,” Journal of Mathematics and Mechanics, 17: 59–87.

Mach, E. (1919). The Science of Mechanics: A Critical and Historical Account of ItsDevelopment, 4th edition. T. J. McCormack (trans.). Chicago: Open Court.

Maudlin, T. (1995). “Three measurement problems,” Topoi, 14: 7–15.Maudlin, T. (2010). “Can the world be only wave-function?” pp. 121–143 in S. Saunders,

J. Barrett, A. Kent, and D. Wallace (eds.), Many Worlds? Everett, Quantum Theory,and Reality. Oxford: Oxford University Press.

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337–348.Pylkkänen, P., Hiley, B. J., and Pättiniemi, I. (2015). “Bohm’s approach and individu-

ality,” pp. 226–246 in A. Guay and T. Pradeu (eds.), Individuals Across the Sciences.Oxford: Oxford University Press.

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244 Michael Esfeld

13

Beyond Loophole-Free Experiments:A Search for Nonergodicity

alejandro a. hnilo

13.1 Introduction

Quantum mechanics (QM) has been controversial since its very inception. In theclimactic point of a famous debate with Niels Bohr, Albert Einstein argued that thecorrelations between measurements performed on distant entangled particles dem-onstrated that the description of physical reality provided by QM was incomplete(note the subtle difference: There was no objection on the completeness of thetheory, it was the description of physical reality that was claimed to be incom-plete). Bohr answered that the idea of a “physical reality” independent of anobserver was meaningless. In my opinion, Einstein’s argument was impeccableand Bohr’s reply was dangerously close to Saint Bellarmino’s refutation toGalileo’s observation of mountains in the moon. Yet, quite incomprehensibly tome, physicists’ public opinion gave the reason to Bohr. Using a football analogy, itwas a beautiful Einstein’s goal disallowed by off-side (the most arcane of foot-ball’s rules, see Figure 13.1). The history of QM would have been very different ifBohr had replied in 1935: “Wow Albert, you have a good point. I don’t know. Butlet’s use this new theory. There is a lot of exciting work to do. Perhaps, byapplying the theory to new problems, this issue of completeness will becomeclearer.”

Bohr’s actual answer or point of view (the so-called Copenhagen interpretationof QM) opens the door to paradoxes that even have negative social consequences.I quote E. T. Jaynes’ opinion (Jaynes 1980):

Defenders of the (quantum) theory say that this notion (“real physical situation”) isphilosophically naïve, a throwback to outmoded ways of thinking, and that recognitionof this constitutes deep new wisdom about the nature of human knowledge. I say that itconstitutes a violent irrationality, that somewhere in this theory the distinction betweenreality and our knowledge of reality has become lost, and the result has more the characterof medieval necromancy than science.

(Jaynes 1980: 42)

245

In fact, there is an increase of what can be called “quantum mysticism” in socialmedia and even in politics, supporting a magical conception of the world derivedfrom the claimed influence of the consciousness in the results of observations. Ofcourse, many serious scientists who defend with intellectual honesty the Copen-hagen interpretation dislike this unexpected, and undesired, outcome. And ofcourse, social consequences have nothing to do with the truth of an idea in physics.

But the ontological debate is not the only controversial feature of QM. Other,more technical issues are: (i) the failure of the “principle of correspondence” forchaotic (classical) trajectories, (ii) the collapse of quantum field theory if a (eveninfinitesimal) deviation from the principle of superposition is introduced (Gisin1990), and (iii) the well-known “measurement problem,” i.e., QM is confessedlyunable to describe what happens in a single measurement. Note that these issuesare related, more or less directly, to nonlinear effects.

Before going on, I would like to emphasize the importance I give to the issue(ii). In my opinion, the essential features of a good theory must be robust against asmall change in any of its parameters. Where it is not, the resulting “structuralinstability” of the theory is a clue that some new physics may be lurking there. My

Figure 13.1 Einstein’s goal in the “EPR paradox” argument was disallowed by thereferee of public opinion. By the way, Bohr did play as goalkeeper (and a verygood one) in his youth.

246 Alejandro A. Hnilo

favorite example is extracted from a commentary in Goldstein’s superb textbook(Goldstein 1950) and refers to Hamilton’s principle: If Hamilton had speculatedthat the variation of the Lagrangian time integral was not exactly zero, but a smallquantity h, he would have “discovered” QM a century earlier. Goldstein sensiblyargues that Hamilton had no experimental reason, in the first half of the ninteenthcentury, to suppose that variation to take any value but zero. However, Hamiltonmight have noted that assuming that value to be infinitesimally nonzero led to atheory completely different from what he knew. In other words, he would havefound that classical mechanics was structurally unstable at the point (in parameters’space) h = 0. If he had used, as a general principle, that a satisfactory theory mustbe structurally stable, he would have revealed classical mechanics as the limit of amore general theory (QM) that “embeds” it. Following the same reasoning,I believe the structural instability of quantum field theory in the point μ = 0 (whereμ is some general nonlinear parameter) to be a clue that new physics are hiddenthere. But, as Goldstein wisely remarked, there is no experimental evidence (yet) tostep into that more general, nonlinear theory. Essentially, this chapter deals withone of the possible ways to find that evidence.

An appealing solution to the previously mentioned controversial features isinterpreting QM as a statistical theory. According to this interpretation, it is notphysical reality that depends on the information available to the observer, but thepredictions of the QM theory. This dependence is an expected feature of anystatistical, probability-based, incomplete-knowledge description. It follows thatthere must exist a more complete and still unknown description or theory, fromwhich QM is able to provide only the statistically averaged results. Hence, QMwould not provide a complete description of physical reality, in agreement withEinstein’s opinion. The statistical interpretation has been developed in recent timesby Ballentine (1998). Yet, the Copenhagen interpretation of QM is widelyaccepted.

There is a sound reason for this acceptance nowadays (although not in 1935).An important step forward in the debate was the derivation by J. S. Bell ofmeasurable boundaries (Bell’s inequalities) that any classical theory must obeyand that are violated by some QM predictions (see e.g., Clauser and Shimony1978). This result moved a purely theoretical discussion into the realm of experi-mental physics. Bell’s experiments measure the correlations between the results ofobservations performed in two remote entangled particles. If a reality independentof an observer exists and there are no instantaneous interactions at a distance (thesetwo assumptions together often receive the name of local realism, or LR), then thecorrelations cannot be higher than a certain number, which depends on theparticular inequality used. The mere derivation of the inequalities demonstratesthat QM (as we know it, at least) is incompatible with LR. Early experiments

Beyond Loophole-Free Experiments: A Search for Nonergodicity 247

measured a violation of the inequalities, confirming QM predictions and hencerefuting the validity of LR in nature. This is a most relevant result, for LR isassumed not only in everyday life, but also in all the scientific practice (except ifQM is involved, of course). Those experiments disproved or, at least, reducedmuch of the appealingness of the statistical interpretation. For some time, experi-mental imperfections (generally called logical loopholes) left some room forhoping that the inequalities were not violated in nature after all. In recent years,a series of experiments with improved techniques practically closed all the loop-holes (for a critical review, see Hnilo 2017a). Yet the consequences of abandoningLR in nature are so serious and deep, that it is sensible to double-check if somehypothesis, additional to LR, has slipped inadvertently into the reasoning.

In fact, a detailed analysis on how Bell’s inequalities are measured (rather thanderived) shows that there is at least one such additional hypothesis involved, whichI call “ergodicity” (as shorthand) in what follows. In order to not to interrupt theflow of this introduction, that analysis is reviewed in the next section. The issue isalso discussed in detail in Hnilo (2013, 2014, 2017b); see also Khrennikov (2017).In few words: It is commonly believed that the following logical relationshipholds:

Locality + Realism ) Bell’s inequalities are valid.

The analysis of how measured numbers are inserted into Bell’s inequalities showsthat the actual logical relationship is:

Locality + Realism + “ergodicity” ) Bell’s inequalities are valid.

That naturally leads to speculating that it is “ergodicity,” and not LR, that has beendisproved by the recent loophole-free experiments. It is convenient to recall herethat the ergodic hypothesis means that the average of the dynamical variables (ofthe system being considered) calculated over the phase space, which is calledensemble average, is equal to the average obtained over the actual evolution of thesystem, or time average. The interest of the ergodic hypothesis is that ensembleaverages are far easier to calculate than time averages. The former can be deducedfrom the system’s symmetry or from laws of conservation. The latter requires thecomplete solution of the equations of motion of the system. But the ergodichypothesis is not always valid. The Fermi-Pasta-Ulam system (a series of couplednonlinear oscillators) is the best known example of a nonergodic system. The ideathat ergodicity may be involved in the QM vs. LR controversy is not new. It wasindicated long ago by V. Buonomano (1978). It has probably passed mostlyunnoticed because he linked the nonergodic possibility (which is a general argu-ment) to specific mechanisms, or loopholes, which always have a conspiratorialflavor. The issue has been updated by Khrennikov (2017 and references therein).

248 Alejandro A. Hnilo

I stress that I use “ergodicity” here as a shorthand for naming a set of hypothesesthat retrieve the validity of the usual form of the Bell’s inequalities. In this set,some hypotheses are weaker and others stronger than strictly speaking ergodicity.The origin and meaning of the hypotheses in the set have been discussed in detailin Hnilo (2013). In that paper, an example of a LR model relevant to the Bell’sexperiment, which is able to violate ergodicity (in its broadest sense), is alsopresented. The main conclusion is that nonergodicity (in its broadest sense) is acondition necessary, but not sufficient, to violate Bell’s inequalities without violat-ing LR. Be aware that that model was devised as an example of how ergodicity canbe reasonably violated, but that it does not survive (it was not intended to survive)the performed loophole-free Bell’s experiments.

The pertinent question now is: How to determine whether ergodicity is violatedor not in a Bell’s experiment? Pragmatically speaking, randomness ) ergodicity,hence nonergodicity ) nonrandomness (I leave aside the precise definition of“randomness,” which is a difficult issue). In consequence, the study of the devi-ations from randomness in a time series of measurements in a Bell’s experimentmay reveal that it is ergodicity (instead of LR) that has been refuted in the recentloophole-free experiments. It is worth noting here that a Bell’s experiment isequivalent to the “quantum link” of a quantum key distribution (QKD) setup.Therefore, such deviation from randomness may imply a QKD vulnerability of afundamental origin. This possibility has practical consequences.

In summary, the key to test this way to save the validity of LR in nature is thesearch for evidence of nonergodic dynamics in Bell’s experiments. This evidencecannot be found by measuring average magnitudes, as it was done in almost allBell’s experiments performed until now, but by analyzing the time evolution ofthese magnitudes. This requires time-resolved acquisition of data, a procedure alsoknown as “stamping” or “tagging” the time values when the entangled particles aredetected. If the analysis of the time series obtained in this way showed theexistence of nonergodic dynamics, QM could be then interpreted as the steady-state approximation of a more general theory, still unknown. Recall that ergodicity) steady-state and that the states of QM belong to a Hilbert space, where theergodic hypothesis is valid. In any case, such revelation would entail rewritingthe basis of microscopic physics. It would also give the reason to Einstein in thefamous debate with Bohr.

In the next section, the reason why the ergodic hypothesis is necessary to insertmeasured data into the Bell’s inequalities is reviewed. A brief analysis of the caseof an experiment using Greenberger-Horne-Zeilinger (GHZ) states is presented. InSection 13.3, the basic idea of the experiment to detect nonergodic behavior ispresented, together with the discussion of some existing antecedents. In Section13.4, I indulge myself in some free considerations.

Beyond Loophole-Free Experiments: A Search for Nonergodicity 249

13.2 An Unnoticed Hypothesis

It is convenient to review the derivation of Bell’s inequalities. Here it is done forthe Clauser-Horne one, which includes the Eberhardt’s inequality. The discussionon the necessity of the ergodic hypothesis for the other experimentally relevantinequality, Clauser-Horne-Shimony and Holt (CHSH), is similar and can be foundin Hnilo (2017b).

13.2.1 Review of the Derivation of the Clauser-Horne Inequality (CH)

Consider the usual experiment with photons entangled in polarization, sketched inFigure 13.2. Assume that the probability to detect a photon after an analyzerA oriented at an angle α is PA(α,λ), where λ is an arbitrary “hidden” variable.

The strength of the Bell’s inequalities is that λ can be anything: real or complex,a vector, a tensor, etc. The only necessary assumption is that the integrals overthe λ-space exist and are “well behaved.” The observable probability of detectionis then:

PA αð Þ ¼ðdλ:ρ λð Þ:PA α; λð Þ (13.1)

where ρ(λ) is a normalized probability distribution in the λ-space (ρ(λ) � 0,Ðdλ.ρ(λ) = 1). These assumptions are in compliance with realism. Consider now

two photons carrying the same value of λ. The probability that both photons aredetected after analyzers A and B set at angles {α,β} is, by definition, PAB(α,β,λ).Locality implies that PAB(α,β,λ) = PA(α,λ) � PB(β,λ), so that the probability toobserve a double detection is:

PAB α; βð Þ ¼ðdλ:ρ λð Þ:PA α; λð Þ:PB β; λð Þ: (13.2)

L

α β

S

A B(+)

(–) (–)

(+)

Figure 13.2 Scheme of a typical Bell’s experiment (actually, Einstein-Podolsky-Rosen-Bohm setup). The source S emits two photons entangled in polarizationtowards two remote stations A and B, where analyzers are oriented at angles α andβ. Detectors after the analyzers count single photons. Relevant measured numbersare the rate of “singles” (detections at each station) and “coincidences” (detectionssimultaneous at both stations). The efficiency is defined as the ratio betweencoincidences and singles.

250 Alejandro A. Hnilo

Locality also implies that {α,β,λ} are statistically independent variables: PA(α,λ) =PA(α)�PA(λ), and that ρ(λ) is independent of {α,β}. The set of all these propertiesoften receive the specific name of measurement independence. To enforce it in thepractice, outstanding experiments have been performed. I point out the ones byGiustina et al. (2015), Shalm et al. (2015), and Weihs et al. (1998). In theseexperiments, the angle setting {α,β} is randomly changed in a time shorter thanL/c. In what follows, measurement independence is taken for granted.

Given {x,y � 0, X � x’, Y � y’} the following equality holds: �1 � xy – xy’ +x’y + x’y’ – Xy – Yx’ � 0. Choosing x = PA(α,λ), x’ = PA(α’,λ), y = PB(β,λ), y’ =PB(β’,λ) and X = Y = 1, where {α,β,α’,β’} are different analyzers’ orientations, andafter integration over the space of the hidden variables applying

Ðdλ.ρ(λ) and

Eq. (13.2), we get:

�1� PAB α;βð Þ� PAB α;β’� �þ PAB α’;βð Þþ PAB α’;β’

� �� PB βð Þ� PA α’ð Þ �� J� 0

(13.3)

which is the Clauser-Horne (CH) inequality. The QM predictions violate it. That is,for the entangled state jφ+i = (1/√2){jxa,xbi + jya,ybi}, PAB(α,β) = ½.cos2(α,β), andchoosing {α,β,α’,β’} = {0, π/8, π/4, 3π/8}, we get: J = 0.427 – 0.073 + 0.427 +0.427 – ½ – ½ = 0.208, violating the right-hand side (r.h.s.) of the inequality. It isconcluded that QM is incompatible with at least one of the assumptions (i.e.,locality and/or realism).

13.2.2 The Necessity of a Hypothesis Additional to Local Realism

Note that all real measurements are made in time. The expression for any observ-able probability is (e.g., for PA):

PA αð Þ ¼ 1=Δtð ÞðθþΔt

θdtρ tð ÞPA α; tð Þ (13.4)

This equation represents the result of the following real process: Set A = α duringthe time interval [θ, θ + Δt], sum up the number of photons detected after theanalyzer A, and obtain PA(α) as the ratio of detected over incident photons. But theintegrals in Eq. (13.1) and Eq. (13.4) are different: The former is an average overthe possible states of the hidden variables; the latter is an average over time. Theyare not necessarily equal. Assuming they are equal means assuming the ergodichypothesis valid. In all Bell’s experiments to date, this assumption has been(implicitly) made to insert measured numbers into derived inequalities. If thisassumption is not made, the insertion of measured data into the derived Bell’s

Beyond Loophole-Free Experiments: A Search for Nonergodicity 251

inequalities implies nothing about the validity of LR in nature; regardless, theinequalities computed in this way are violated, or not.

The usual logical value of the inequality can be formally retrieved by integratingover the total measuring time (in the same way than the integral over λ in theprevious subsection). But, the result turns out to include time integrals for settingangles, say {α,β}, calculated over times when the actual setting angles weredifferent, say {α’,β}. That is, we have to deal with the results of counterfactualmeasurements (see Hnilo 2013, 2014, 2017b for details). In the LR framework,Bell’s inequalities are derived and counterfactual measurements are assumed tohave definite outcomes (what is called counterfactual definiteness). Therefore, nohypothesis additional to LR is needed to legitimately deal with counterfactuals.Yet, a “possible world” must still be defined, in addition to counterfactual definite-ness, to ensure logical consistency and to assign numerical values to the counter-factual terms (see d’Espagnat 1984). Depending on the “possible world” chosen,the values taken by the counterfactual terms are different, and the Bell’sinequalities have a different form. In other words, assuming counterfactual defin-iteness is not enough. There still remains the problem of assigning numericalvalues to the counterfactual terms.

There is no mystery in this situation, simply lack of information. Let us examinean example from everyday life: If when I go to the cafeteria I have 30% probabilityof finding my friend Alice there, then what is the probability for me to find Alice inthe cafeteria when I don’t go there? If the question is strictly considered, there is noanswer. If it is assumed that Alice and the cafeteria have a well defined existenceeven when I do not go there (roughly speaking, if counterfactual definiteness isassumed), the question does have an answer, say, q. But the only available infor-mation at this point is that the probability that I find Alice is 30%, so that thenumerical value of q is definite, but unknown. More information is needed aboutwhat happens in the cafeteria when I don’t go there (i.e., a “possible world” mustbe defined) to assign a numerical value to q. That is, if an ergodic “possible world”is defined, then the counterfactual terms take values such that the usual meaning ofBell’s inequalities is retrieved. Yet this definition means a hypothesis additional toLR. Several conceivable “possible worlds” are explored in Hnilo (2013), leading todifferent inequalities. Some of them are not violated by experiments and not evenby QM.

In summary, a “possible world” must be defined (it may be ergodic or not) toassign numerical values to the counterfactual terms. That definition unavoidablymeans one assumption additional to LR, thus weakening the consequences of theobserved violation of the Bell’s inequalities. Note that this weakening does notarise from an experimental imperfection, as it is in the case of the loopholes. Thesetup is assumed ideally perfect. The weakening arises only from the fact that real

252 Alejandro A. Hnilo

measurements are performed during time and that it is impossible to measure withtwo different angle settings simultaneously. It is impossible to travel in time tomeasure again, at the same value of time, with a different angle setting.

13.2.3 The Case of Greenberger-Horne-Zeilinger States

Professor Lev Vaidman put forward the interesting question of whether the ergodichypothesis is also necessary in the case of tests using GHZ states. Here I show thatthe answer is “yes.” Note that this subsection is independent from the rest of thetext, and it can be skipped by the reader with no further consequences.

The question arises from the widespread belief that GHZ states can provide a“single-shot” disproval of LR, hence making the time integral in Eq. (13.4) irrele-vant. Therefore, let first see why this belief is erroneous. Consider, e.g., the GHZstate of three photons:

jϕ 3ð Þi ¼ 1=ffiffiffi2

p� �jx1; x2; x3f i þ ijy1; y2; y3ig (13.5)

where x,y are the planes of linear polarization and 1, 2, 3 label the entangledphotons. The Pauli operator σl, which acts: σljxi = jyi, σljyi = jxi, representsa polarization analyzer that fully transmits a photon linearly polarized at 45o of thex,y axes. The Pauli operator σr, which acts: σrjxi = ijyi, σrjyi = �ijxi, does thesame with circularly polarized photons (rightwards: transmitted, leftwards:reflected). These operators are applied to each particle of the state and are chosenin a random and independent way in three remote stations. The configuration of thewhole setup can be lll (the three photons find linear polarization analyzers), llr(photons 1 and 2 find linear analyzers, the third one a circular one), etc. with equalprobability. There are eight possible configurations. A transmitted photon means aresult +1, a reflected one�1, in each station. The result of a complete measurementis the product of the results obtained at the three stations. That is, if the photons aretransmitted at the first two stations and reflected at the third one, the result of themeasurement is: (+1) � (+1) � (�1) = �1. The state jϕ(3)i is an eigenstate witheigenvalue +1 for the configurations rll, lrl, and llr, and with eigenvalue �1 for theconfiguration rrr. For the other four possible configurations (the ones having aneven number of r) jϕ(3)i is not an eigenstate, and the result of a measurement maybe �1 or +1 with equal probability. In a long run, they average to zero. Let call thefour elements of the set {rll, lrl, llr, rrr} “words,” and the other four possibleconfigurations “strings.”

A simple hidden variables theory can be constructed in the form of a set of 2� 3matrices. The matrix determines the result of detection for each photon of the trio,depending what type of analyzer it finds. For example, the following matrix yields

Beyond Loophole-Free Experiments: A Search for Nonergodicity 253

the result (+1) if the configuration is lrl or llr and (�1) if it is rrr, all in agreementwith QM predictions. However, it yields (�1) for rll, contrarily to QM prediction(see Table 13.1).

There are 26 = 64 of these matrices. Each matrix reproduces the QM predictionsfor three of the four “words” and for all the “strings” (in the average). Let’s call“bad word” the configuration for which a given matrix cannot reproduce the QMprediction (in the example stated earlier, the “bad word” is rll). There are eightmatrices sharing the same “bad word,” and each configuration is the “bad word” ofeight matrices. The probability that a (randomly chosen) configuration is the “badword” of the (randomly chosen) matrix carried by the entangled trio is hence:8/64 = 1/8. This is the probability of the matrix theory to not reproduce the QMpredictions. In other words, the probability of the matrix model to reproduce theQM predictions for a single trio is 7/8. In order to disprove LR with a reliability>99%, 35 trios (all of them showing results coincident with the QM predictions)must be detected in an ideal setup. There are actually 70 trios, because half of theconfigurations are “strings,” in a setup where the bases are randomly and independ-ently chosen in each station. There is, in consequence, no single-shot disproval ofLR with GHZ states, but a statistical one, as in the usual two-particles Bell’s case.What can be single-shot disproved is QM. It suffices to observe (+1) for rrr, or(�1) for any of the other four “words,” to get a result that refutes QM. Of course,this is for an ideal setup. In a real setup the imperfections must be taken intoaccount, and the conditions to discriminate QM from LR become much like in thetwo-particles case. The issue is discussed in detail in Hnilo (1994) for GHZ stateswith arbitrary numbers of particles. The main conclusion of that paper is, in short,that experiments with GHZ states do not provide any significant advantage (for thetests of LR) over the two-particles Bell states, even if the difficulty in thepreparation of the states is not taken into account.

Now that we have seen that a single-shot disproval of LR is not provided byGHZ states, let go back to the main question. Mermin (1990) has demonstrated thatclassical theories are limited by the inequality:

Fn ¼ Im

ðdλ:ρ λð Þ:Π El

j þ iErj

� � � 2n=2 n evenð Þ, or 2 n-1Þð =2 n oddð Þ: (13.6)

Table 13.1: Example of a matrix that yields the result (+1) if the configuration is lrl or llr,and (�1) if it is rrr, all in agreement with QM predictions; but, it yields (�1) for rll,contrarily to QM predictions.

Analyzer found Result in station 1 Result in station 2 Result in station 3

l + + �r + � +

254 Alejandro A. Hnilo

where n is the number of particles in the GHZ state, λ is the hidden variable, theproduct goes from j = 1 to n, and El

j � (Nl+ � Nl

�)/(Nl+ + Nl

�), where Nl+ (Nl

�) isthe number of detections that produced a result “+1” (�1) in the j-station when thesetting was l (analogously for r). In the case of jϕ(3)iMermin’s inequality takes theform:

F3 ¼ðdλ:ρ λð Þ: El

1:El2:Er

3 þ El1:Er

2:El3 þ Er

1:El2:El

3 � Er1:Er

2:Er3

� � � 2

(13.7)

Note that the settings are randomly and independently chosen, so that in anyexperiment (even in an ideal one) the “string” configurations (say, lrr) also appearin the set of measured data. However, they average to zero, so that they aredropped in Eq. (13.7). The QM prediction in the case of jϕ(3)i is F3 = 4, whichviolates the inequality. In the case of the matrix model, the integral over λ isreplaced by a sum over the 64 matrices. The matrix model saturates the inequality(F3 = 2), because all the matrices fail to reproduce at least one of the “words” (e.g.,“bad word” of each matrix).

The definition of Fn involves an integral over the hidden variables, so that it isclear that the ergodic hypothesis is necessary in an experiment using GHZ states,too. To be specific, what is actually measured for n = 3 is:

Fmeas: ¼ðdt:ρ tð ÞEl

1:El2:Er

3 þðdt:ρ tð Þ El

1:Er2:El

3 þðdt:ρ tð Þ Er

1:El2:El

3

�ðdt:ρ tð Þ Er

1:Er2:Er

3 (13.8)

where each integral spans over a different period of time (regardless of whetherthey are continuous or divided in many randomly chosen, separate small intervals).If we make λ = t and integrate over the whole measuring time, we are faced againwith the problem of assigning values to the counterfactual integrals, as before.Therefore, F3 6¼ Fmeas., unless the ergodic hypothesis (or something like it) isassumed. The involved inequality is different (Mermin’s instead of Bell’s), but thesituation is the same, as in the case of the experiment with two entangled particles.

13.3 The Search for Nonergodicity

As it was shown, ergodicity must be assumed to validly insert measured valuesinto the mathematical expression of Bell’s inequalities. Taking for granted that theperformed loophole-free experiments have demonstrated the violation of Bell’sinequalities, then locality, realism, or ergodicity, must be invalid in nature. It is myintention to save LR, so let us see how to demonstrate that it is ergodicity that is

Beyond Loophole-Free Experiments: A Search for Nonergodicity 255

violated in the experiments. The direct way cannot be taken, for it is impossible tomeasure an average over the space of the hidden variables (to check whether it isequal to the time average or not). Therefore, one must follow an indirect approach.As was discussed in the introduction, the key may be deviations from randomnessin time series of data obtained in a Bell’s experiment. This means measuring notonly the average values of magnitudes (say, J, SCHSH, or concurrence), as in almostall experiments performed to date, but also the time evolution of these or othermagnitudes.

13.3.1 Experiments to Detect Nonergodicity

In a chaotic system, the dynamical variables are linked through nonlinear equationsin such a way that the evolution is very complex and apparently random. Yet, thedynamics are described by few degrees of freedom. This is a difference with “true”random processes, which can be thought of as having a very high (eventually,infinite) number of degrees of freedom. Nonlinear analysis (Abarbanel 1983)allows, under favorable circumstances, measuring the number of dimensions ofthe phase space where the system evolves (this is called the dimension of embed-ding dE), and hence, discriminating chaos from true randomness. In the case ofinterest here, it may allow detecting the existence of the hypothesized nonergodicdynamics, as opposed to the fundamental randomness assumed by QM.

However, revealing chaotic dynamics from observations is not an easy task. Inprinciple, one should see a complex time behavior involving quasi-periodicitiesand long transients. In order to reconstruct the (hypothesized) underlying object inphase space, one must record a time series with a density of data capable ofdetecting oscillations at some basic frequency f. The value of f in the case ofinterest here is unknown. Nevertheless, assuming that locality holds (recall that thepurpose here is to save LR), it is intuitive to expect f � c/L, where L is the physicaldistance between the two stations and c is the speed of light (see Figure 13.2). Thecapacity of recording oscillations at a frequency as high as c/L is hence the maincondition to reveal nonergodic behavior. No experiment performed until now hasfulfilled it. In order to get an idea of what that condition means, let us useShannon’s criterion of two samples in the period of interest. A sample of, e.g.,the value of the probability of coincidence P++(α,β) with the necessary resolutionrequires a minimum of 19 coincidences before the analyzers (0.052 � 1/19 is thelargest difference between the QM prediction for P++(α,β) and the limit imposed byBell’s inequalities for a maximally entangled state). Therefore, at least 38 coinci-dences detected in a period L/c are needed. Let see now how far performedexperiments are from this figure. The highest reported rate obtained in a laboratoryenvironment is � 3 � 105 s�1 coincidences (see Kurtsiefer, Oberparleiter, and

256 Alejandro A. Hnilo

Weinfurter 2001). This number cannot be much improved nowadays, for thefastest currently available single-photon detectors (avalanche photodiodes) cannotbe used reliably if the rate approaches 106 s�1 because of the high number ofsecondary (false) counts. In consequence, to detect 38 pairs with 3�105 s�1 coinci-dences in a time L/c, one must have L> 38 km. Bell’s setups with L = 13 km (Penget al. 2005) and even 144 km (Scheidl et al. 2010) have been performed, but theachieved coincidence rate was much lower than necessary: 50 and 8 s–1 (scaledvalues are 2 � 10–3 c/L and 4 � 10–3 c/L). The recent loophole-free experiment inVienna (Giustina et al. 2015) reached�200 s–1 for L = 58m, or 4� 10–5 c/L; the onein Boulder (Shalm et al. 2015) �5 s–1 for L = 185 m, or 3 � 10–6 c/L. The rate ofdetected pairs should be thus increased by several orders of magnitude to enter therange where the basic oscillations are expected to be detectable. A direct search fornonergodic dynamics seems beyond the current technical capacity.

Yet, the task is reachable under a “stroboscopic” approximation. That is, bysupposing that the system decays to a “ground state” in a time unknown, but finiteτdecay after the source of entangled states is turned off. The hypothesized picture isthen as follows: Once the source of entangled states is turned on, the nonclassicalcorrelation between measurements in the remote stations starts to evolve in anonergodic way. After the source is turned off, the correlation decays with timeτdecay (unknown, but finite). We have studied a general form that the dynamicsmay take in this picture in Hnilo (2012) and Hnilo and Agüero (2015) if locality isimposed. Noteworthy, oscillations with period � 4L/c are predicted for a broadregion in parameters’ space.

If the hypothesized picture is correct, a stroboscopic reconstruction of thesystem’s evolution is possible by using a pulsed source of entangled states. Thetime between pulses is adjusted longer than τdecay. The pulse duration is sliced inperiods shorter than L/2c, and the number of photons detected in each time slice isrecorded. Due to technical reasons (see Agüero, Hnilo, and Kovalsky 2014), lessthan one photon per pulse must be recorded in the average but, after millions ofpulses are detected, the time slices are gradually “filled” with data and the evolu-tion of the correlation during the pulse duration can be reconstructed with arbitraryprecision. The value of τdecay is unknown, but the pulse repetition rate can belowered as much as necessary, at the only cost of increasing the total duration ofthe experimental run. In summary, the stroboscopic approach allows the search fornonergodic dynamics with accessible means. There is, however, a risk of failure: Ifthe system did not decay to the same ground state after each pump pulse, then theinitial condition before each new pulse would not always be the same, jamming thereconstruction. This is an unavoidable risk in any stroboscopic observation.Anyway, even an imperfect reconstruction of the dynamics may provide a valuableantecedent to consider the realization (or not) of the “complete,” and much more

Beyond Loophole-Free Experiments: A Search for Nonergodicity 257

difficult, L > 38 km experiment. Note that the “complete” experiment would be, inany case, a simpler setup than, e.g., a laser interferometer gravitational-waveobservatory (LIGO) or a supercollider.

The specific experiment proposed (Figure 13.3) uses a pulsed laser to pump thenonlinear crystals to generate entangled (in polarization) states of photons. Boththe pump repetition rate and the pulse length are adjustable. This is to explore theunknown value of τdecay and of the time of evolution of the dynamics. Theentangled photons are inserted into single-mode optical fibers and transmitted toremote stations. The time-stamped files allow the calculation of the variables ofinterest (say, concurrence, efficiency, etc.) after the experimental run has ended.The distance between the stations is adjustable. Varying the value of L allowsdiscarding artifacts in the case some dynamics are actually observed, for L issupposed to define the timescale of the problem. It is relevant to mention here thatthe effect of polarization mode dispersion limits the use of optical fibers to aboutL � 1 km. Photons are detected at the stations with avalanche photodiodes, anddetections’ time values stored in time-stamped files with a resolution of 1 or 2 ns.A better resolution is meaningless, because of the intrinsic time jitter of thephotodiodes. A sample of the pump pulse is sent to each station to synchronizethe time-stamping devices, to set a uniform and stable starting point for thestroboscopic reconstruction, and to avoid the drift of the clocks that apparentlyoccurred in some previous experiments (see the next section).

Pump laser

(pulsed)

Time stamper B

TriggerC+C–

SPCM SPCM

SPCM SPCM

PhotodiodePhotodiode

“Bat-ears” “Bat-ears”

Single-mode fiber Single-mode fiber

Frequency downconversion crystals

HWP HWP

L

Time stamper A

Trigger C+ C–

Figure 13.3 Sketch of the proposed experiment. SPCM: single-photon countingmodules (avalanche photodiodes). The “bat ears” allow the compensation ofbirefringence in the optical fibers. HWP (“half wave plate”) adjust the anglesetting before each analyzer. Samples from the pump laser are used to trigger(synchronize) the time stampers. The time values of detection of each singlephoton and of the trigger signals are saved for further analysis.

258 Alejandro A. Hnilo

The stroboscopically obtained time series (Figure 13.4) are then analyzed,looking for the existence of a low-dimension object in phase space.

13.3.2 Some Antecedents

An early search of nonergodic dynamics in a Bell’s experiment was completed byour group (Hnilo, Peuriot, and Santiago 2002) using the raw data of the Innsbruckexperiment (Weihs et al. 1998). This experiment had implemented time stampingfor reasons different from the test of ergodicity. The focus of our study was to findtime series with a measurable value of dE. The theorem of embedding ensures thatdE can be measured using the time series of any dynamical scalar variable. Inpractice, favorable experimental conditions and choosing the appropriate observedvariable are necessary. Our group developed skills and techniques to deal withthese problems for the experimental study of the chaotic dynamics of Kerr-lensmode locked femtosecond lasers (Kovalsky and Hnilo 2004) and the formation ofoptical rogue waves in lasers with a saturable absorber (Bonazzola et al. 2015).

Our search in the data of the Innsbruck experiment involved dozens of files ofraw time-stamped data, generously provided by professor Gregor Weihs, andseveral possible observables. The hope to find evidence of nonergodic dynamicswas dim, for the detection rate in the Innsbruck experiment was too low to detectoscillations at c/L (the scaled coincidence rate was 2 � 10–3 c/L << 38 c/L). Theresult of our search was that no definite value of dE was reliably measured in any of

Entanglement

Efficiency

Pump pulse

Time

Figure 13.4 Sketch of a possible result of the proposed experiment. Here, it issupposed that the efficiency increases monotonically (as observed in Hnilo andAgüero 2015). It is also assumed that the entanglement, which is now measuredwith sufficient time resolution, displays a complex behavior. Be aware that theseresults would be a stroboscopic reconstruction obtained from the record of severalmillions of pump pulses separated, from each other, by a time longer than τdecay.The basic oscillation would have a frequency f < c/L. The nonlinear analysis ofthis time series may reveal the presence of a compact object in phase space, i.e., adeviation from randomness, and hence, the existence of nonergodic dynamics.

Beyond Loophole-Free Experiments: A Search for Nonergodicity 259

the data files, except one. The exception was the longest run in real time (6 minutes,rather than 10 to 30 seconds of most files), and the result was dE = 10 with fourpositive Lyapunov exponents, which meant that the series was hyperchaotic. Basedon the reconstructed attractor, we were able to “predict” the future outcomes in theseries, with satisfactory precision, up to an average of five to six pairs, that roughlycorresponded to the inverse of the largest positive Lyapunov exponent. As therewere four possible settings in the series, about 20 bits of the key were predictable.The importance of this finding for the security of QKD was remarked. When wefound this result, the Innsbruck experiment had been dismantled, so that the causeof the detected dynamics is impossible to know for sure. It is believed to be a driftbetween the clocks in each station. This belief is supported by the fact that a time-stamped experiment performed by our group some years later (Agüero et al. 2009),with an even longer session of data recording (>30 min) but a single clock,produced no measurable value of dE. Regardless whether the cause of the chaoticdynamics was instrumental or fundamental, the nonlinear analysis approach wasable to reveal it in one file of the Innsbruck experiment. This result proves thecapacity and power of the approach.

Another antecedent is the stroboscopic reconstruction of the evolution ofentanglement achieved by our group (Agüero, Hnilo, and Kovalsky 2012). Theexperiment’s aim was not to test ergodicity, but to close the time-coincidenceloophole. No evolution of entanglement was observed during the pulse duration.Entanglement was constant during the pulse, and was born and dead “instantan-eously” (strictly speaking, in a time shorter than the time resolution of the device,12.5 ns). There was only an increase of statistical errors at the pulse’s edges,naturally caused by scarcer data. Nevertheless, the time resolution of photondetectors and time-stamping devices was insufficient to detect f for the small valueof L used which was, to make things even worse, fixed. After the setup wasdismantled, we realized there was a linear increase of efficiency with time duringthe pulse duration. This result is consistent with the predictions of a simple LRhidden variables theory, but we consider it in no way conclusive (see Hnilo andAgüero 2015). A repetition of the experiment is in order to discard possibleartifacts.

Finally, in my opinion, the proposed stroboscopic experiment is worth doing forthe following reasons:

1) It is new. Instead of measuring Bell’s inequalities over and over again withdifferent random number generators (as in the recent “Big Bell Test,” or byusing light from remote stars) and/or improved detectors, this proposal involvesanalyzing the time evolution of the system, not only the average values. I do notmean those experiments are worthless. On the contrary, they are formidable

260 Alejandro A. Hnilo

technical achievements – useful and meaningful. However, you cannot keepdoing the same thing and expect different results.

2) It is important. I find it difficult to imagine an experiment with consequencesdeeper and broader than one whose scope is to reveal one of the limits to thevalidity of QM. That the limits exist, that QM is only an approximate, incom-plete description of physical reality should be evident to everybody. The fate ofall human knowledge is to be incomplete and provisory. Claiming the oppositeis not only methodologically wrong (recall Popper) but, to my taste, of anunbearable arrogance.

3) It is at hand, and some encouraging antecedents exist. Besides, even if theexperiment were not fully successful (say, because a failure of the stroboscopicassumption or a value of τdecay too long, such that it is impossible to be handledin practice), it may still provide clues useful to consider (or to discard) the“complete” version of the test with L > 38 km.

13.4 Some Personal Views and Conclusions

As famously stated by Schrödinger, “entanglement is the characteristic trait ofQM.” All performed quantum optical experiments can be explained by semiclas-sical theories, except for the ones involving spatially spread entangled states (seeScully and Zubairy 1997). The key to describing these experiments in QM isinterference of waves, yet not in real space, but in an abstract space. Now I wouldlike to share with the reader some imprecise (yet hopefully funny) thoughts.

Interference is, in itself, a formidable phenomenon. Although not often noted, itis alien to our intuitive way of thinking. It sets us apart from the ancients’philosophical realm. Parmenides stated that “something” cannot arise from “noth-ing” nor to vanish into “nothing,” then nihil novum sub sole. The eternity ofDemocritus’ atoms follows. But, in the interference phenomenon, two (or more)“somethings” can become “nothing,” and vice versa. The natural way to describeinterference is by using arrows. Here I use the term “arrow” instead of “vector”because the latter is a specific mathematical entity. A vector is an element of(obviously) a vector space, where the principle of superposition holds, i.e., thelinear combination of vectors is also a vector. Arrow, instead, is a more generalentity; e.g., the solutions of nonlinear equations can sum up to zero (they caninterfere) but their linear combination is not necessarily a solution of the equation(so, they do not form a vector space). Vectors rotating in time are solutions oflinear equations of physics, like Maxwell’s equations, and are convenientlydescribed with complex numbers. In this way we get a complex Hilbert (vector)space, the realm of QM, where ergodicity is valid.

Beyond Loophole-Free Experiments: A Search for Nonergodicity 261

Now I would like to remark that arrows and vectors are really strange things. Ifone thinks in the familiar terms of sets, the properties common to different sets arefound by their intersection. The same for arrows is found by projecting one arrowinto the other. Orthogonal arrows have “nothing” in common. The intersection ofsets is associative and commutative. The projection of arrows is not. In a popularchildren’s game, one has to find a person in a set by asking whether the person is aman or a woman, if he or she is blond or not, wears glasses or not, etc. The result isthe same regardless of the order in which the questions are posed. If played witharrows, however, the game may have a different result depending the order of thequestions. Some centuries-old mysteries are impossible to grasp by thinking interms of sets; e.g., Christian Trinity – three different persons, but only one God.Considered in terms of sets, it has no solution other than faith. In terms of arrows,God can be thought as the sum of three orthogonal (i.e., completely different)entities: Son, Father, and Holy Spirit. This picture also explains the Jesuits’commandments to study nature (to get close to the Father), to do charitable actions(to get close to the Son), and to exercise introspection (to get close to the HolySpirit). Failing to complete any of these three commandments means to fall shortof reaching God by a distance 1/√3 (in God’s space, assumed to be Euclidean).

The QM description of Bell’s experiments requires interference of waves lying inremote positions in real space. It is interference in an abstract, nonlocal space. This isnot only anti-intuitive (interference in real space has already led us far from intuition,so that this is not too serious) but, much more important in my opinion, structurallyunstable. For nonlocal interference and nonlinearity (even if infinitesimally small)lead to the possibility of transmission of information at infinite speed and quantumfield theory collapses. Someway to include nonlinear terms without leading to faster-than-light signaling (or something even stranger, see Polchinski 1991 and referencestherein) would be most welcome, for nonlinearities exist everywhere. They exist atthe large scale of the universe; the equations of general relativity are nonlinear. Theyalso exist at our scale – we can describe the evolution of the surrounding world withlinear equations only as an approximation. Yet, QM claims nonlinearities to do notexist at the microscopic scale, not even infinitesimal ones. I prefer thinking thatnonlinearities also exist at the microscopic scale, but that they cannot lead toinstantaneous transmission of information, because the strong correlations character-istic of entanglement appear only after a time L/c has elapsed. This is an effect that isnot predicted by QM. In other words, I believe entanglement to be an averageproperty of the setup’s symmetries and, for spatially spread systems, it requires atime longer than L/c to grow. In shorter times, transient deviations from the highcorrelations predicted by QM should be observed. I find this the most economicalway to save quantum field theory, the statistical interpretation of QM, and LR.

QM claims that only probabilities can be predicted, that information is physical,and that an independent physical reality does not exist. This has tome the smell of an

262 Alejandro A. Hnilo

approximation used beyond its range of validity. Consider the alternative: QM is astatistical approximation, a theory to be used when the knowledge we have on thesystem is incomplete. Therefore it isQM theorywhich can only predict probabilities.Information does not change physical reality, but affectsQM theory predictions (fora change in the available information changes the predictions of a probabilistictheory, as it is well known). An independent physical reality does exist; QM is anincomplete (statistical) description of this physical reality. This is a very good anduseful description, beyond any doubt, but, as Einstein stated, it is an incomplete one.And, as one of the LIGO’s leaders recently said: Don’t bet against Einstein.

Historically, the laws of evolution of particles (whose properties obey the logic ofsets) were developed before their statistical approximation. Classical statistical mech-anics describes the collective, average behavior of a very large number of particles(say, a mesoscopic volume of ideal gas), where Newton’s equations are impossible(and unnecessary) to solve. Extending its concepts and results to the evolution of twoparticles leads to paradoxes. However, everyone knows that they are only apparentparadoxes. They are the mere consequence of having used the theory outside therange of validity of the statistical approximation. Historically, we have entered therealm of arrows through the opposite door (for good, practical reasons, no doubt). Wehave got the statistical mechanics of vectors (i.e., QM) before developing the“mechanics of single arrows” (strictly speaking, the mechanics of entities whoseproperties obey, in the statistical approximation, the logic of vectors). This missing(hopefully LR) theory, where the principle of superposition does not necessarily holdand from which QM is the statistical, steady-state, ergodic approximation, is thephantom whose vanishing tracks the proposed experiments are intended to find.

Acknowledgments

I would like to offer many thanks to the participants on the workshop Identity,indistinguishability and non-locality in quantum physics (Buenos Aires, June2017), especially to Federico Holik, Nino Zanghì, Olimpia Lombardi, Lev Vaid-man, and Sebastian Fortin, for so many exciting and fruitful discussions on thesubject of this contribution. This research was partially supported by the grantPIP11–077 of CONICET, Argentina.

References

Abarbanel, H. (1983) “The analysis of observed chaotic data in physical systems,” Reviewsof Modern Physics, 65: 1331–1392.

Agüero, M., Hnilo, A., Kovalsky, M., and Larotonda, M. (2009). “Time stamping in EPRBexperiments: Application on the test of non-ergodic theories,” European PhysicalJournal D, 55: 705–709.

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Agüero, M., Hnilo, A., and Kovalsky, M. (2012). “Time resolved measurement of theBell’s inequalities and the coincidence-loophole,” Physical Review A, 86: 052121.

Agüero, M., Hnilo, A., and Kovalsky, M. (2014). “Measuring the entanglement of photonsproduced by a nanosecond pulsed source,” Journal of the Optical Society of America B,31: 3088–3096.

Ballentine, L. E. (1998). Quantum Mechanics. A Modern Development. Singapore: WorldScientific Publishing.

Bonazzola, C., Hnilo, A., Kovalsky, M., and Tredicce, J. (2015). “Features of the extremeevents observed in an all-solid-state laser with saturable absorber,” Physical Review A,92: 053816.

Buonomano, V. (1978). “A limitation on Bell’s inequality,” Annales de l’Institut HenriPoincaré, 29A: 379–394.

Clauser, J. and Shimony, A. (1978). “Bell’s theorem: Experimental tests and implications,”Reports on Progress in Physics, 41: 1881–1927.

d’Espagnat, B. (1984). “Nonseparability and the tentative descriptions of reality,” PhysicsReports, 110: 201–264.

Gisin, N. (1990). “Weinberg’s non-linear quantum mechanics and superluminal communi-cations,” Physics Letters A, 143: 1–2.

Giustina, M., Versteegh, M. A. M., Wengerowsky, S., Handsteiner, J., Hochrainer, A.,Phelan, K., . . . Zeilinger, A. (2015). “A significant loophole-free test of Bell’stheorem with entangled photons,” Physical Review Letters, 115: 250401.

Goldstein, H. (1950). Classical Mechanics. Reading, MA: Addison-Wesley.Hnilo, A. (1994). “On testing objective local theories by using GHZ states,” Foundations

of Physics, 24: 139–162.Hnilo, A. (2012). “Observable consequences of a hypothetical transient deviation from

Quantum Mechanics,” arXiv/quant-ph/1212.5722.Hnilo, A. (2013). “Time weakens the Bell’s inequalities,” arXiv/quant-ph/1306.1383v2.Hnilo, A. (2014). “On the meaning of an additional hypothesis in the Bell’s inequalities,”

arXiv/quant-ph/1402.6177.Hnilo, A. (2017a). “Consequences of recent loophole-free experiments on a relaxation of

measurement independence,” Physical Review A, 95: 022102.Hnilo, A. (2017b). “Using measured values in Bell’s inequalities entails at least one

hypothesis additional to Local Realism,” Entropy, 19: 80.Hnilo, A. and Agüero, M. (2015). “Simple experiment to test a hypothetical transient

deviation from Quantum Mechanics,” arXiv/abs/1507.01766.Hnilo, A., Peuriot, A., and Santiago, G. (2002). “Local realistic models tested by the EPRB

experiment with random variable analyzers,” Foundations of Physics Letters, 15:359–371.

Jaynes, E. T. (1980). “Quantum beats,” pp. 37–43 in A. Barut (ed.), Foundations ofRadiation Theory and Quantum Electrodynamics. New York: Plenum Press.

Khrennikov, A. (2017). “Buonomano against Bell: Nonergodicity or nonlocality?”, Inter-national Journal of Quantum Information, 8: 1740010.

Kovalsky, M. and Hnilo, A. (2004). “Different routes to chaos in the Ti: Sapphire laser,”Physical Review A, 70: 043813.

Kurtsiefer, C., Oberparleiter, M., and Weinfurter, H. (2001). “High efficiency entangledphoton pair collection in type II parametric fluorescence,” Physical Review A, 64:023802.

Mermin, D. (1990). “Extreme quantum entanglement in a superposition of macroscopic-ally distinct states,” Physical Review, 65: 1838–1840.

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Peng, C., Yang, T., Bao, X., Zhang, J., Jin, X., Feng, F., . . . Pan, J. W. (2005). “Experi-mental free-space distribution of entangled photon pairs over 13km: towards satellite-based global quantum communication,” Physical Review Letters, 94: 150501.

Polchinski, J. (1991). “Weinberg’s nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox,” Physical Review Letters, 66: 397–400.

Scheidl, T., Ursin, R., Kofler, J., Ramelow, S., Ma, X. -S., Herbst, T., . . . Zeilinger, A.(2010). “Violation of local realism with freedom of choice,” Proceedings of theNational Academy of Sciences of the United States of America, 107: 19708–19713.

Scully, O. M. and Zubairy, M. S. (1997). Quantum Optics. Cambridge: CambridgeUniversity Press.

Shalm, L. Meyer-Scott, E., Christensen, B., Bierhorst, P., Wayne, M., Stevens, M., . . .Nam, S. W. (2015). “A strong loophole-free test of local realism,” Physical ReviewLetters, 115: 250402.

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Part IV

Symmetries and Structure in Quantum Mechanics

14

Spacetime Symmetries in Quantum Mechanics

cristian lopez and olimpia lombardi

14.1 Introduction

In the last decades, the philosophy of physics has begun to pay attention to themeaning and the role of symmetries, an issue that has, however, had a greatrelevance in physics since, at least, the middle of the twentieth century. Notwith-standing this fact, this increasing interest in symmetries has not yet been trans-ferred to the field of the interpretation of quantum mechanics. Although it isusually accepted that the Galilean group is the group of invariance of the theory,discussions about interpretations of quantum mechanics, with very few exceptions,have not taken into account symmetry considerations. But the invariance of atheory under a group does not guarantee the invariance of its interpretations, asthey usually add interpretive assumptions to the formal structure of the theory.Symmetry considerations should thus be seriously taken into account in the field ofthe interpretation of quantum mechanics.

For this reason, in this chapter we shall focus on the spacetime symmetries ofquantum mechanics. After briefly introducing certain terminological clarifications,we shall focus on two aspects of spacetime transformations. First, we shallconsider the behavior of nonrelativistic quantum mechanics under the Galileangroup, aiming at assessing its Galilean invariance in relation to interpretive con-cerns. Second, we shall analyze the widely-accepted view about the invariance ofthe Schrödinger equation under time reversal, in order to unveil some implicitassumptions underlying such a claim.

14.2 The Concepts of Invariance and Covariance

The meaning of the term ‘symmetry’ is rooted in ordinary language: Symmetry is ageometrical property of a body whose parts are equal in a certain sense. Inmathematics, the term acquires a precise meaning in terms of invariance – an

269

object is symmetric with respect to a certain transformation when it is invariantunder such transformation; that is, it remains unchanged under its application. Theconcept of group, originally proposed by Galois in the first half of the nineteenthcentury, comes to supplement the notion of symmetry – a group clusters differenttransformations into a specific structure.

Despite its mathematical precision, in physics, the concept of symmetry hasgiven rise to some disagreements on the meaning and the scope of the concept ofinvariance and of the closely-related concept of covariance. Commonly, theproperty of invariance only applies to mathematical objects and, derivatively, tothe physical items to which they refer, and the property of covariance is reservedfor equations and, derivatively, for the physical laws they express. However, someauthors claim that the difference between invariance and covariance not onlymakes sense but is also relevant when applied to laws (Ohanian and Ruffini1994, Suppes 2000, Brading and Castellani 2007). Hans Ohanian and RemoRuffini (1994), for instance, claim that an equation is said to be covariant whenits form is left unaltered under a certain transformation, and it is said to be invariantwhen it is covariant and its content, that is, its absolute objects (constants andnondynamical quantities) are also left unchanged by the transformation. Althoughinspired in this idea, we will not follow it in every detail. Here we will consider thatan equation is invariant under a certain transformation when it does not changeunder the application of such transformation, and it is covariant under that trans-formation when its form is left unchanged by it (Suppes 2000). From this perspec-tive, the invariance of a law does not imply the invariance of the objects containedin its representing equation.

Once one accepts that the concept of invariance makes sense in its application tolaws, the conceptual implications both of the invariance of the law and of theinvolved objects under a particular group of transformations deserve to be con-sidered. Moreover, when a law is covariant under a transformation and all theobjects it contains are also invariant under the same transformation, the law isinvariant under the transformation as well. Nevertheless, this is not the only wayfor a law to be invariant – if a law is covariant under a certain transformation, it canturn out to be invariant under the transformation even in the case that some of theobjects it contains are not invariant under the same transformation (we will comeback to this point in the next section, when discussing the invariance of theSchrödinger equation).

On the basis of these conceptual clarifications, some formal definitions can nowbe introduced.

Def. 1 Let us consider a set A of objects ai 2 A, and a group G of transformationsgα 2 G, where the gα : A ! A act upon the ai as ai ! ~ai. An object ai 2 A is

270 Cristian López and Olimpia Lombardi

invariant under the transformation gα if, for that transformation, ~ai ¼ ai. In turn, theobject ai 2 A is invariant under the group G if it is invariant under all the trans-formations gα 2 G.

In physics, the objects to which transformations apply are usually those represent-ing states s, observables O, and differential operators D, and each transformationacts upon them in a particular way. In turn, those objects are combined in equationsrepresenting the laws of a theory. Then,

Def. 2 Let L be a law represented by an equation E s;Oi;Dj

� � ¼ 0, where srepresents a state, the Oi represent observables, and the Dj represent differentialoperators, and let G be a group of transformations gα 2 G acting upon the objectsinvolved in the equation as s ! ~s, Oi ! ~Oi, and Dj ! ~Dj. L is covariant under thetransformation gα if E ~s; ~Oi; ~Dj

� � ¼ 0, and L is invariant under the transformation gαif E ~s;Oi;Dj

� � ¼ 0. Moreover, L is covariant� invariant� under the group G if it iscovariant � invariant � under all the transformations gα 2 G.

On this basis, it is usually said that a certain group is the symmetry group of a theory:

Def. 3 A group G of transformations is said to be the symmetry group of a theory ifthe laws of the theory are covariant under the group G.

This means that the laws preserve their validity even when the transformations ofthe group are applied to the involved objects.

Some authors prefer to talk about symmetry instead of covariance. This is the caseof John Earman (2004), who defines symmetry in the language of model theory:

Def. 4 Let M be the set of the models of a certain mathematical structure, and letML � M be the subset of the models satisfying the law L. A symmetry of the lawL is a map S : M ! M that preserves ML, that is, for any m 2 ML, S mð Þ 2 ML.

When L is represented by a differential equation E s;Oi;Dj

� � ¼ 0, each modelm 2 ML is represented by a solution s ¼ F Oi; s0ð Þ of the equation, correspondingto a possible evolution of the system. Then, the covariance of L under a transform-ation g – that is, the fact that E ~s; ~Oi; ~Dj

� � ¼ 0 – implies that if s ¼ F Oi; s0ð Þ is asolution of the equation, ~s ¼ ~F ~Oi; s0

� �is also a solution and, as a consequence, it

represents a model S mð Þ 2 ML. This means that the definition of covariance givenby Def. 2 and the definition of symmetry given by Def. 4 are equivalent.

In turn, the covariance of a dynamical law – represented by a differentialequation – does not imply the invariance of the possible evolutions – representedby the solutions of the equation. In fact, the covariance of the law L, represented bythe equation E s;Oi;Dj

� � ¼ 0, implies that s ¼ F Oi; s0ð Þ and~s ¼ ~F ~Oi; s0� �

are bothsolutions of the equation, but does not imply that s ¼ ~s. In the model-theory

Spacetime Symmetries in Quantum Mechanics 271

language, the symmetry of L does not imply that S mð Þ ¼ m. By contrast, invarianceis a stronger property of the law: The invariance of Lmeans that E ~s;Oi;Dj

� � ¼ 0; inthis case s ¼ ~s ¼ F Oi; s0ð Þ or, in the model language, S mð Þ ¼ m.

Def. 5 Let M be the set of the models of a certain mathematical structure, and letML � M be the subset of the models satisfying the law L. Let a transformation be amap S : M ! M that preserves ML. The law L is invariant under the transform-ation S if, for any m 2 ML, S mð Þ ¼ m.

The general definitions just described can be applied to the Schrödinger equationso as to explicitly state the conditions of covariance and invariance for quantummechanics. Here we will focus on the evolution equation of the theory, leavingaside the collapse postulate, since it is an interpretive postulate in orthodoxquantum mechanics. Given a transformation g acting as φj i ! ~φj i, O ! ~O,d=dt ! ~d=dt, and i ! ~i (considering i as the shorthand for the operator i I), bymaking ℏ ¼ 1 the Schrödinger equation is covariant under g when

~d ~φj idt

¼ �~i ~H ~φj i (14.1)

and it is invariant under g when

d ~φj idt

¼ �iH ~φj i: (14.2)

14.3 Quantum Mechanics and the Galilean Group

14.3.1 The Galilean Group

As time is represented by the variable t 2 R and position is represented by thevariable r ¼ x; y; zð Þ 2 R3, the Galilean group G ¼ Tαf g, with α ¼ 1 to 10, is agroup of continuous spacetime transformations Tα : R3 � R ! R3 � R such that

• t ! t0 ¼ t þ τ (time-displacement)

• r ! r0 ¼ rþ ρ (space-displacement)

• r ! r0 ¼ Rθr (space-rotation)

• r ! r0 ¼ rþ ut (velocity-boost)

where τ 2 R is a real number representing a time interval, ρ ¼ �ρx, ρy, ρz

� 2 R3 is

a triple of real numbers representing a space interval, Rθ 2 M3�3 is a 3� 3 matrix

representing a space rotation by an angle θ, and u ¼ ux; uy; uz� � 2 R3 is a triple of

real numbers representing a constant velocity.For the Galilean group, G is a Lie group, the Galilean transformations Tα

can be represented by unitary operators Uα over the Hilbert space, with the

272 Cristian López and Olimpia Lombardi

exponential parametrization Uα ¼ eiKαsα , where sα is a continuous parameterand Kα is a hermitian operator independent of sα, the generator of the trans-formation Tα. Then, G is defined by 10 group generators Kα: one time-displacement Kτ, three space-displacements Kρi , three space-rotations Kθi ,and three velocity-boosts Kui , with i ¼ x, y, z. Therefore, by taking ℏ ¼ 1 asusual, the Galilean group is defined by the commutation relations between itsgenerators:

að Þ Kρi ;Kρj

h i¼ 0 fð Þ Kui ;Kρj

h i¼ iδijM

bð Þ Kui ;Kuj

� � ¼ 0 gð Þ Kρi ;Kτ� � ¼ 0

cð Þ Kθi ;Kθj

� � ¼ iεijkKθj hð Þ Kθi ;Kτ½ � ¼ 0

dð Þ Kθi ;Kρj

h i¼ iεijkKρk ið Þ Kui ;Kτ½ � ¼ iKρi

eð Þ Kθi ;Kuj

� � ¼ iεijkKuk

(14.3)

where εijk is the Levi-Civita tensor. Strictly speaking, in the case of quantummechanics the symmetry group is the group corresponding to the central extensionof the Galilean algebra, obtained as a semi-direct product between the Galileanalgebra and the algebra generated by a central charge, which in this case is themass operator M ¼ mI, where I is the identity operator and m is the mass. Themass operator as a central charge is a consequence of the projective representationof the Galilean group (see Bose 1995, Weinberg 1995). However, in order tosimplify the presentation, we will use the expression “Galilean group” from nowon to refer to the corresponding central extension.

In a closed, constant-energy system free from external fields, the generators Kα

are given by the basic magnitudes of the theory: the energy H ¼ ℏKτ, the threemomentum components Pi ¼ ℏKρi , the three angular momentum componentsJi ¼ ℏKθi , and the three boost components Gi ¼ ℏKui . Then, in this case thecommutation relations turn out to be

að Þ Pi;Pj

� � ¼ 0 fð Þ Gi;Pj

� � ¼ iδijM

bð Þ Gi;Gj

� � ¼ 0 gð Þ Pi;H½ � ¼ 0

cð Þ Ji; Jj� � ¼ iεijkJk hð Þ Ji;H½ � ¼ 0

dð Þ Ji;Pj

� � ¼ iεijkPk ið Þ Gi;H½ � ¼ iPi

eð Þ Ji;Gj

� � ¼ iεijkGk

(14.4)

.The rest of the physical magnitudes can be defined in terms of these basic ones. Forinstance, the three position components are Qi ¼ Gi=m, the three orbital angularmomentum components are Li ¼ εijkQjPk, and the three spin components are

Spacetime Symmetries in Quantum Mechanics 273

Si ¼ Ji � Li. In the Hilbert formulation of quantum mechanics, each Galileantransformation gα 2 G acts upon states and upon observables as

φj i ! ~φj i ¼ Usα φj i ¼ eiKαsα φj i (14.5)

O ! ~O ¼ Usα OU�1sα

¼ eiKαsα Oe�iKαsα (14.6)

The invariance of an observable O under a Galilean transformation Tα amounts tothe commutation between O and the corresponding generator Kα:

~O ¼ eiKαsαOe�iKαsα ¼ O , O;Kα½ � ¼ 0 (14.7)

It is worth bearing in mind that there are operators that are invariant under allthe transformations of the group, and thereby, commute with all the generators ofthe group – the Casimir operators of a group. In the case of the Galilean group, theCasimir operators are the internal energyW ¼ H � P2=2M, the square of total spinS2 ¼ J � 1

M G� P� �2

, and the mass M, which are multiples of the identity in anyirreducible representation.

14.3.2 The Covariance of the Schrödinger Equation

Given the Schrödinger equation, let us begin by (i) premultiplying its two membersby U ¼ eiKs, (ii) adding and subtracting dU=dtð Þ φj i to its first member, and (iii)using the property U�1U ¼ I:

Ud φj idt

þ dU

dtφj i � dU

dtU�1U φj i ¼ �UiU�1UHU�1U φj i (14.8)

Then, by recalling the transformations of states and observables of Eq. (14.5) andEq. (14.6), we obtain

d ~φj idt

� dU

dtU�1 ~φj i ¼ �~i ~H ~φj i (14.9)

This shows that covariance obtains when the time-derivative operator transformsas

d

dt!

~d

dt¼ D

Dt¼ d

dt� dU

dtU�1 )

~d ~φj idt

¼ �~i ~H ~φj i (14.10)

This means that the transformed differential operator ~d=dt is a covariant time-derivative D=Dt, which makes the Schrödinger equation to be Galilean-covariantin the sense of Eq. (14.1).

In a closed, constant-energy system free from external fields, H is time-independent and the Pi and the Ji are constants of motion (see Eq. (14.4g) and

274 Cristian López and Olimpia Lombardi

Eq. (14.4h)). Then, for time-translations, space-translations and space-rotations,dU=dt ¼ deiKs=dt ¼ 0, where K and s stand for H and τ, Pi and ρi, and Ji and θi,respectively. As a consequence, the time-derivative is invariant under time-displacements, space-displacements, and space-rotations (see Eq. (14.10)):d=dt ! ~d=dt ¼ d=dt. But for boost-transformations this is not the case: Thecovariance of the Schrödinger equation implies the transformation of the differen-tial operator as d=dt ! D=Dt. This means that covariance under boosts amounts toa sort of “nonhomogeneity” of time, which requires the covariant adjustment of thetime-derivative. This conclusion should not be surprising since, when the system isdescribed in a reference frame ~F at uniform motion corresponding to a velocity uxwith respect to the original frame F, the boost-transformed state depends on agenerator that is a linear function of time:

Gx ¼ mQx ¼ m Qx0 þ Vxtð Þ ¼ mQx0 þ Pxt (14.11)

Then, if the Schrödinger equation is to be valid in ~F , where the state is ~φj i, thetransformed time-derivative has to be adjusted to compensate the time-dependingtransformation of the state.

14.3.3 The Invariance of the Schrödinger Equation

As we have seen in the previous section, in a closed, constant-energy system free fromexternal fields, H is time-independent and the Pi and the Ji are constants of motion.Then, for time-translations, space-translations, and space-rotations, it follows thatdU=dt ¼ deiKs=dt ¼ 0 and d=dt ! ~d=dt ¼ d=dt. Moreover, for those transform-ations,~i ¼ i follows trivially, and ~H ¼ H because (see Eq. (14.7)) (i) H;H½ � ¼ 0, (ii)Pi;H½ � ¼ 0 (Eq. (14.4g)), and (iii) Ji;H½ � ¼ 0 (Eq. (14.4h)). When these results applyto Eq. (14.9), it is easy to see that the Schrödinger equation is invariant under time-displacements, space-displacements, and space-rotations in the sense of Eq. (14.2).

The case of boost-transformations is different from the previous cases because,although ~i ¼ i still holds, the Hamiltonian is not boost-invariant even when thesystem is free from external fields (the same happens in classical mechanics, seeButterfield 2007: 6). In fact, under a boost-transformation corresponding to avelocity ux, H changes as (see Eq. (14.4i): Gx;H½ � ¼ iPx 6¼ 0)

~H ¼ eiGxuxHe�iGxux 6¼ H (14.12)

Since Gx is not time-independent, dU=dt ¼ deiGxux=dt 6¼ 0, and Eq. (14.9) yields

d ~φj idt

¼ �i ~H þ ideiGxux

dte�iGxux

� �~φj i (14.13)

Spacetime Symmetries in Quantum Mechanics 275

In order to know the value of the bracket in the right-hand side (r.h.s.) of Eq.(14.13), the two terms in the bracket must be computed. When the task isperformed, it can be proved that the terms added to H in ~H cancel out with thosecoming from the term containing the time-derivative (see Lombardi, Castagnino,and Ardenghi 2010: appendices). Therefore, Eq. (14.2) is again obtained and theinvariance of the Schrödinger equation is proved to hold also for boost-transformations.

The case of boost-transformations illustrates a claim previously mentioned inSection 14.2: Even though a law is invariant under a transformation when it iscovariant and all the involved objects are invariant, this is not the only way toobtain invariance. When the quantum system is free from external fields, theSchrödinger equation is invariant under boost-transformations, in spite of the factthat the Hamiltonian and the differential operator d=dt are not boost-invariantobjects.

14.3.4 Galilean Group and External Fields

As explained in the previous subsection, when there are no external fields acting onthe system, the Hamiltonian is invariant under time-displacements, space-displace-ments, and space-rotations, but not under boost-transformations. Despite this fact,the Schrödinger equation is completely invariant under the Galilean group, andthis conceptually means that the state vector φj i does not “see” the effect ofthe transformations – the evolutions of φj i and ~φj i are identical. In other words,the time-behavior of the system is independent of the reference frame used for thedescription.

When the system is under the action of external fields, the fields modify theevolution of the system. But, in nonrelativistic quantum mechanics, fields are notquantized: They do not play the role of quantum systems that interact with othersystems. For this reason, the effect of the fields on a system must be included in itsHamiltonian, because it is the only observable involved in the time-evolution law.It can be proved that the most general form of the Hamiltonian in the presence ofexternal fields is (see, e.g., Ballentine 1998)

H ¼ P� A Qð Þð Þ22M

þ V Qð Þ (14.14)

where A Qð Þ is a vector potential and V Qð Þ is a scalar potential. The covariance ofthe Schrödinger equation, as expressed in Eq. (14.9), fixes the way in which thepotentials A Qð Þ and V Qð Þ must transform under the Galilean group. The electro-magnetic field may be derived from a vector potential and a scalar potential; thus, afully Galilean-covariant quantum theory of the Schrödinger field interacting with

276 Cristian López and Olimpia Lombardi

an external electromagnetic field is possible. However, the electric and magneticfields that transform as required to preserve Galilean covariance, although relatedto the scalar and vector potentials in the usual way, are ruled by one of two sets ofelectromagnetic “field” equations. Those equations can be considered the nonre-lativistic limits of Maxwell’s equations in cases where either (i) magnetic effectspredominate over electric ones (“magnetic limit”: c Bj j >> Ej j), or (ii) electriceffects predominate over magnetic ones (“electric limit”: c Bj j << Ej j) (see Brownand Holland 1999, Colussi and Wickramasekara 2008). Nevertheless, A Qð Þ andV Qð Þ should not necessarily be identified with the electromagnetic potentials,because they are arbitrary functions that need not satisfy Maxwell’s equations;for example, the Newtonian gravitational potential can also be included in thescalar V Qð Þ (Ballentine 1998).

At this point, a relevant issue must be stressed. Space-displacements and space-rotations are purely geometric operations of displacing and rotating the systemself-congruently to another place and to another direction, respectively. Analo-gously, time-displacements are purely geometric operations of displacing thesystem self-congruently to another time, and they may agree or not with dynamicalevolutions. The commutation of two transformation generators means that thecorresponding geometric operations can be performed in either order with thesame result; for instance, the commutation ½Kρi ,Kρj � ¼ 0 (see Eq. (14.3a)) means

that the order in which space-displacements in different directions are performeddoes not modify the result. In particular, the validity of the Galilean group impliesthat time-displacements commute both with space-displacements and with space-rotations (see Eq. (14.3g) and Eq. (14.3h)). When there are no external fields actingon the system, this feature is given by the commutation relations involving theHamiltonian, the three momentum components Pi, and the three angular momen-tum components Ji, Pi;H½ � ¼ 0 and Ji;H½ � ¼ 0 (Eq. (14.4g) and Eq. (14.4h)),because here the Hamiltonian is the time-displacement generator. But in thepresence of external fields, since the action of the fields is incorporated into theHamiltonian of the system, the Hamiltonian is no longer the generator of time-displacements: It only retains its role as the generator of the dynamical evolution(see Laue 1996, Ballentine 1998). For this reason, the commutation with themomentum components and with the angular momentum components gets broken:Pi;H½ � 6¼ 0 and Ji;H½ � 6¼ 0. However, to the extent that the covariance of theSchrödinger equation is retained, the commutation of time-displacements withboth space-displacements and space-rotations still holds, and is still representedby Kρi ;Kτ

� � ¼ 0 and Kθi ;Kτ½ � ¼ 0, respectively (see Eq. (14.3g) and Eq. (14.3h)),where the momentum components are still the generators of space-displacements,Pi ¼ ℏKρi , and the angular momentum components are still the generators of

Spacetime Symmetries in Quantum Mechanics 277

space-rotations, Ji ¼ ℏKθi , but the Hamiltonian is no longer the generator of time-displacements, H 6¼ ℏKτ (we will come back to this point in the next section, abouttime-reversal invariance).

14.3.5 The Relevance to Interpretation

In principle, there are two possible interpretations of a transformation: active andpassive. Under the active interpretation, the transformation corresponds to achange from a system to the transformed system; for instance, a system translatedin space with respect to the original one. Under the passive interpretation, thetransformation consists in a change of the viewpoint – the reference frame – fromwhich the system is described; for instance, the space-translation of the observerthat describes the system. In the case of continuous spacetime transformations,both active and passive interpretation are equally allowed; but such a situation isnot so clear in the case of discrete transformations. In general, it is accepted thatonly the active interpretation makes sense in the case of discrete transformations(Sklar 1974: 363). Nevertheless, no matter which interpretation is adopted, thecovariance of the fundamental law of a theory under its continuous symmetrygroup implies that the law still holds when the transformations are applied. In theactive interpretation language, the original and the transformed systems are equiva-lent; in the passive interpretation language, the original and the transformedreference frames are equivalent.

As is typically accepted, the Galilean group is the symmetry group of continuousspacetime transformations of classical and quantum mechanics. In the language ofthe passive interpretation, the covariance of the dynamical laws amounts to theequivalence among inertial reference frames (time-translated, space-translated,space-rotated, or uniformly moving with respect to each other). In other words,Galilean transformations do not introduce anymodification in the physical situation,but only express a change in the perspective from which the system is described.

These remarks are related to the fact that certain quantities are physicallyirrelevant in the light of theory’s symmetries. For instance, the space-translationsymmetry of a dynamical law means that the specific place where the system islocated in space is irrelevant to its evolution governed by such law: “A globalsymmetry reflects the irrelevance of absolute values of a certain quantity: onlyrelative values are relevant” (see Brading and Castellani 2007: 1360). In classicalmechanics, for example, space-translation invariance implies that absolute positionis irrelevant to the system’s behavior – the equations of motion do not depend onabsolute positions, only relative positions matter. The physical irrelevance ofcertain quantities is strongly linked with the issue of objectivity.

278 Cristian López and Olimpia Lombardi

The intuition about a strong link between invariance and objectivity is rooted ina natural idea: What is objective should not depend on the particular perspectiveused for the description. When this intuition is translated into the group-theoreticallanguage, it can be said that what is objective according to a theory is what isinvariant under the symmetry group of the theory. This idea appeared in thedomain of formal sciences in Felix Klein’s “Erlangen Program” of 1872, withthe attempt to characterize all known geometries by their invariants (see Kramer1970). This idea passed to physics with the advent of relativity, regarding theontological status of space and time (Minkowski 1923). The claim that objectivitymeans invariance becomes a main thesis of Hemann Weyl’s book Symmetry(1952). Max Born also very clearly expressed his conviction about the strong linkbetween invariance and objectivity: “I think the idea of invariance is the clue to arational concept of reality” (Born 1953: 144). In recent times, the idea has stronglyreappeared in several works. For instance, in her deep analysis of quantum fieldtheory, Sunny Auyang (1995) makes her general concept of “object” to be foundedon its invariance under transformations among all representations. The assumptionof invariance as the root of objectivity is also the central theme of Robert Nozick’sbook Invariances: The Structure of the Objective World (2001). In the same vein,David Baker (2010) has argued that symmetries are a guide to finding out whichquantities represent fundamental natural properties in a physical theory.

If the ontological meaning of symmetries is accepted, it is easy to see thatsymmetries must play an active role in the understanding of a physical theory. Inthe particular case of quantum mechanics, the consideration of its Galileancovariance cannot be overlooked in the discussions about interpretation.

As it is well known, the Kochen-Specker theorem (Kochen and Specker 1967)establishes a barrier to any realist classical-like interpretation of quantum mechan-ics: It proves the impossibility of ascribing definite values to all the physicalquantities (observables) of a quantum system simultaneously, while preserving thefunctional relations between commuting observables. This result is a manifestationof the contextuality of quantum mechanics – the ascription of definite values to theobservables of a quantum system is always contextual. As a consequence of theKochen-Specker theorem, any realist interpretation of quantum mechanics is com-mitted to selecting a subset of definite-valued observables from the set of all theobservables of the system (or a preferred basis from all the formally equivalent basesof the Hilbert space). The observables of that subset will be those that acquiredefinite values without violating quantum contextuality. It is at this point that thesymmetry group of the theory becomes a leading character. As noticed by HarveyBrown, Mauricio Suárez, and Guido Bacciagaluppi (1998), any interpretation thatselects the set of the definite-valued observables of a quantum system in a given stateis committed to considering how that set is transformed under the Galilean group.

Spacetime Symmetries in Quantum Mechanics 279

However, now the link between invariance and objectivity comes into play. Thestudy of the role of symmetries is particularly pressing in the case of realistinterpretations of quantum mechanics, which conceive a definite-valued observ-able as a physical magnitude that objectively acquires an actual definite valueamong all its possible values: The fact that a certain observable acquires a definitevalue should be an objective fact that should not depend on the descriptiveperspective. Therefore, the set of the definite-valued observables of a systempicked out by the interpretation should be left invariant by the Galileantransformations. From a realist viewpoint, it would be unacceptable that such aset changed as result of a mere change in the perspective from which the system isdescribed.

In his article “Aspects of objectivity in quantum mechanics,” Harvey Brown(1999) explicitly tackles the problem in discussing the objectivity of “sharpvalues.” In particular, he focuses on interpretations that specify state-dependentrules for assigning sharp values to some of the self-adjoint operators representingquantum magnitudes, such as the interpretations whose value-assignment rulescoincide with the eigenstate-eigenvalue link, or the modal interpretations that makethe set of definite-valued observables to depend on the instantaneous state of thesystem. Brown clearly explains the difference between the classical and thequantum case. In classical mechanics, Galilean noninvariant magnitudes modifytheir values with the change of reference frame; for this reason, if their objectivityis to be retained, they must be regarded not as intrinsic properties but as relationalproperties. For instance, the values of classical position and momentum can beconceived as relational properties that link the system and the reference frame. Inquantum mechanics, by contrast, the relational nature acquires a further degree;whereas, in the classical case the sharp value of a magnitude depends on thereference frame, in the quantum case the very sharpness of an observable’s valuemust be relational in order to preserve its objectivity. For instance, the fact that theposition of a system has a sharp (definite) value in a certain reference frame, and,as a consequence, that the system can be conceived as a localized particle, is itselfrelational. In a different reference frame, the system may have an unsharp value ofposition and, then, may behave as a delocalized particle. Brown also correctlystresses that it is not just boosts that produce this kind of situation; passive spatialtranslations can cause that some sharp-valued observables to become unsharp. Onthe basis of this analysis, he concludes that

If, in the hope of providing an ontological interpretation of quantum mechanics, weintroduce state-dependent rules for assigning sharp values to magnitudes associated witha specific quantum system, we should recognise that the objective status of such sharpvalues is relational, not absolute.

(Brown 1999: 66–67)

280 Cristian López and Olimpia Lombardi

In the same article, Brown considers an interpretation in which the rule ofdefinite-value ascription is not state-dependent; he analyzes the problem of covar-iance in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics, buthe discards it because, although no privileged frame is picked out by the hiddendynamics of the corpuscles, the forces acting upon the corpuscles and generated bythe guiding wave are Aristotelian, not Newtonian, because they produce velocities,not accelerations. This seems to lead him to interpretations that introduce state-dependent rules of definite-value ascription as the only alternative. This maneuveris opposed to that which led Jeffrey Bub (1997) to advocate for Bohmian mechan-ics, conceived as a modal interpretation whose rule of definite-value ascriptionpicks out the position observable: The difficulties of the original modalinterpretations to deal with nonideal measurements due to their state-dependentrules turns Bohmian mechanics into a natural alternative. But what both Bub andBrown seem to overlook is that there are other interpretive strategies beyondBohmian mechanics and traditional modal interpretations that make the definite-valued observables to depend on the state of the system. One of them is even morenatural than the Bohmian proposal when the aim is to preserve the objectivity ofdefinite-valuedness (or of sharpness, in Brown’s terms) in the light of Galileansymmetry. In fact, the natural way to reach this goal, without making the objectivestatus of definite-valuedness relational, is to appeal to the Casimir operators of theGalilean group: If the interpretation has to select a Galilean-invariant set ofdefinite-valued observables, such a set must depend on those Casimir operators,insofar as they are invariant under all the transformations of the Galilean group. Aninterpretation that has adopted this interpretive strategy is the modal-Hamiltonianinterpretation in its Galilean invariant version (Ardenghi, Castagnino, and Lom-bardi 2009, Lombardi, Castagnino, and Ardenghi 2010), which has been success-fully applied to many well-known physical situations and has proved to beeffective for solving the measurement problem, both in its ideal and its nonidealversions (Lombardi and Castagnino 2008).

Considering that the Casimir operators of the Galilean group represent thedefinite-valued observables of a quantum system has the advantage of being verygeneral. When the system is free from external fields and the Galilean group isdefined by the commutation relations Eq. (14.4), the Casimir operators correspondto the observables mass M, squared-spin S2, and internal energy W . Yet theCasimir operators of the Galilean group can always be defined, even when Eq.(14.4) does not hold. The group must thus be defined in the completely generalway as expressed by Eq. (14.3). Therefore, when there are external fields applied tothe system, and such fields do not break the covariance of quantum mechanicsunder the Galilean group, the strategy of defining the definite-valued observablesin terms of the Casimir operators remains valid, whatever they represent.

Spacetime Symmetries in Quantum Mechanics 281

Furthermore, the strategy admits a further generalization in its application torelativistic quantum theories, such as relativistic quantum mechanics and quantumfield theory: The interpretive postulate of endowing the Casimir operators withdefinite-valuedness and objectivity is retained, but the relevant group structure isreplaced by changing the Galilean group by the Poincaré group.

14.4 Quantum Mechanics and Time Reversal

14.4.1 A General Notion of Time Reversal

What was said in the previous section seems not to straightforwardly apply to thequestion about time-reversal symmetry. To begin with, time reversal is a trans-formation that does not belong to the Galilean group. Furthermore, while allGalilean transformations are continuous, time reversal is a discrete transformationthat, at first glance, simply performs the transformation t ! �t. Notwithstandingthese facts, time reversal encloses even more subtle features as it is somehowrelated to the nature of time and its unavoidable differences with respect to space;whereas one can move freely in all directions of space, it seems that one “moves”in just one direction of time, from past to future and never the other way around.From early twentieth century, the very notion of time reversal was quite relevantnot only for many physicists working on the foundations of physics, but also formany philosophers aiming at getting a grasp of the nature of time.

Unfortunately, time is not the kind of thing one can experiment on: Unlikeelectrons, pendulums or electromagnetic fields, one cannot directly find out time’sproperties by running an experiment. However, physicists and philosophers havemanaged this setback by devising a formal way to dig into time’s properties – thenotions of time reversal and time-reversal invariance have been keystones for thefamous problem of the arrow of time in physics (problem lying on the borderlinebetween the physics and the metaphysics of time). In fact, the arrow of time haslargely been introduced in terms of time-reversal symmetry: If physical lawssomehow fail to be time-reversal invariant, then one might come to the conclusionthat time is headed according to such a physical law. To put it differently, time-reversal symmetry is supposed to shed light on the structure of time according to agiven theory. Quoting Jill North:

If the fundamental laws cannot be formulated without reference to a particular kind ofstructure, then this structure must exist in order to support the laws – “support” in the sensethat the laws could not be formulated without making reference to that structure.

(North 2009: 203)

The idea is that, by knowing how dynamical equations (standing for physical laws)behave under time reversal, one can learn about the nature of time according to a

282 Cristian López and Olimpia Lombardi

theory. This kind of principle is well-seated in the literature (see also Earman 1974,Sklar 1974, Arntzenius 1997), and it is the key element that links time’s properties,physical laws, and the time-reversal transformation.

However, here nothing has yet been said about the time-reversal transformationin itself, other than that it minimally performs the transformation t ! �t. Astypically noted in the literature, the topic is somewhat tricky as there is no sharedunderstanding of what time reversal is exactly supposed to do nor of what proper-ties it should instantiate (see Savitt 1996 for a careful analysis of varied time-reversal operators; see Peterson 2015 for an updated approach). Furthermore, timereversal’s properties seem to change from theory to theory, to the extent thatdynamics also changes. This remark already assumes a strong premise – that thestructure of time, in particular, regarding its time-reversal symmetry property, isclosely tied up to the theory’s dynamics.

14.4.2 Time Reversal in Quantum Mechanics

Independently of the considerations discussed previously, the community ofphysicists has reached a wide consensus about the appropriate time-reversaltransformation for standard quantum mechanics. The traditional procedure startsout by arguing that time reversal can no longer be considered as in Hamiltonianclassical mechanics. As it is well known, the time-reversal operator in classicalcontexts is typically defined as the operator that changes the sign of the variable t.But in quantum mechanics the rationale seems to be different; in fact, Ballentine(1998) warns us:

One might suppose that time reversal would be closely analogous to space inversion, withthe operation t ! �t replacing x ! �x. In fact, this simple analogy proves to bemisleading at almost every step.

(Ballentine 1998: 377)

Quantum mechanics textbooks rarely offer a thorough justification for such a claimand commonly go on by formally introducing the “proper” way to reverse time inquantum mechanics. In some cases, the only justification is based on a classicalanalogy: The transformation t ! �t does not lead to the transformation ofmomentum as P ! �P, which is expected because this is the way in whichmomentum transforms under time reversal in classical mechanics. But mereanalogy does not seem to be a sufficiently good argument; for this reason, BryanRoberts (2017) has very recently brought up an updated and purely quantum-mechanic-based reasoning for defending the standard procedure. Here we will notanalyze in detail those arguments; rather, we will consider the problem in the lightof the symmetries of the Schrödinger equation related to the reversal of time.

Spacetime Symmetries in Quantum Mechanics 283

Let us use θ to call a generic time-reversal operator, which performs at least thetransformation t ! �t but whose precise form is not defined yet. As explained inSection 14.2, this operator acts as φj i ! ~φj i, O ! ~O, d=dt ! ~d=dt, and i ! ~i; inparticular, ~φj i ¼ θ φj i, ~O ¼ θOθ�1, ~d=dt ¼ θd=dt θ�1, and~i ¼ θ iθ�1. The Schrö-dinger equation is covariant/invariant under θ when Eq. (14.2)/(14.3) holds,respectively. So, let us begin by premultiplying the two members of the Schrö-dinger equation by θ and using the property θ�1θ ¼ I:

θd

dt

� θ�1θ φj i ¼ �θiθ�1θHθ�1θ φj i (14.15)

As long as θ is not a function of t, it is easy to prove that the Schrödinger equationis covariant under the application of θ:

~d ~φj idt

¼ �~i ~H ~φj i (14.16)

Now, in order to know if the Schrödinger equation is also invariant under theapplication of θ, it is necessary to define the precise form of θ to see how it actsupon d=dt, i, and H. As in the case of the Galilean group, the situation of a closed,constant-energy system will be considered.

Case (i): If θ ¼ T only performs the transformation t ! �t, then

~d

dt¼ T

d

dt

� T�1 ¼ � d

dt~i ¼ TiT�1 ¼ i ~H ¼ THT�1 ¼ H (14.17)

Introducing these equations into Eq. (14.16) leads to the conclusion that theSchrödinger equation is not T-invariant, since

d ~φj idt

¼ iH ~φj i (14.18)

Case (ii): If θ ¼ T∗ performs the transformation t ! �t and the complexconjugation i ! �i, then

~d

dt¼ T∗ d

dt

� T∗�1 ¼ � d

dt~i ¼ T∗iT∗�1 ¼ �i ~H ¼ T∗HT∗�1 ¼ H

(14.19)

Introducing these equations into Eq. (14.16) leads to the conclusion the Schrödingerequation is T∗-invariant, since

d ~φj idt

¼ �iH ~φj i (14.20)

284 Cristian López and Olimpia Lombardi

Summing up, independently of any interpretation, the results given by Eq. (14.18)and Eq. (14.20) show that the Schrödinger equation is not invariant under the unitaryoperatorT , and it is invariant under the antiunitary operatorT∗. The conceptual questionis which of the two operators, T or T∗, represents the operation of time reversal.

14.4.3 Time-Reversal Invariance: Between Petitio Principiiand a Priori Truth

A common answer to that conceptual question is saying that the unitary operator T isunacceptable as the time-reversal operator because it breaks the requirement that theenergy of the system must be bounded from below. In Jun John Sakurai’s words:

Consider an energy eigenket nj i with energy eigenvalue En. The corresponding time-reversed state would be Θ nj i [where Θ stands for our T], and we would have, because of(4.4.27) [�HΘ ¼ ΘH]

HΘ nj i ¼ �ΘH nj i ¼ �Enð ÞΘ nj i (4.4.28)

This equation says that Θ nj i is an eigenket of the Hamiltonian with energy eigenvalues�En. But this is nonsensical even in the very elementary case of a free particle. We knowthat the energy spectrum of the free particle is positive semidefinite – from 0 to +∞. Thereis no state lower than a particle at rest (momentum eigenstate with momentum eigenvaluezero); the energy spectrum ranging from �∞ to 0 would be completely unacceptable.

(Sakurai 1994: 272–273)

Why does the unitary operator T not meet that requirement? It is not unusual to readthat the reason is that the unitary operator T transforms the Hamiltonian asTHT�1 ¼ �H. This sounds very strange because, by performing only the transform-ation t ! �t, the operator T should leave the time-independent Hamiltonian invari-ant. So, now the question is: Why does the Hamiltonian transform as H ! �H?Although not always explicit, a typical answer is that offered by StephenGasiorowicz:

we find that [the equation of motion] can be invariant only if

THT�1 ¼ �H

This, however, is an unacceptable condition, because time reversal cannot change the spectrumof H, which consists of positive energies only. If T is taken to be anti-unitary [our T∗], the* operator changes the i to �i [in the equation of motion] and the trouble does not occur.

(Gasiorowicz 1966: 27; italics added)

In a similar vein, Robert Sachs clearly explains that

we require that the [time-reversal] transformations leave the equations of motion invariantwhen all forces or interactions vanish.

(Sachs 1987: 7; italics added)

Spacetime Symmetries in Quantum Mechanics 285

In other words, under time reversal, the Hamiltonian should transform as H ! �Hin order to preserve the time-reversal invariance of the Schrödinger equation; forthis reason, the unitary T is unacceptable as time-reversal operator, and the rightone is the antiunitary T∗.

At this point, one seems to be caught in the dilemma between petitio principiiand a priori truth. If our original question was whether the Schrödinger equation istime-reversal invariant, an argument that selects the right operator describing timereversal by taking the time-reversal invariance of the equation as one of itspremises clearly begs the question. However, some authors do not fall in petitioprincipii by considering that certain symmetries of the physical laws have an apriori status:

A symmetry can be a priori, i.e., the physical law is built in such a way that it respects thatparticular symmetry by construction. This is exemplified by spacetime symmetries,because spacetime is the theater in which the physical law acts . . . and must thereforerespect the rules of the theater.

(Dürr and Teufel 2009: 43–44)

From this perspective, the invariance under the Galilean group must be built intothe Schrödinger equation due to the homogeneity and the isotropy of space and thehomogeneity of time. This view may sound reasonable to the extent that those arefeatures of space and time that we, in a certain sense, can experience. But, whyshould we impose time-reversal invariance? We have no experience of the isotropyof time since we cannot travel backwards in time. Despite this, time-reversalinvariance must be introduced as a postulate:

One should ask why we are so keen to have this feature in the fundamental laws when weexperience the contrary. Indeed, we typically experience thermodynamic changes whichare irreversible, i.e., which are not time reversible. The simple answer is that our platonicidea (or mathematical idea) of time and space is that they are without preferred direction,and that the “directed” experience we have is to be explained from the underlying timesymmetric law.

(Dürr and Teufel 2009: 47)

Challenging the most widely-held position about time reversal in the fieldof quantum mechanics, a few authors have raised their voices against it byappealing to philosophical reasons: The antiunitary operator T∗ would fail tooffer a conceptually sound and clear-cut representation of time reversal. Onthe one hand, a far-reaching tradition, which tracks back to the work of GiulioRacah (1937) and Satosi Watanabe (1955), pleads for a unitary time-reversaloperator in quantum theories. Oliver Costa de Beauregard (1980) has arguedfor such a view by claiming that a unitary time-reversal operator that merelyreverses the direction of time by flipping the sign of the variable t goes more

286 Cristian López and Olimpia Lombardi

naturally along with relativistic contexts and is more naturally akin to theFeynmann zig-zag philosophy. On the other hand, philosophers such as CraigCallender (2000) and David Albert (2000) have claimed that the Schrödingerequation should actually be considered as nontime-reversal invariant, sinceit is not invariant under T , which more fairly represents what one meansby “reversing the direction of time.” Without any further ado, Jill Northclaims:

What is a time-reversal transformation? Just a flipping of the direction of time! That is allthere is to a transformation that changes how things are with respect to time: change thedirection of time itself.

(North 2009: 212)

For these philosophers, if the time-reversal invariance of a theory will tell ussomething about the structure of time, time reversal should only reverse thedirection of time without extra additions. In particular, if the question at issue isthe problem of the arrow or time and the time-reversal invariance of the theory isconsidered relevant to this problem, imposing the time-reversal invariance of theSchrödinger equation as a requirement that the theory must satisfy is a circularstrategy.

14.4.4 Wigner’s Definition

The line of argumentation sketched in the previous section seems to be notcompletely convincing for adopting the antiunitary operator T∗ as the adequaterepresentation of time reversal. Yet a thorough argument can be introduced byappealing to the authority of Eugene Wigner, who defines time reversal as atransformation such that:

The following four operations, carried out in succession on an arbitrary state, will result inthe system returning to its original state. The first operation is time inversion, the secondtime displacement by t, the third again time inversion, and the last on again timedisplacement by t.

(Wigner 1931/1959: 326)

In other words,

time reversal � displacement by Δt � time reversal � displacement by Δt = identity

This requirement is commonly interpreted in formal terms as follows:

UΔt θ�1UΔt θ s ¼ s (14.21)

where s is the arbitrary state and UΔt is the evolution operator for Δt. Thisrequirement is precisely Sakurai’s starting point in the argument that led him to

Spacetime Symmetries in Quantum Mechanics 287

the conclusion stated previously. Under the assumption that the evolution oper-ators form a group (an assumption that is not always satisfied, see Bohm andGadella [1989], where the time evolution is represented by a semigroup), a U�Δt

exists such that U�Δt UΔt ¼ I. In this case, Eq. (14.21) becomes

θ�1UΔt θ s ¼ U�Δt s (14.22)

In quantum mechanics, the argument continues, UΔt ¼ e�iHΔt:

θ�1e�iHΔt θ φj i ¼ eiHΔt φj i (14.23)

Therefore,

θ�1 �iHð Þθ ¼ iH (14.24)

So, when θ is the unitary operator T , this leads to

�H ¼ THT�1 (14.25)

which is unacceptable because it leads to values of energy that are unbounded frombelow. Since Wigner (1931/1959) also proved that any symmetry transformation isrepresented by a unitary or an antiunitary operator, the argument concludes that theright time-reversal operator is the antiunitary operator T∗.

This argument in favor of T∗ is certainly much better than the previous one,which takes the time-reversal invariance of the Schrödinger equation as one of thepremises. Nevertheless, as we will see, this second argument imposes the time-reversal invariance of the dynamical equation beforehand as well, even if in a moresubtle way.

Let us begin by noticing that, when Eq. (14.21) is used to formalize Wigner’srequirement, the “displacement by Δt” is represented by the time evolution of thesystem by Δt according to the dynamical law of the theory – the Schrödingerequation, here expressed as s ¼ UΔt s0. However, as stressed in Section 14.3.4,when considering the Galilean group, time displacement is not time evolution.Time evolution is ruled by the dynamical law of the theory, in this case quantummechanics: According to the Schrödinger equation, the Hamiltonian is the gener-ator of the dynamical evolution. By contrast, spacetime transformations arepurely geometric operations of displacing or rotating the system self-congruently.In particular, time-displacement is a purely geometric operation that displaces thesystem self-congruently to another time, and may agree or not with time evolu-tion. In fact, as Hans Laue (1996) and Leslie Ballentine (1998) stress, in a genericcase, the Hamiltonian is not the generator of time displacements and only retainsits role as the generator of the dynamical evolution. This clearly shows thattime displacement and time evolution are different concepts. Hence, insofar as

288 Cristian López and Olimpia Lombardi

Wigner’s definition involves time displacement and not time evolution, it must beformally expressed as (recall how Galilean transformations act upon states,Eq. (14.5))

Uτ θ�1Uτ θ s ¼ eiKτΔt θ�1eiKτΔt θ s ¼ s (14.26)

where Kτ is the generator of time-displacement (see Eq. (14.3)). Now, since U�τ

exists such that U�τUτ ¼ I, an analogue of Eq. (14.23) can be obtained:

θ�1eiKτΔt θ φj i ¼ e�iKτΔt φj i (14.27)

where the difference in sign between Eq. (14.27) and Eq. (14.22) is due to theinverse relation between transformations on function space and transformations oncoordinates (see Ballentine 1998: 67). Eq. (14.27) expresses what time reversalmeans: It should be a transformation such that

time reversal � displacement by Δt � time reversal = displacement by �Δt

More explicitly, take an arbitrary state, time-reverse it, time-displace it by Δt in agiven time-displacement direction, and time-reverse it again; these three operationsmust be equivalent to time-displace the original state the same time interval Δt inthe opposite time-displacement direction. Eq. (14.27) represents formally thiscondition, but no conclusion about how the Hamiltonian is transformed by timereversal can be drawn from it.

Even if accepting the conceptual difference between time displacement and timeevolution in Wigner’s definition of time reversal, somebody might retort by sayingthat there are cases in which time displacement amounts to time evolution: asdiscussed in Section 14.3, in those cases the generator Kτ of time displacement isequal to the generator H of time evolution. If Kτ is replaced by H in Eq. (14.27),when θ is the unitary operator T the relation �H ¼ THT�1 of Eq. (14.25) obtainsagain, and this is sufficient to discard the unitary operator T as the adequaterepresentation of time reversal.

Although the argument just discussed seems to be conclusive, when con-sidered in detail, the implicit assumptions come to light. In fact, the argumentequates time displacement and time evolution, both in the case of Δt and in thecase of �Δt. We have good empirical reasons to accept that, in certain cases, thetime displacement toward the future by Δt is equivalent to the time evolutiongiven by UΔt ¼ e�iHΔt. But we do not know how the system would evolve intime toward the past, as we have no experience at all of such an evolution; this isthe specific feature that makes time so different than space. If we impose that,when time displacement is time evolution toward the future, this is the case

Spacetime Symmetries in Quantum Mechanics 289

toward the past too, then we are introducing the time-reversal invariance of thedynamical law by hand.

In other words, the question about the time-reversal invariance of a law isprecisely the question of whether the time displacement of the system toward thepast is also ruled by the dynamical law, that is, whether it is also a time evolution. Ifthe answer is positive, the law is time-reversal invariant, if the answer is negative, thelaw is not time-reversal invariant. Therefore, supposing from the very beginning thatany time displacement toward the past is a dynamical evolution amounts to puttingthe cart before the horse.

But, then, what do Eq. (14.21) and Eq. (14.22) mean? Actually, those equationsexpress the conditions that define what can be called motion reversal:

motion reversal � evolution by Δt � motion reversal � evolution by Δt = identity

motion reversal � evolution by Δt � motion reversal = evolution by �Δt

To put it precisely, the motion-reversal operator is the operator that reverses thedirection of a lawful motion of the system so as to obtain another lawful motion.Then, the argument that, starting by Eq. (14.21), concludes with discarding theunitary operator T is a proof of the fact that the antiunitary operator T∗ is the rightmotion-reversal operator for quantum mechanics.

Even though the difference between motion reversal and time reversal has notbeen sufficiently stressed, it is acknowledged by some authors. For example,Ballentine clearly states:

In the first place, the term “time reversal” is misleading, and the operation that is thesubject of this section would be more accurately described as motion reversal. We shallcontinue to use the traditional but less accurate expression “time reversal”, because it is sofirmly entrenched.

(Ballentine 1998: 377; italics in original)

Sakurai also emphasizes the point just at the beginning of the section devoted totime reversal:

In this section we study another discrete symmetry operator, called time reversal. This is adifficult topic for the novice, partly because the term time reversal is a misnomer; it remindsus of science fiction. Actually what we do in this section can be more appropriatelycharacterized by the term reversal of motion. Indeed, that is the terminology used byE. Wigner, who formulated time reversal in a very fundamental paper written in 1932.

(Sakurai 1994: 266; bold and italics in original)

Summing up, it is quite clear that the antiunitary operator T∗ is the motion-reversaloperator in quantum mechanics. But the initial question still remains: which is theright quantum time-reversal operator?

290 Cristian López and Olimpia Lombardi

14.5 Conclusions

In this chapter we have focused on the spacetime symmetries of quantum mechan-ics under the assumption that exploring the meaning of those symmetries isrelevant to the interpretation of the theory.

In the first part, we have considered the behavior of nonrelativistic quantummechanics under the Galilean group. We have shown that the Schrödinger equa-tion is always covariant under the Galilean group, but its Galilean invariance canonly be guaranteed when it is applied to a closed system free from external fields.We have also discussed the relevance of symmetries to interpretation; in particular,any realist interpretation that intends to select a Galilean-invariant set of definite-valued observables should make that set to depend on the Casimir operators of theGalilean group, since they are invariant under all the transformations of the group.In future works, these conclusions can be extended in two senses. On the one hand,they can be transferred to quantum field theory by changing the symmetry groupaccordingly: The definite-valued observables of a system in quantum field theorywould be those represented by the Casimir operators of the Poincaré group. Sincethe mass operator M and the squared-spin operator S2 are the only Casimiroperators of the Poincaré group, they would always represent definite-valuedobservables, a view that stands in agreement with a usual physical assumption inquantum field theory. On the other hand, if invariance is a mark of objectivity,there is no reason to focus only on spacetime global symmetries. Internal or gaugesymmetries should also be considered as relevant in the definition of objectivityand, as a consequence, in the identification of the definite-valued observables ofthe system.

In the second part of the chapter, we have carefully disentangled the differentnotions involved in the issue of the time-reversal invariance of the Schrödingerequation. We have assessed the usual claim about the matter, according to whichthe Schrödinger equation is time-reversal invariant and the quantum time-reversaloperator is antiunitary. We have argued that the antiunitary operator is actually amotion-reversal operator and that the question about the right time-reversal oper-ator in quantum mechanics is still an open question. Those who think that time isontologically independent of and prior to the processes in it will stress thedifference between time reversal and motion reversal and, consequently, may tendto prefer a time-reversal operator that only flips the direction of time. Others, bycontrast, may claim that the very concept of time as independent of motion has nomeaning. From this relationalist-like view, distinguishing between time reversaland motion reversal as different operations makes no sense and, as a consequence,the right time-reversal operator is necessarily a motion-reversal operator. Thisshows that the question about the time-reversal invariance of quantum mechanics

Spacetime Symmetries in Quantum Mechanics 291

involves deep issues about the very nature of time. But the further development ofthis aspect of the problem will be the subject of future work.

Acknowledgments

We want to thank the participants of the workshop Identity, indistinguishabilityand non-locality in quantum physics (Buenos Aires, June 2017) for their contribu-tion to a philosophically exciting and fruitful time. This work was made possiblethrough the support of Grant 57919 from the John Templeton Foundation andGrant PICT-2014–2812 from the National Agency of Scientific and TechnologicalPromotion of Argentina.

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15

Symmetry, Structure, and Emergent Subsystems

nathan harshman

15.1 Introduction

One of the most evocative results in the whole history of mathematical physics isthat there are exactly five polyhedra with perfect symmetry, i.e., all faces, edges,and vertices are congruent. Starting from the intuitive and reasonable definitionsand axioms of Euclidean geometry and applying the constraint of symmetry, thesefive structures are logically inevitable. Book 13, the climax of Euclid’s Elements,constructs these solids and proves they exhaust the possibilities. While the beautyof symmetric geometry provokes attention, I believe it is as much their “fiveness”that inspires. Plato, Kepler, and others sought five-fold explanations for thephysical structures of the world (see Wilczek 2015). Those chimerical hopesnotwithstanding, Platonic solids do appear throughout science and technologyfor natural and practical reasons: They are fundamental structures.

Starting in the nineteenth century, combining generalized notions of geometryand symmetry yielded a bounty of new structures. In turn, these structures pro-vided a framework for interpreting, classifying, and generating mathematicalmodels of physical reality. Glossing over the technical details (upon which hangcritical distinctions and academic careers), the same structural system that classifiesPlatonic solids and their generalizations in any dimension also classifies Liegroups, dynamical catastrophes, symmetric manifolds, random matrices, topo-logical insulators, gauge quantum field theories, and so on. The imposition ofsymmetry on vector spaces is surprisingly rigid, often giving finite or countablepossibilities. And then, more often than not, it seems we find these possibilitiesmanifest in our models of systems and dynamics. This coincidence of logicallyinevitable mathematical structures with elements of physical reality remains asseductive now as it did to Plato and Kepler. Some theoretical physicists spend theircareer chasing these beautifully symmetric models and enumerating their qualities,hoping one day the model will play an important role in the next “paradigm shift.”

294

My proposal is more modest. Symmetry is not the answer to every question, andthe universe may have contingent features that no model can predict or explain.However, symmetries seem to exist in reality and certainly exist in many effectiveand productive models of reality. Within a framework like quantum mechanics, thepresence of symmetry entails the existence of mathematical structures that areprivileged by their relation to the symmetry. The focus of this chapter is theparticular structures called irreducible representations of symmetry groups. Similarto how a Hamiltonian gives a spectrum in quantum mechanics, the irreduciblerepresentations (irreps) of a group of symmetry transformations form a kind ofbasis for possible manifestations of a symmetry on vector spaces. Like the Platonicsolids, these irreps are the denumerable “atoms” of the Hilbert spaces of quantummechanical models with symmetry.

For example, any quantum model with special unitary, or SU(2) symmetry has adescription in terms of units that are often called spins, even when they have norotational origin. These spin units are Hilbert subspaces that carry irreps of SU(2),and the total Hilbert space is reducible into products and sums of these “atoms”of spin.

Similarly, quantum field theory is built by associating free particles to irreps ofthe Poincaré group of symmetry transformations of Minkowski spacetime (see, forexample, Streater and Wightman 1964, Haag 1992, Weinberg 1995). As EugeneWigner (1939) discovered, the irreps of the Poincaré group are convenientlylabeled by three invariants: mass, internal energy, and spin. Irreps of the Poincarégroup are equivalent to relativistic particles for most technical and interpretationalpurposes. A similar construction with the Galilean group and nonrelativisticparticles is discussed later.

Let us recall that, technically, relativistic particles correspond to irreps of theuniversal covering group of the Poincaré group. This is necessary to account forthe fact that, in quantum mechanics, we are typically interested in projectiverepresentations, i.e., representations up to a phase. Projective representations aremore general than precise representations, and they preserve the interpretation ofpure states as rays in the Hilbert space. This technicality is mostly swept under therug for the rest of this chapter.

These examples make it clear that using irreps to analyze symmetry in quantummechanics is an old story, full of many bold successes and productive technicaldetails. Why rehash it here? The purpose of this contributed chapter is to explorethe connections between the mathematical units of symmetry embodied by irreps,arguably the “most inevitable” symmetric structures of quantum mechanics, andthe conceptual units of reality that form the basis for interpretation of quantumtheories. Since irreps are symmetric structures that have the appealing properties ofbeing denumerable, they hold the same appeal as Platonic “fiveness.” Their logical

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inevitability (given the standard formulation of quantum mechanics) makes themthe natural vocabulary for asking and answering questions about the fundamentalnature of quantum reality, whether a more epistemic or ontic interpretation isadvanced. At the very least, any interpretation of quantum mechanics must providesome justification for the “unreasonable effectiveness” of these mathematicalstructures as conceptual units. At the same time, the conceptual overreach perpet-rated by natural philosophers enamored by the Platonic solids should serve as acautionary tale.

With this motivation, Section 15.2 gives the local definitions of quantummechanical models and symmetries. Section 15.3 and Section 15.4 subject thereader to a technical presentation about “Hilbert space arithmetic”. The Hilbertspace can be built “bottom-up” out of products and sums of irrep spaces, orreduced “top-down” into products and sums of subspaces that carry irreps.I argue that Hilbert space arithmetic, and the identification of how symmetrymanifests itself in that arithmetic, provide a lens for understanding quantumconcepts like energy levels, entanglement, locality (and its generalization, specifi-city), distinguishability, and emergence.

But the view through that lens is not yet clear, so I also offer to the skepticalreader an application of symmetrized Hilbert space arithmetic. In Section 15.5,I investigate whether the mathematical structures of irreps have interpretativepower when considered as conceptual units of reality within a model of quantumfew-body systems in one dimension. This choice is partially due to the charmingcoincidence that such systems can carry realizations of the same symmetries as thePlatonic solids and their generalizations. More meaningfully, quantum few-bodysystems in one dimension are at the knife’s edge in terms of dynamical regimes forclosed quantum systems. Depending on the symmetry of the Hamiltonian, suchsystems can have dynamics that are regular, integrable, and solvable or irregular,chaotic, and ill-conditioned. I argue that irreps form conceptual units to interpretthis rich physics of few-body systems.

The stakes of this analysis have been raised by recent experiments with ultracoldatoms in effectively one-dimensional optical traps (see Serwane et al. 2011, Zürnet al. 2012, 2013, Wenz et al. 2013). These offer the possibility of implementingcontrollable few-body models, and they provide a relevant milieu for evaluatingclaims of interpretative utility of irreps as conceptual units. Because irreps arederived from symmetries of the Hamiltonian of a few-body model, they can beused to find useful observables to describe the collective degrees of freedom offew-body systems (Harshman 2016a, 2016b). Further, the connections betweenreducibility into irreps and solvability, while not completely clear, can still beproductive in the quest to identify new solvable models in few-body physics (seeHarshman 2017a, Andersen, Harshman, and Zinner 2017 for recent examples).

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Finally, the connections between solvability and controllability provide opportun-ities for technological applications with ultracold atoms. Controllability is a keyrequirement for the relativity of entanglement pioneered by Zanardi (see Section15.4). As a practical concern for developing quantum information processingdevices, it motivates the search for emergent subsystems (Zanardi and Lloyd2004).

15.2 Quantum Mechanical Models and Symmetry

Symmetries are often invoked as grand, unifying principles at the foundations ofquantum mechanics. For example, the standard model of particle physics ispresented as though it emerges as a logical consequence of combining the sym-metries of Minkowski spacetime with gauge symmetries of internal flavors. Such apresentation is an exaggeration, as the numerous unexplained free parameters inthe standard model attest. Here, I want to present symmetries in the narrowerscope – symmetries as defined within the context of a specific quantummechanical model.

15.2.1 Model Definition and Scope

Following Haag (1992), I define a quantum mechanical model as a set of observ-ables A that are represented by Hermitian operators acting on a Hilbert space H .This rather airless definition of quantum mechanical model makes minimal onto-logical claims beyond the standard framework of quantum mechanics: Pure statesare unit vectors in H , measurements are expectation values of observables, BornRule, and all that. The origin of the observables and their representation within themodel could be established by scientific utility alone, such as making predictions,or by other epistemic concerns. Using this definition of model, I am consciouslyattempting to restrict my ontological commitment (as much as possible) to astructure of relatively defined mathematical objects while still doing “normal”quantum mechanics. To those who have ontological commitments in their inter-pretations of quantum mechanics, I believe that this framework for models isgeneral enough to encompass those theories, and so I ask indulgence in thisexercise in structuralism.

The simplest nontrivial example of a model is given by the algebra of Paulimatrices acting on H � C2. Any two-level system can be described in thisframework, and although there is an analogy to a spin-1/2 system, the modelmakes no ontological commitment to any fundamental particle or other unit ofreality. The model could describe an isolated discrete degree of freedom, like theisospin of a nucleon, or could emerge as an effective theory, like the lowest two

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energy levels of a more complicated system. It could also be derived from thesmallest nontrivial projective representation of an underlying three-dimensionalrotation group, i.e., a true spin. In any case, the model as a mathematical structurecan be studied without reference to any physical embodiment, ontological com-mitment, or larger theoretical framework.

Nonrelativistic few-body models often do start with an ontological commitment,such as N particles on an underlying d-dimensional Euclidean space X � Rd. TheHilbert space is realized as H � L2 RNd

� �, Lebesgue square-integrable functions

on the configuration space X�N constructed from the N-fold Cartesian product ofsingle-particle spaces. The corresponding set of observables is generated by theHeisenberg algebra of 2Nd canonical position and momentum operators which arerepresented as variable multiplication and differentiation operators on L2 RNd

� �.

There are certainly technical demons (like unbounded operators) hidden in thismodel, but for now let us assume they can be exorcised.

The subsequent analysis presumes the existence of a model, whether it has beendefined in the abstract or derived from underlying particles or other units of reality.The goal is then to use symmetry to identify structures within the model that aidinterpretation of empirical phenomena without making further commitments. Notethat in both of the examples presented earlier, the set of observables was actuallyan algebra of observables, closed under addition and multiplication. That will notalways be the case for models I consider. For models with finite-dimensionalHilbert spaces the distinction is minor, but it is important for infinite-dimensionalspaces.

In order to make progress, I restrict my attention to models with two additionalproperties. First, I assume that within the set of observables there is a particularobservable H 2 A called the Hamiltonian. The Hamiltonian generates time trans-lations by the unitary operator U tð Þ ¼ exp �iHt=ℏð Þ defined on all of H . Thisassumption thereby excludes models that describe systems with time-dependentHamiltonians or open dynamics. These are obviously cases of physical interest, butthe complications they introduce require attention to detail beyond the scope of thischapter.

Second, I assume that the set of observables is represented on a Hilbert spacethat is separable in the topological sense. A separable Hilbert space has a countableorthonormal basis (see Streater and Wightman 1964). Equivalently, it implies theexistence of a decomposition of the Hilbert space H of the system into a direct sumof a countable (but still possibly infinite) set of one-dimensional subspaces H i:

H ¼ ⊕iH i (15.1)

Restricting to models with separable Hilbert spaces is a convenience that (to myknowledge) is not much of a restriction for models of quantum mechanical systems

298 Nathan Harshman

with a finite number of particles. However, there are many-body systems andquantum field theories where this kind of separability cannot be assumed (again,see Streater and Wightman 1964).

15.2.2 Model Symmetries

Following Wigner (1959), a symmetry of a model is a group of unitary (andpossibly antiunitary) operators on H . Such operators preserve the magnitude ofinner products and therefore of probabilities. For finite dimensional Hilbert spacesH � CD, the symmetry group of the model must be a subgroup of the unitarymatrices U Dð Þ. For infinite dimensional separable Hilbert spaces, the unitary groupU ∞ð Þ can be defined by the inductive limit of U Dð Þ (Olshanski 2003) and anysubgroup of U ∞ð Þ could be a symmetry group. But most of these subgroups do nothave any physical interpretation; they are purely formal. So how do we distinguishand classify useful or meaningful symmetry groups?

Within the context of a formal model, the empirical content of a symmetry isinferred from its relation to the set of observables, specifically, how the unitarygroup of symmetry operators transform the Hilbert-space representations of theobservables. For example, the symmetry identifies invariant operators of themodel, or more generally, can be used to classify operators based on theirtransformation properties (Lombardi and Fortin 2015). A familiar example isclassifying operators by their tensor rank under a representation of a matrix group,e.g., scalar, vector, pseudovector, and so on for the rotation group.

However, starting from a formal model with Hilbert space and algebra and thenidentifying meaningful symmetries is not always how physical analysis proceeds.The logic is sometimes reversed and the model is constructed from the symmetryof an underlying space or spacetime. The group of unitary operators on the Hilbertspace is a representation of the universal covering group of some spacetimesymmetry. The set of observables is the enveloping algebra built from the gener-ators of the symmetry representation. Three relevant examples are: (1) spin modelsfrom rotational symmetry; (2) free nonrelativistic particle models from Galileansymmetry (Lévy-Leblond 1967); and (3) free relativistic particle models fromPoincaré symmetry (Wigner 1939). In each of these cases, the Hilbert space of asingle spin or a single particle corresponds to an irreducible representation space ofthe spacetime symmetry group. As mentioned before, technically “particles”correspond to irreps of the universal covering groups of these spacetime symmet-ries. This distinction allows for projective representations. Further, in order to getmassive representations, an observable corresponding to mass must be added as acentral element to the Galilean algebra. Bottom-up approaches to quantum

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mechanics often take these basic models as starting units and build more compli-cated models out of their irreps spaces and algebras of observables.

In practice, identifying the symmetries of a nonrelativistic few-body modeltakes place using a hybrid of formal, top-down and reductionist, bottom-upapproaches. Symmetries in physical space are important, but so are symmetriesin the spaces of configuration space and phase space. Although these auxiliaryspaces can be derived from the bottom-up approach, starting with free particlemodels, the symmetries of these derived spaces may not be easily reducible intofree-particle symmetries and their corresponding irreps can describe collective oremergent degrees of freedom. Additionally, few-body models can have symmet-ries that are defined by unitary operators on the Hilbert space itself withoutreference to a symmetry of an underlying space. For example, these kind ofsymmetries are present when there are accidental degeneracies (Harshman 2017a).

15.2.3 Kinematic and Dynamic Symmetries

The symmetry groups with which I am primarily concerned are those representedby unitary operators that leave the Hamiltonian invariant. Call these kinematicsymmetries of the model. These unitary operators transform stationary states of theHamiltonian, i.e., energy eigenstates, into other energy eigenstates with the sameenergy. Sometimes instead of considering a single Hamiltonian, it is profitable toconsider the kinematic symmetry of a family of Hamiltonians that (for example)depends on a parameter or parameters but exist within the same model. This can bea tricky business, because sometimes as the parameter is varied the Hamiltonianloses the property of self-adjointness on the original Hilbert space of the model.A notable example is when parameter variation makes configuration space effect-ively disconnected (Harshman 2017a).

As opposed to the kinematic symmetry group of a Hamiltonian, a dynamicsymmetry group does not commute with the Hamiltonian as a whole (althoughsome elements may). This definition encompasses a wide class of possibilities. Intheir simplest manifestation, dynamic symmetries provide algebraic relationshipsbetween the group and the Hamiltonian that map energy eigenstates into otherenergy eigenstates (like ladder operators) and induce algebraic relationships amongexpectation values of noncommuting observables (Wybourne 1974). Poincaré andGalilean transformations are also dynamic symmetries in this sense; boosts changethe energy of a state in an algebraic way. Dynamic symmetries can also describemaps among Hamiltonians within a parametrized family, like a scale transform-ations (Jackiw 1972). They can even serve as maps between models and Hamilto-nians that on the surface seem radically different, like supersymmetric partnerHamiltonians in quantum mechanics (see Cooper, Khare, and Sukhatme 1995).

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15.3 Hilbert Space Arithmetic: Direct Sum Decompositions

The definition of model provided here seemingly relies on a the Hilbert space as anessential feature, and this section and the next are going to outline how Hilbertspaces can be decomposed into direct sums and factored into tensor products usingobservables and symmetries as the starting point. Topological separability of theHilbert space is assumed throughout this vector space arithmetic, and the goal is tosee how much interpretive power this kind of analysis can provide. Carving modelsinto subspaces by decomposing into direct sum and into submodels by factoring intotensor products has a long history in quantum mechanics for practical reasons ofmathematical analysis. I argue that it also forms units of conceptual analysis that arecalled different names in different contexts but are all manifestations of the sameunderlying structure. The tentative claim is that these structures are as valid asconceptual units of reality as are the more tangible concepts like “particles.”

However, before proceeding, I must declare that I am not a Hilbert spacefetishist. The specific topology of the Hilbert space is at once too loose and toorestrictive for some purposes. For example, the domains on which the set ofobservables are bounded and self-adjoint may not be the entire Hilbert space,and their eigenspaces may not be contained in the Hilbert space. This technicalaspect can be rigorously handled using Gel’fand triplets and rigged Hilbert spaces(Bohm and Gadella 1989, Bohm and Harshman 1998), and so for the rest of thissection when I say “Hilbert space,” assume that I am talking about decomposingand factoring some kind topological vector space with suffciently nice properties.

15.3.1 Hamiltonian-Induced Direct-Sum Decomposition

Observables can be used to partition the Hilbert space into subspaces. Each sub-space is associated to an eigenvalue of the operator, and on that subspace thatoperator acts like a multiple of the identity. The most familiar example is theHamiltonian H. Assume for simplicity that the spectrum of the Hamiltonian σ Hð Þis discrete, as it would be for most systems with finite extent. Then the Hilbertspace for the system is decomposed on the spectrum of H as

H ¼ ⊕E2σ Hð Þ

HE (15.2)

On the surface, this looks like Eq. (15.1), but here, instead of a decomposition intoone-dimensional spaces, each energy eigenspace HE is realized by a finite-dimensional space Cd Eð Þ with dimension equal to the degeneracy d Eð Þ of theenergy E.

If the Hamiltonian under consideration is time-independent and describes aclosed system, then the system has time-translation symmetry and the

Symmetry, Structure, and Emergent Subsystems 301

decomposition Eq. (15.2) has another interpretation in terms of irreps. Timetranslation symmetry is an abelian symmetry, i.e., a system translated in time byintervals t then t0 is the same as if translated by t0 then t. Abelian symmetries haveone-dimensional irreps, and the unitary operators are just phases. Different irrepsof time translation are distinguished by a scale-setting parameter (called theenergy) that determines how fast the phase advances:

U t0ð ÞU tð ÞHE ¼ e�iEt0=ℏe�iEt0=ℏHE ¼ e�iE t0þtð Þ=ℏHE ¼ U t0 þ tð ÞHE (15.3)

When HE has more than one dimension, it is therefore a sum of multiple time-translation irrep spaces. This implies the existence of at least one other operatorthat commutes with H and can be used to diagonalize the subspace HE. Formally,one can always construct a single operator on the Hilbert space that commutes withH and whose eigenvalues uniquely distinguish every vector in every degeneratespace. Such an operator can be chosen as block diagonal, one block for each HE

consisting of a d Eð Þ-dimensional diagonal matrix with, for example, the numbers1 through d Eð Þ on the diagonal. Call this operator D. Then H can be reduced to adirect sum of one-dimensional subspaces on the joint spectrum of H and D. In thissense, H and D are a complete set of commuting observables. However, theobservable D is defined by its construction as an operator on H and has nofundamental origin as, for example, an observable defined by a measuring appar-atus or the generator of a symmetry transformation. It is an example of a formalmathematical structure without physical interpretation. One goal, when analyzing asystem, is to find operators that perform the same diagonalizing function as D, buthave some other physical meaning, perhaps from kinematic symmetries.

For a particular Hamiltonian, the decomposition of H into energy subspaces HE

is fixed. However, consider a family of Hamiltonians from the same model that canbe characterized by a variable parameter g. The subspaces HE may coalesce or splitas g is varied. The energy eigenspaces are called “levels,” and I argue they functionas interpretational units that are treated as “real” objects. One speaks of level“dynamics,” for example, levels “shifting,” “splitting,” “diverging,” etc., but whatis doing these actions is quite abstract – a subspace built from irreps in a modelwith a family of Hamiltonians.

15.3.2 Observable-Induced Direct-Sum Decomposition

Any observable Λ with a discrete spectrum λ 2 σ Λð Þ can serve as the origin of adecomposition like Eq. (15.2), not just the Hamiltonian:

H ¼ ⊕λ2σ Λð Þ

H λ (15.4)

302 Nathan Harshman

As before, the dimensionality of H λ is the degeneracy of the eigenvalue λ. Todiagonalize this degeneracy, one can add additional observables and arrive at a setof k commuting observables Λ ¼ Λ1; . . . ;Λkf g with discrete eigenvalues denotedλ ¼ λ1; . . . ; λkf g 2 σ Λð Þ. Of course the Hamiltonian could be one of these Λi.

Note that σ Λð Þ is the joint spectrum of all k observables. For many models, thejoint spectrum cannot be decomposed into the product of spectra. When it can (seeHarshman 2016c for examples of separable three-body Hamiltonians), then thedecomposition can be further reduced into independent sums

H ¼ ⊕λ12σ Λ1ð Þ

. . . ⊕λk2σ Λkð Þ

H λ (15.5)

As discussed in (Harshman 2016c), this kind of spectral separability distinguishesHamiltonians with “silver” and “gold” separability from “bronze” separability (thisis a different notion of separability than topological separability; see more discus-sion in Subsection 15.4.4).

15.3.3 Symmetry-Induced Direct-Sum Decomposition

The two previous subsections used observables to decompose the Hilbert spaceinto a direct sum of observable eigenspaces. Groups of symmetry transformationscan be used for the same purpose, but now the subspaces are not necessarilyeigenspaces. Instead, the subspaces are irreducible representation spaces, alsoknown as modules, and carry a linear representation of the transformation group.A representation that is irreducible means that, within the corresponding module,there are no invariant subspaces.

For specificity (and to avoid details that are important but technical) consider agroup G that has only finite-dimensional representations. Discrete finite groups,compact Lie groups, and their combinations have this property. Denote the labelfor an irrep by a Greek letter in parentheses like μð Þ and the corresponding irrepspace with dimension d μð Þ by V μð Þ � Cd μð Þ. Then the Hilbert space can be brokeninto sectors labeled by μð Þ:

H ¼ ⊕μð ÞH μð Þ (15.6a)

Each sector H μð Þ is a tower of irreps spaces

H μð Þ ¼ ⊕iV μð Þi (15.6b)

This may seem all a bit abstract, so here are a few examples. Perhaps thesimplest and most familiar is the case of parity. Parity is realized by a finite groupof two elements Z2. This abelian group has two irreps denoted + for even states

Symmetry, Structure, and Emergent Subsystems 303

under parity and � for odd states. So the Hilbert space can be divided into sectorsof even and odd states H ¼ H þ ⊕H �. Because the group Z2 has only one-dimensional irreps, nothing more can be said except that H þ and H � are builtout of one-dimensional subspaces that are invariant under parity. If parity is akinematic symmetry of a model, then all energy eigenstates are in one of those twosectors and the expectation value of parity is a dynamical invariant. If parity is akinematic symmetry of a family of Hamiltonians, then varying the parameters ofthe family mixes states within a sector, but not across sectors, i.e., changing thecontrol parameter does not change the parity of a state.

Another familiar example is rotational symmetry in three dimensions. Theeigenvalue of the operator representing angular momentum squared ℏ2s sþ 1ð Þcan be used to characterize irreps and irrep spaces. The spin s comes in two infiniteseries: nonnegative integers and nonnegative half-integers. A startling fact, called asuperselection rule, is that a decomposition into rotation group irreps only consistof one of those two types, either integer or half-integer irreps. There can never be asuperposition of states with integer and half-integer total angular momentum. Thatmeans in any quantum model, the Hilbert space only has sectors of integer or half-integer irrep spaces.

A final example is the symmetry of particle permutations for N identicalparticles, realized by the symmetric group SN . Here the irrep decompositionprovides a conceptual unit for analyzing the meaning of identical particles. Irrepsof SN are labeled by positive integer partitions of N. For example, there are fourpartitions of N = 4: [4], [31], [22]�[22], [211]�[212], and [1111]�[14]. Thatmeans the Hilbert space for a model with four identical particles can be decom-posed into five sectors:

H ¼ H 4½ �⊕H 31½ �⊕H 22½ �⊕H 212½ �⊕H 14½ � (15.7)

The sectors have transformation properties dependent on their corresponding irrep.The irrep labeled [4] is the trivial one-dimensional representation in which allpermutations are represented by multiplication by +1. So this sector H 4½ � isappropriate for representing bosons; in fact, it is the full scope of possibilities forfour identical bosons. The irrep labeled [14] is the one-dimensional totally anti-symmetric representations where odd permutations are represented by�1 and evenpermutations by +1. The corresponding sector H 14½ � is therefore where the fer-mions live. The three other sectors become necessary when considering particleswith parastatistics (an exotic generalization between fermions and bosons), or moreprosaically, when considering particles with spin and spatial degrees of freedom,but fixing or tracing over either the spin or spatial degrees.

A sector like H 4½ � or H 14½ � is not a monolithic space; it is a tower of irrep spaces.Alternatively, it can be decomposed into energy subspaces, parity subspaces, etc.

304 Nathan Harshman

The mathematical structure of decomposition into irreps provides not just atechnical tool for solving problems with identical particles, but also a key concep-tual unit for a minimal interpretation of what identical particles mean. This isdiscussed in more detail later, in the specific discussion of the few-body model.

As noted earlier, the Hamiltonian itself is the generator for time translationsymmetry, so the decomposition into energy subspaces Eq. (15.2) is also an exampleof the irrep decomposition Eq. (15.6). Kinematic symmetries add more structure.Then each energy eigenspace is a sum of irrep spaces of the kinematic symmetry:

HE ¼ ⊕μð ÞiV μð ÞE, i (15.8)

A familiar example is the reduction of hydrogen energy levels into subspaces withfixed orbital angular momentum, doubled by the presence of spin. In principle, thedirect sum in Eq. (15.8) extends over all irreps of the kinematic group and there canbe multiple copies of the same irrep. However, when there are multiple irreps withmultiplicity, that usually signifies the presence of additional kinematic symmetries.In the hydrogen atom example, there is the larger kinematic symmetry group SO(4)that explains why subspaces with different orbital angular momentum have thesame energy.

Define the maximal kinematic symmetry group GH of the Hamiltonian H as thegroup such that every HE corresponds to a single irrep μð Þ of GH . When this groupcan be found, energy levels are irreps of the maximal kinematic symmetry group,and this is a powerful tool for the analysis of the model. It allows the physics ofdegeneracy to be handled in a systematic, algebraic fashion because the symmetrygroup provides all invariant observables necessary to diagonalize degeneracies.Other observables in the model can be characterized by their transformationproperties under the group, simplifying calculations of expectation values, transi-tion rates, and perturbation theory. Further, if H is part of a family of Hamiltonians,then how GH changes with varying parameters determines how the energy levels(irreps) split and merge and how invariants are broken and reformed.

To close this section on decomposition, consider a dynamic symmetry group Gwith irrep labeled by μð Þ representations. Since the symmetry group does notcommute with the Hamiltonian, there is no necessary relationship between irrepsof G and HE. However, one possibility is that each irrep of G is decomposable in asum of energy eigenspaces, i.e., the reversal of Eq. (15.8):

V μð Þi ¼ ⊕

E2σ μð Þi Hð Þ

HE (15.9)

where σ μð Þi Hð Þ is a purely symbolic shorthand for the spectrum of energies E

corresponding to the irrep V μð Þi and depends on the Hamiltonian H and the

Symmetry, Structure, and Emergent Subsystems 305

symmetry group G in a model-specific way. In this case, G is called a spectrum-generating group for the Hamiltonian H (see Wybourne 1974).

15.4 Hilbert Space Arithmetic: Tensor Product Factorizations

The other method of partitioning a model is through factoring the model Hilbertspace into a tensor product structure. Any finite-dimensional Hilbert spaceH � Cd can be factorized into a k-fold tensor product

H ¼ H 1⊗H 2⊗ . . .⊗H k � Cd1⊗Cd2⊗ . . .⊗Cdk (15.10)

as long as d1 � d2 � . . .� dk is a factorization of the positive integer d. Such afactorization is not unique; any unitary matrix in U dð Þ that cannot be factorizedinto U d1ð Þ � U d2ð Þ � . . .� U dkð Þ defines another factorization with the samestructure but different subspaces H i. The situation with a separable but infinite-dimensional Hilbert space is even wilder, where at least formally, Hilbert sub-spaces of any finite dimension can be factored off willy-nilly.

However, as with the generic decomposition Eq. (15.1), such factorizations donot necessarily have any physical meaning, even within the limited ontology of aformal model. It is the allowed set of observables that distinguishes which factor-izations have functional, conceptual, or interpretational value. In this section,several methods are presented in which a factorization of a model Hilbert spacehas an operational meaning in terms of observables.

15.4.1 Models of Independent, Distinguishable Subsystems

The most transparent case is when a model describes a system that is composed ofdenumerable, independent, distinguishable systems. This is the bottom-upapproach to building a model. Consider each of these systems as a submodel witha Hilbert space H i and set of observables Ai. The total Hilbert space is the tensorproduct of k subspaces H i

H ¼ ⊗k

i¼1H i (15.11)

Pure states in H can be classified as to whether they are separable or not separablewith respect to this factorization. In this context, separable is used in the algebraicsense that a separable pure state ψj i can be written at the tensor product of statesψij i 2 H i as

ψj i ¼ ψ1j i⊗ ψ2j i⊗ . . .⊗ ψkj i (15.12)

An entangled pure state is not separable and does not admit a factorization likeEq. (15.12). To be clear, this is separable in a totally different sense than the

306 Nathan Harshman

topological notion of a separable Hilbert space admitting a decomposition intodenumerable one-dimensional subspaces Eq. (15.1), and it is also not equivalent tothe separability of a differential equation discussed later.

The set of observables denoted A⊕ is constructed using the Kronecker sum ofthe subsets of observables Ai. The Kronecker sum of two operators in differentalgebras A 2 A1 and B 2 A2 is

A⊕B ¼ A⊗I2 þ I1⊗B (15.13)

where I1 is the identity operator in H i. This definition of the Kronecker sum canbe generalized to more factors. (Unfortunately, note that the same symbol istypically used for the direct sum of vector spaces and the Kronecker sum ofoperators.) For observables in A⊕, expectation values of measurements factorinto a sum of expectation values in each subsystem, and this holds for bothseparable and entangled pure states. I argue this can be taken as the definition ofa model of independent systems. Note that A⊕ is not an algebra of observables,even if each Ai were algebras. The set A⊕ is closed under sums, but not underproducts.

Technically, a distinction is necessary between the submodel observables Ai

and the representation of those observables in A⊕. The representation of Ai in A⊕

is found by taking the Kronecker product of elements of Ai with the identityelements in all other subspaces H i. In other words, the element A 2 A1 is mappedinto the element A⊗I2⊗ . . .⊗Ik in the set A⊕. It is in this sense that we can saythat the subsets of operators in the model commute with each other, anotherindication of their independence in this model.

Within each submodel there is a Hamiltonian Hi 2 Ai. Considered as operatorsof the total space H , the sub-Hamiltonian H1 is realized by H1⊗I2⊗ . . .⊗Ik and soon for all k sub-Hamiltonians. The total Hamiltonian

H ¼ ⊕k

i¼1Hi (15.14)

describes a system composed of subsystems that are noninteracting. When H isexponentiated exp �iHt=ℏð Þ to generate time translations, it factors into a productof unitary operators on each subspace H i

U tð Þ ¼ exp �iH1t=ℏð Þ⊗ exp �iH2t=ℏð Þ⊗ . . .⊗ exp �iHkt=ℏð Þ (15.15)

Because time translation factors into operators that do not mix subspaces, entangle-ment is time-invariant. The only entanglement in the system is present in the initialstate and remains unchanged. This definition can be lifted to describe the tensorproduct structure in Eq. (15.11) – it is a dynamically invariant tensor productstructure (see Harshman and Wickramasekara 2007a).

Symmetry, Structure, and Emergent Subsystems 307

One physical interpretation of such a model is that the subsystems are onisolated patches of space. For example, in this case each Hilbert subspace H i

might be a spin (i.e., an irrep of the rotation group). Even if these subsystems areinternally identical, they can be distinguished by their location. In this sense, thefactorization of the tensor product corresponds to the intuitive notion of locality,the observables in A⊕ are local observables, and the unitary operator is a localunitary. However, the subsystems could just as well be in the same location, butnoninteracting and distinguishable. In this case, one could still use the term localityto refer to operators in A⊕ that are local with respect to the tensor product of thestructure. In fact, following Zanardi (see later), I have used this generalization ofthe term locality in numerous talks, including at the workshop that inspired thischapter, and received angry rebukes. A significant portion of the audience alwaysseems to prefer that local retain its original meaning in terms of space (or space-time for relativistic systems). In response to the persistent headwind I have finallyconceded and tentatively propose the terms specific and specificity to replace localand locality in this context.

15.4.2 Models of Interacting Subsystems

A standard assumption of quantum mechanics is that the model for an interactingsystem can be built from the models of the subsystems. The construction of theHilbert space by tensor product is the same as before, and therewith follow thesame notions of specificity and entanglement among subsystems. The difference isnow that the set of observables is extended to include operators not in A⊕. In themost extreme case, the total set of observables could be the algebra of observablesis constructed as the tensor product of the subsets Ai as

A ¼ ⊗k

i¼1Ai (15.16)

More generally, the setA includes at least some observables that are not specific tothe tensor product structure induced by the subsystems.

For an interacting system, the Hamiltonian must be nonspecific, i.e., an operatorthat cannot be constructed by Kronecker products of sub-Hamiltonians, like Eq.(15.14). As a result, time evolution no longer factors into a specific unitaryoperator like Eq. (15.15). Entanglement of a state evolving in time is typicallyno longer a dynamical invariant with respect to the tensor product structure Eq.(15.11). However, there may be other observables that are dynamical invariants.There may even be other tensor product structures besides the original constructionthat are dynamically invariant. The question becomes: Can one exploit theseobservables to find an alternate factorization and an alternate notion of specificity?

308 Nathan Harshman

15.4.3 Zanardi’s Theorem and Virtual Subsystems

For systems with finite-dimensional Hilbert spaces, one step towards answeringthis question is addressed by Zanardi’s theorem (Zanardi 2001, Zanardi, Lidar, andLloyd 2004; see also Harshman and Ranade 2011). It provides criteria for whethera partition of the observables leads to a tensor product structure and a notion ofspecificity.

A version of the theorem can be stated as follows:

Given a state space Φ 2 H � Cd and a collection of subalgebras A1;A2; . . .f g of the totalalgebra of observables A acting on H , the subalgebras induce a tensor product structureH ¼ ⊗iH i if they satisfy the following criteria:

• Subsystem independence: The subalgebras commute A1;A2½ � ¼ 0 for all i, j.

• Completeness: The subalgebras generate the total algebra of observables A ¼ ⊗iAi.

• Specific (née local) accessibility: Each subalgebra corresponds to a set of controllableobservables.

In this statement, the first two requirements on the subalgebras of observables aremathematical in nature, and they could be assessed from within the model as trueor false for any particular partition of the observables. However, the third require-ment is a physical criterion about empirical accessibility of measurement andcontrol. There could be partitions of the observables that satisfy the first two, butare inadmissible based on a physical limitation of reality or some other constraintfrom outside the model.

An extension of Zanardi’s theorem, called the “tailored observables theorem,”demonstrates the flexibility provided by the first two requirements in constructingsubalgebras that factor the Hilbert space into “virtual” subsystems. For a finite-dimensional Hilbert space H � Cd, one can construct subalgebras of observablesthat induce a tensor product structure from a finite basis of operators, such that anyknown pure state can have any entanglement that is possible for any factorization of d(Harshman and Ranade 2011). The proof relies on the unitary equivalence of Hilbertspaces with the same dimension, and it is constructive in the sense that a procedure isgiven to construct the generators for the subalgebras in a finite number of steps(depending on the factorization of d). A consequence of this theorem is that, for anypure state, observables can be found that will detect as much or as little entanglementas is possible in a Hilbert space with dimension d. The only hitch is that entanglementis completely relative onlywhen the control of the system is absolute, and therefore thethird criterion of Zanardi’s theorem is unrestrictive. For example, in a system of linearquantum optics (i.e., using only mirrors, phase shifters, and beam splitters) any finite-dimensional unitary operator can be implemented (Reck et al. 1994). Combined withMach-Zehnder interferometers, this system has enough control to extract anyobserver-relevant entanglement from any pure state.

Symmetry, Structure, and Emergent Subsystems 309

15.4.4 Top-Down Approach to Emergent Tensor Product Structures

As stated and proved, Zanardi’s theorem and the tailored observables theorem holdfor finite-dimensional Hilbert spaces and algebras of observables. It should havegeneralizations to separable Hilbert spaces and more general sets of observables,but I am unaware of any work in that direction.

Nonetheless, there are other cases of greater scope where partitions of observ-ables provided by extra-model considerations lead to novel tensor productstructures. Decoherence can select preferred tensor structures and thereby notionsof subsystems (Jeknić-Dugić, Arsenijević, and Dugić 2013). This approach tobuilding the classical-quantum correspondence using decoherence and emergenttensor product structures has been referred to as the “top-down approach” (Fortinand Lombardi 2016).

Another category of top-down virtual subsystems is provided by yet anothernotion of separability – separation of variables and separation of integrationconstants. The Hamiltonian can be represented as the Schrödinger operator formodels with an underlying D-dimensional space or configuration space. This kindof separability describes the existence of an orthogonal coordinate system thattotally separates the Schrödinger equation into D one-dimensional differentialequations (or partially separates it into d < D differential equations). There aredifferent levels of separability depending on how the separated differential equa-tions depend on the separation constants.

The “gold standard” is when each differential equation only depends on a singleseparation constant λi (Harshman 2016c). Then each differential operatorΛi defines aZanardi-like subsystem and the whole Hilbert space can be decomposed as

H ¼ ⊕σ λ1ð Þ

. . . ⊕σ λDð Þ

H λ1⊗ . . .⊗H λD (15.17)

where σ λið Þ is the spectrum of the differential operator Λi. This is a more robustseparability than “silver” separability described in Eq. (15.5), where only thespectrum was separable, not the Hilbert space. For “bronze” separability, thespectrum of each differential operator depends on the values of other separationconstants, and so spectral separability is lost. Only for “gold” separability doesdifferential separability correspond to a tensor product structure and therewith toan algebraic notion of separability.

A final method for identifying top-down tensor product structure is requiringthat the tensor product structure be invariant with respect to a symmetry group ofthe model. A general theory of when this is possible has not been developed, buttwo examples from nonrelativistic physics illustrate the idea.

Consider a model whose Hilbert space is an irrep space of the Galilean groupand whose algebra of observables is the Galilean algebra extended by a central

310 Nathan Harshman

mass observable, i.e., a model of a single nonrelativistic particle (Lévy-Leblond1967). Elements of this irrep space are wave functions in three-dimensional spacewith an internal spin degree of freedom. The Hilbert space has a naturalfactorization

H ¼ H space⊗H spin � L2 R3� �

⊗C2sþ1 (15.18)

Because it is an irrep of the Galilean group, no vector in this space is invariantunder all transformations. However, this factorization of spatial and spin degrees offreedom is invariant. The unitary operator that represents any Galilean transform-ation factors into a product of a specific unitary operator on each subspace (Harsh-man and Wickramasekara 2007b). In contrast, irrep spaces of Poincarétransformations, corresponding to relativistic particles, do not have an invariantfactorization between spatial and spin degrees of freedom, although they mayunder a subgroup of transformations (Harshman 2005).

Another example is a model built from two Galilean irreps H ¼ H 1⊗H 02 and aHamiltonian with interactions that is not specific to that factorization, but for whichthe center-of-mass degrees of freedom still separate (in the sense of a differentialequation). Then there is an alternate tensor product structure

H ¼ H com⊗H rel (15.19)

between the center-of-mass and relative degrees of freedom. This tensor productstructure is symmetry-invariant with respect to Galilean transformation, and fur-ther, is it a dynamically-invariant tensor product structure. The unitary operatorsfor a general Galilean transformation and for time-translation with the interactingHamiltonian both factor into specific unitaries (Harshman and Wickramasekara2007a). As a result, entanglement between center-of-mass and relative degrees offreedom is invariant under transformation of coordinate systems and invariant intime. This kind of entanglement between center-of-mass and relative degrees offreedom is not peculiar; it is present even in typical initial states of a scatteringexperiment where the particles are not originally entangled with respect to theinterparticle tensor product structure H ¼ H 1⊗H 2.

15.5 Context: The Simplest Quantum Few-Body Problem

The purpose of this section is to apply the techniques of Hilbert space decom-position and factorization to a specific quantum mechanical model describing a fewinteracting particles in one dimension. I consider distinguishable and indistinguish-able particles without internal degrees of freedom like spin (or if they have spin,the spin is fixed and not dynamical). It is arguably the simplest quantum modelthat exhibits the full range of complexity of quantum dynamical systems.

Symmetry, Structure, and Emergent Subsystems 311

Understanding the model requires assessing the interplay of interaction,indistinguishability, identity, integrability, solvability, and entanglement.A consistent, coherent synthesis of these issues provides a challenge for anyinterpretation of quantum mechanics in terms of units of reality. I claim thatcombining the bottom-up and top-down structural perspectives of model symmet-ries and Hilbert space arithmetic responds to this challenge with surprising depth.

15.5.1 Single-Particle Hamiltonian

A first step in building the interacting few-body model is describing the single-particle model. The Hamiltonian for a single particle in one dimension experi-encing a static externally generated potential:

H1 ¼ � 12m

∂2

∂x2þ V xð Þ (15.20)

Here is the first place the restriction to one-dimensional systems pays dividends. First,one-dimensional systems are always integrable. An integrable system has as manyalgebraically independent, globally defined constants of the motion as the number ofdegrees of freedom. (The classical definition of integrability is usually formulated interm of operators generating flows on phase space. In a classical one-dimensionalsystem, the constraint provided by this integral ofmotion reduces the two-dimensionalphase space to a one-dimensional manifold, i.e., the trajectory of the particle. There issome ambiguity in the quantum case. See Caux and Mossel 2011 for a review of thedifficulties). For a one-dimensional system, the Hamiltonian itself is the singleconserved integral of motion necessary for integrability. This does not mean that thesystem is necessarily solvable, in the sense that the eigenvalues and eigenstates of theHamiltonian can be expressed in closed-form analytic expressions. However, formoderately well-behaved trapping potentials (Harshman 2017a), the Sturm-Liouvilletheory guarantees that the energy spectrum of Eq. (15.20) is a denumerable tower ofsingly-degenerate states bounded from below. The wave function with lowest energyε0 has no nodes, and each successive state with energy εn has n nodes.

Again assuming a reasonable trapping potential, theHilbert space of the one-particlesystemH 1 is the space of Lebesgue-square-integrable functions L2 Rð Þ on the real line.This space carries an irrep of the one-dimensional Galilean group, although thetrapping potential breaks that symmetry. Each energy eigenstate εnj i spans a one-dimensional subspaceH 1

εn 2 H 1, leading to the decomposition ofH 1 into a direct sumof energy eigenspaces (or equivalently, time translation symmetry irrep spaces):

H 1 ¼ ⊕∞

n¼0H 1

εn (15.21)

312 Nathan Harshman

Note that in two and more dimensions, integrability is not guaranteed withoutmore knowledge of the potential and its symmetries. Although states can still belabeled by a spectrum of energies and a decomposition in energy subspaces likeEq. (15.21) is still possible, the subspaces are not necessarily one-dimensional andmuch less can be inferred about the properties of the wave functions.

15.5.2 N Noninteracting, Distinguishable Identical Particles

The next step in constructing this minimal model is to combine N noninteractingparticles. The Hilbert space for the system is constructed by the tensor product ofsingle-particle Hilbert spaces

H N ¼ ⊗N

i¼1H 1

i (15.22)

One realization of this Hilbert space is as Lebesgue square-integrable functions onconfiguration space L2 RN

� �. The Hamiltonian can be written as a sum of differen-

tial operators acting on functions of N one-dimensional coordinates xi:

HN0 ¼

XNi¼1

� 12m

∂2

∂x2iþ V xið Þ

� �(15.23)

The noninteracting dynamical model with Hamiltonian Eq. (15.23) is separable.Each degree of freedom xi is independent and integrable; each single-particleHamiltonian is an integral of the motion. For distinguishable particles withoutinteractions, the N coordinates xi remain dynamically uncoupled. Though occupy-ing the same physical space, the particles might as well be scattered throughout thegalaxy as far as the dynamics are concerned. The Hamiltonian HN

0 is specific to thetensor product structure Eq. (15.22), and any entanglement among the particles is adynamical invariant.

The kinematic symmetry group of HN0 includes the finite group of particle

permutations isomorphic to SN . The transformations in this group are representedby operations on H N , but they can also be realized as orthogonal transformations,i.e., reflections and rotations, on the configuration space RN (Harshman 2016a).This realization of particle permutations by geometrical point transformationsconnects back to the Platonic solids of the introduction. For example, the realiza-tion of S3 in configuration space R3 is the point group of a triangle, the realizationof S4 in R4 is the point group of a tetrahedron, and so on for N-dimensionalsymmetries of (N-1)-simplices. If the single-particle system has parity symmetry(i.e., reflection symmetry about some point) then, in addition, a sequence of cubic-type symmetries appears (Harshman 2016b).

Symmetry, Structure, and Emergent Subsystems 313

An energy eigenstate basis for the total system is formed by all tensor productsof single-particle basis vectors like nj i � εn1j i⊗ εn2j i⊗ . . .⊗ εnNj i. The energy En

of a basis state is the sum of the single-particle energies. Most energies are nolonger singly degenerate, but they are still denumerable and provide a decom-position of H N into a tower of energy eigenspaces

H N ¼ ⊕En

HEn (15.24)

where the direct sum is over all possible energies constructed as sums of N single-particle energies εn. Note that each space HEn has a complete basis that isunentangled with respect to the tensor product structure Eq. (15.22).

The dimension of HEn is determined by the number of ways the set of single-particle energies that sum to En can be permuted. For example, for three particlesthe spaces HEn can have one, three, or six dimensions. Irreps of S3 either have oneor two dimensions and that signals the presence of additional kinematic symmet-ries beyond S3 (Leyvraz et al. 1997, Fernández 2013).

In fact, the decomposition Eq. (15.24) is a reduction into the irrep spaces of thekinematic symmetry group SN o Tt, there Tt is the time translation group of asingle-particle system and o is the wreath product (Harshman 2016b). There couldbe an even larger kinematic symmetry group of the same form incorporatingadditional single-particle symmetries, like parity.

Note that I have not made the claim that all the spaces HEn correspond todifferent energies. That depends one whether each energy can be uniquely associ-ated to a set of N single-particle energies. If several energy sums coincide, thenthere must be an even larger kinematic symmetry group. I call this an emergentkinematic symmetry, because it cannot be generated by single-particle symmetriesand particle permutations. One example is when the trapping potential is a har-monic trap and then the maximal kinematic symmetry is U Nð Þ and can be realizedas symmetry transformations on phase space (Baker 1956, Louck 1965). For thissystem, the degeneracies of the energy eigenspaces grow like a factorial in theenergy but can be reduced into spaces like HEn . Another example of an emergentkinematic symmetry is when the trapping potential is an infinite square well. Thenthere are “pythagorean degeneracies” that do not appear to have a description as agroup of transformations realized on configuration space or phase space (Shaw1974).

15.5.3 N Noninteracting Indistinguishable Particles

If the particles are indistinguishable fermions or bosons, then the model of Nnoninteracting particles described earlier contains states and observables that

314 Nathan Harshman

cannot physically be realized or measured. There are several approaches to refiningthe model to account for indistinguishability, some more “bottom-up” and othersmore “top-down.”

One traditional bottom-up approach starts with the overcomplete Hilbert spaceH N and uses the symmetry group of particle exchanges to decompose it intosectors. Each sector is labeled by an irrep of the symmetric group SN , and eachsector is a tower of irrep spaces like Eq. (15.6). In the example given earlier withfour particles Eq. (15.7), there were five kinds of irreps. More generally, there areas many irreps as there are positive integer partitions of N. The number P Nð Þ ofinequivalent irreps for a given N, or equivalently, the number of partitions of N, is acombinatoric problem that does not have an algebraic expression. However, one ofthese is always the totally symmetric irrep [N] which is one-dimensional and onwhich every particle permutation is represented by multiplication by +1. Anotherof these is the totally symmetric irrep 1N

� �which is also one-dimensional, but now

every odd particle permutation is represented by multiplication by�1 and the evenpermutations by +1.

To summarize, the Hilbert space H N can always be decomposed as

H N ¼ H NN½ �⊕H N

1N½ �⊕H Neverything else (15.25)

The space H NN½ � contains all the allowed bosonic states, and the space H

N1N½ � contains

the allowed fermionic states. Further, because particle permutations are a kinematicsymmetry of HN

0 , all of the energy eigenspaces H En can be similarly decomposed.For one-dimensional systems, there is a single bosonic state in each H En , but there isonly a fermionic state in H En when all the single-particle states are different.

The sectors H NN½ � and H

N1N½ � do not inherit the tensor product structure Eq. (15.23)

from the distinguishable particle construction. In fact for fermions, there are nopure states that are separable with respect to the tensor product structure Eq. (15.23).That sounds exciting on the surface, but because of indistinguishability, there arealso no observables that can detect the implied interparticle entanglementcorrelations of pure states (Benatti, Floreanini, and Titimbo 2014). One method todefine entanglement in bosonic and fermionic systems is to find several completecommuting subalgebras of observables, a la Zanardi, and use them to partition thesymmetrized spaces H N

N½ � and H N1N½ �.

A distinct, somewhat more “top-down” approach to indistinguishable particlesbuilds the model on the symmetrized few-body configuration space (Leinaas andMyrheim 1977). In the case of N one-dimensional particles, one takes the quotientof the configuration space RN with the symmetric group RN=SN . This means thatpoints in configuration space x¼ x1; x2; . . . ; xNf g and x0 ¼ x01; x

02; . . . ; x

0N

� , which

Symmetry, Structure, and Emergent Subsystems 315

are equivalent up to a permutation of coordinates, are identified as the same point.In this way, indistinguishability becomes a topological notion. The topologicallytrivial manifold RN is warped into a topologically nontrivial orbifold RN=SN . Forthree-dimensional systems in Euclidean space, this approach leads to an equivalentformulation containing bosons and fermions. However, for two-dimensionalsystems, or other systems with topologically nontrivial one-particle spaces, thistopological approach to identical particles can yield new physics, most famouslyanyons in two dimensions (Wilczek 1982).

15.5.4 Contact Pairwise Interactions

The final piece of the model adds contact (or zero-range) interactions between eachpair i; jh i:

HN ¼XNi¼1

� 12m

∂2

∂x2iþ V xið Þ

� �þ g

Xi;jh i

δ xi � xj� �

(15.26)

The contact interaction is given functional form by the one-dimensional delta-function, weighted by the interaction parameter g. The contact interaction isappropriate for modeling physical scenarios where the range of the pairwiseinteraction is much shorter than any other length scale in the problem, e.g., thelength scale determined by the trap and from the de Broglie wavelengths. The factthat the contact interaction is characterized by a single parameter makes it particu-larly amenable to analysis, as described later.

This model has been well studied for over 50 years, going back at least as far asanalyses for trapped bosonic particles in the g ! ∞ limit by Girardeau (1960), freebosonic particles by Lieb and Liniger (1963), and free fermions by Yang (1967).For reviews, including experimental implementations, see Cazalilla et al. (2011),and Guan, Batchelor, and Lee (2013).

The Hamiltonian HN is no longer specific to the single-particle tensor productstructure. Interactions break the separability and integrability of HN

0 and engendercorrelations among the distinguishable or indistinguishable particles. Howeverthere are two important limiting cases.

• When g ! ∞, the interactions are called “hard-core interactions.” In one dimen-sion, there is no way for particles to move past each other and so distinguishableparticles would remain in a fixed order. In this limit, the system is no longerseparable, but integrability reemerges for any trapping potential V xð Þ. AsGirardeau showed, the wave functions can be expressed as algebraic combin-ations of the noninteracting wave functions restricted to specific orderings (seeGirardeau 1960, Harshman 2017b). However, in this limit HN is not continuous

316 Nathan Harshman

and self-adjoint on all L2 RN� �

, which creates some difficulties for level dynam-ics (Sen 1999).

• When the trapping potential is homogenous but finite in extent, i.e., the infinitesquare well, then the system is integrable for any value of g using a methodcalled the Bethe ansatz (Batchelor et al. 2005, Oelkers et al. 2006). There are Nintegrals of motion that are symmetrized polynomials in the single-particlemomenta.

Both of these limiting cases can be understood as examples when the Yang-Baxterequation holds and there is diffractionless scattering (Sutherland 2004).

One more special case is when the external trap is quadratic in position. Thenoninteracting system HN

0 is equivalent to an isotropic harmonic oscillator in Ndimensions and, as mentioned earlier, has kinematic symmetry group U Nð Þ. Thatsystem is maximally superintegrable, meaning there are 2N � 1 integrals ofmotion, and exactly solvable, meaning that the energy is an algebraic function ofthe quantum numbers, and all excited states are products of the ground state withpolynomials (Post, Tsujimoto, and Vinet 2012). At finite interaction strength, mostof this additional analytical tractability is lost, but there is one extra integral of themotion corresponding to the separable center-of-mass degree of freedom (Harsh-man 2014). This separability survives symmetrization of indistinguishableparticles and, therefore, entanglement between center-of-mass and relative degreesof freedom remains a dynamical invariant.

For general traps and arbitrary g, the model Eq. (15.26) is not integrable, nor is itsolvable except numerically. Then two questions become: How far from integra-bility and deep into chaos and is it? How difficult is it to achieve convergentnumerical solutions? The second question has been investigated exhaustively, bythis author and many others, because of the relevance to current experiments(Serwane et al. 2011, Zürn et al. 2012, 2013, Wenz et al. 2013; for a partial list,see the references of Harshman 2016a). However, I would claim a productivemetatheory of when particular approximation methods work well has not arrived.

One way to answer the first question about chaos is by comparing the spectrumto the Wigner-Dyson distribution of eigenvalues of a random matrix (Gutzwiller1990). According to the Bohigas-Giannoni-Schmit conjecture, this is a universalfeature of systems with quantum chaos. Work on closely related systems suggeststhat chaos is present in these systems (Bohigas, Giannoni, and Schmit 1984), butthe world currently waits for a more detailed analysis, especially one that situatesthe model in the hierarchy of chaos from ergodic, mixing, Kolmogorov, andBernoulli (Ullmo 2016, Gomez, Losada, and Lombardi 2017).

In summary, depending on the trap shape and interaction strength, the modelEq. (15.26) can manifest the full range of possible dynamic behaviors, from (super)

Symmetry, Structure, and Emergent Subsystems 317

integrability to (conjectured) hard chaos. For integrable cases, there are observ-ables privileged by dynamical conservation laws that fully characterize the system,i.e., a complete set of commuting observables. The specific nature of the integralsof motion depends on the trap and interaction strength. For the noninteracting case,the integrals of motion are single-particle observables. In integrable interactingcases like the hard-core limit or Bethe-ansatz solvable cases, the conserved quan-tities are collective observables built from symmetrized polynomials of single-particle observables. In contrast, for chaotic cases, the unique conserved observ-able is the energy itself, and the spectrum matches with a relevant form ofrandomness. A goal of this avenue of research is to see if the approach to chaoscan be understood as the dissolution of structures based on irreps ofsymmetry group.

15.6 Conclusion: A Few Comments on Symmetry, Structures, and Solvability

The previous section introduced in situ certain technical terms of art, like integra-bility and solvability. The reader should not be misled by my breezy usage of theterms to infer that there is universal acceptance among the community of math-ematical physicists on how these terms should be applied. For example, Liouvillianintegrability is naturally defined in classical dynamics on phase spaces, but there isdebate on how this works for quantum systems with a mixture of continuous anddiscrete degrees of freedom (Caux and Mossel 2011). And there is not consensuson the relationship between Bethe-ansatz integrability relevant to the model dis-cussed earlier and Liouvillian integrability, and I know of only a few cases wherethey coexist (e.g., Harshman et al. 2017). Even more egregious is the dissent anddissemblance around the term solvable. Some physicists throw around the termexact solution when they have actually found a somewhere-convergent, asymptot-ically approximate numerical solution to the limiting case of a mathematically ill-conditioned dynamical model.

I argue that solvability should be considered as a property of a dynamical model,and it should be considered as a continuum. At the most solvable extreme are modelswhere one can push analysis deep into the realm of pure algebra. Examples includesuperintegrable systems and exactly solvable systems (which, in fact, are conjecturedto be the same thing). Moving down the spectrum we have systems whose solutionsare formulated in exact analytic relations, but those relations may require (for example)solving transcendental equations for the spectrum or other model parameters.

My argument is that this same continuum, from more solvable to less solvable,coincides roughly with two other features: (1) the amount of symmetry in thesystem, as measured by the size or complexity of the symmetry group and (2) thetractability of Hilbert space arithmetic, meaning the variety of inequivalent ways

318 Nathan Harshman

the system can be decomposed or factorized into subspaces. The structural unit thatunifies these two features is the irreducible representation.

Irreps appear in many guises – as invariant subspaces in direct sums and tensorproducts, as the building blocks of towers for describing identical particles thatgenerate unusable entanglement and frustrate algebraic separability, as the conceptof energy levels that vary across families of Hamiltonians in the same model, andmore. No matter the underlying ontological commitment of an interpretation, anyformulation of quantum mechanics must account for the prevalence and utility ofthese structures. Unlike the Platonic solids, these are not metaphors. They aremathematical building blocks of quantum mechanics and will remain so even whennew or reformed ontologies emerge.

The most egregious oversight of this chapter is that I have not discussed howsymmetry groups partition the set of observables into irreps. This is more technic-ally challenging that the Hilbert space arithmetic I have presented, but I think thepotential rewards are a deeper understanding of the connections among the observ-ables, separability (in all three senses), and integrability.

At this stage, the investigation is still incomplete, but I argue that one notionemerges – the importance of solvability. Solvability is a concept lying in thatawkward place of being a technical term with multiple overlapping and connectingdefinitions in different contexts. One of the unifying themes across these contextsis that solvable systems play a central role in the interpretation of physicalphenomena. Solvable models in mechanics, like coupled harmonic oscillatorsand hydrogenic atoms, are not just touchstones for mathematical analysis. Theyare ubiquitous as direct and approximate models in nature, and they provide thecognitive framework for understanding other physical systems. Is it a coincidencethat solvable models are so useful? Is it just attention bias, i.e., we pay moreattention to things we understand more thoroughly? Or, is there something moredeeply “real” about solvable systems, either in an epistemic or ontic sense?

Acknowledgments

I would like to thank Olimpia Lombardi and the other organizers of the workshopIdentity, indistinguishability and non-locality in quantum physics (Buenos Aires,June 2017) for assembling such a stimulating group of physicists and philosophers.

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322 Nathan Harshman

16

Majorization, across the (Quantum) Universe

guido bellomo and gustavo m. bosyk

16.1 Introduction

In how many ways can one represent a given quantum mixed state as a mixture ofpure states? Why (and in which sense) are separable states more disorderedglobally than locally? Is it possible to transform a given pure state into anotherby means of local operations and classical communication? How should anadequate formulation of the uncertainty principle be? All these questions, asdissimilar as they may seem, share one element in common: They can be answeredby appealing to the notion of majorization partial order. Majorization is nowadaysa well-established and powerful mathematical tool with many and different appli-cations in several disciplines, such as economics, biology, and physics, amongothers. Indeed, the seminal idea of this concept had already been glimpsed byLorenz (1905) while studying the inequality of wealth distribution and developingthe representation of the (nowadays called) Lorenz curves. Moreover, the famousGini coefficient (Gini 1912), widely accepted as a legitimate quantifier of incomedistribution inequality, is merely a ratio between graphical areas defined by aLorenz curve. Other key contributions to the subject were those by Muirhead(1903), Dalton (1920), Schur (1923), and Hardy, Littlewood, and Pólya (1929).The name “majorization,” though, appears first in the prominent book by Hardy,Littlewood, and Pólya (1934).

At present, it is clear that anyone who is interested in the field would find itappropriate to begin by the celebrated book by Marshall and Olkin (1979), whosesecond edition was coauthored by Arnold (2010). Recently, Arnold (2007) pub-lished an article entitled “Majorization: Here, there and everywhere,” in which hepresents a sampling of diverse areas in which majorization has been found usefulin the last few years, such as geometry, probability, statistical mechanics, andgraph theory. However, those contributions do not explore its quantum theoreticalimplications, which are much more extensively covered, for example, in Nielsen’s

323

lecture notes (Nielsen 2002, see also Nielsen and Vidal 2001). In this chapter, weattempt to make a brief review of the subject and then to highlight the mostimportant results of this research line in the quantum realm, in order to offer akind of quantum counterpart of Arnold’s work.

A natural question for our current work regards the roots beneath the wideapplicability of majorization in quantum mechanics. Some authors argue that theconnection arises as a result of two important theorems that link majorization tounitary matrices, Horn’s lemma, and Uhlmann’s theorem, together with theubiquity of unitary matrices in quantum mechanics (Nielsen 2002: 5). Here, wepresent and discuss a variety of situations to show that the spread applicability ofmajorization in the quantum realm emerges as a consequence of deep connectionsamong majorization, partially ordered probability vectors, unitary matrices, andthe probabilistic structure of quantum mechanics. To this end, we review basicaspects of majorization, focusing on its connections with some quantuminformation problems. In particular, we organize our study here into threedifferent facets. The first one consists in the role played by majorization as adisorder signature on quantum states, how quantum operations affect this, and theconnection with quantum entropies. The second one is the study of entanglementtransformations of bipartite pure states, applying local operations, and classicalcommunication. Third, the problem of how to formulate the uncertainty principleis posed, and different proposals of majorization uncertainty relations arereviewed.

In the following section, in order to make the ensuing study more self-contained, we present some elementary definitions and mathematical propertiesrelated to majorization theory.

16.2 Majorization Theory: Definitions and Mathematical Properties

Let us begin with some basic definitions and notation. Hereafter, we consider theset of d-dimensional probability vectors whose components are sorted in nonde-creasing order, namely

Δd ¼ x1; . . . ; xd½ � : xi � xiþ1 � 0 andXdi¼1

xi ¼ 1

( ): (16.1)

We take for granted that any vector that appears below lives in Δd, that is, has itselements sorted in a nonincreasing manner. In order to define the concept ofmajorization between probability vectors, let x and y be two members of Δd. It issaid that x is majorized by y, denoted as x≺ y, if and only if the entire set of d � 1inequalities

324 Guido Bellomo and Gustavo M. Bosyk

Xni¼1

xi �Xni¼1

yi for all n 2 1; . . . ; d � 1f g (16.2)

is satisfied. (Notice that the conditionPd

i¼1 xi ¼Pd

i¼1 yi is trivially satisfied; forthat reason we discard this condition from the definition of majorization). Theintuitive idea is that a probability distribution majorizes another if the former ismore concentrated than the latter. In this sense, majorization provides a quantifica-tion of the notion of disorder for given probability vectors. To fix ideas, let usobserve that any probability vector x 2 Δd trivially satisfies the relation

1d. . .

1d

� �≺ x≺ 1 0 . . . 0½ � (16.3)

where the left-hand side (l.h.s.) corresponds to the most uncertain case (uniformprobability vector), whereas the right-hand side (r.h.s.) represents completecertainty.

It is essential to note that majorization defines a partial order over Δd, meaningthat there exist x, y 2 Δd, such that neither x≺ y nor y≺ x. That is the case, forexample, for x ¼ 0:6; 0:2; 0:2½ � and y ¼ 0:5; 0:4; 0:1½ �.

There is a way to easily visualize whether, given two probability vectors, thereis a majorization relation between them. This is done through the notion of Lorenzcurve (Lorenz 1905). A Lorenz curve of a probability vector x 2 Δd consists of thelinear segments joining the points n;

Pni¼1 xi

� �for n 2 0; . . . ; df g. Therefore, a

Lorenz curve is always concave and has extreme points 0; 0ð Þ and 1; 0ð Þ. In thisway, there is a majorization relation between x, y 2 Δd, if and only if theircorresponding Lorenz curves do not intersect each other, except in the extremepoints. In Figure 16.1, we illustrate these situations.

Figure 16.1 Lorenz curve for four-dimensional probability vectors x ¼1; 0; 0; 0½ � (line with circles), z1 ¼ 0:6; 0:15; 0:15; 0:1½ � (line with triangles),z2 ¼ 0:5; 0:25; 0:2; 0:05½ � (line with diamonds), and y ¼ 0:25; 0:25; 0:25; 0:25½ �(line with squares). Notice that all the Lorenz curves are concave and lie betweenthe corresponding extremes: ones of x and y. Futhermore, notice that there is anintersection between the Lorenz curves of z1 and z2, showing that there is not amajorization relation.

Majorization, across the (Quantum) Universe 325

Remarkably, the partial order defined by majorization can be posed in severalequivalent ways. We are particularly interested in the following two:

• x≺ y if and only if there exists a doubly stochastic matrix B, i.e., Bij � 0 for alli, j and

Pdi¼1 Bij ¼ 1 ¼Pd

j¼1 Bij, such that x ¼ By

• x≺ y if and only ifPd

i¼1 ϕ xið Þ �Pdi¼1 ϕ yið Þ for every concave function ϕ

The first equivalent characterization was originally discussed by Schur (1923).With the aid of Birkhoff’s theorem (Birkhoff 1946), which states that the class ofd � d doubly stochastic matrices coincides with the convex hull of the set of d � dpermutation matrices, this condition turns into a more geometrical one:

• x≺ y if and only if x lies in the convex hull of the orbit of y under the group ofpermutation matrices (see Figure 16.2)

The second equivalent condition, linking majorization with concave functions,relates further to the notions of Schur-concavity and entropy. Functions thatpreserve the majorization order form a large class, which was formerly studiedby Schur (1923). In his honor, we say that a function Φ : Δd ↦R is Schur-concaveif it (anti)preserves the majorization relation, that is,

if x≺ y ) Φ xð Þ � Φ yð Þ for all x, y 2 Δd: (16.4)

For example, the very general family of h; ϕð Þ-entropies satisfies the Schur-concavity (Bosyk, Zozor, Holik, et al. 2016). They are defined as

H h;ϕð Þ xð Þ ¼ hXdi¼1

ϕ xið Þ !

, (16.5)

where the entropic functionals h : R↦R and ϕ : 0; 1½ �↦R are such that either: (i)h is increasing and ϕ is concave, or (ii) h is decreasing and ϕ is convex, togetherwith the conventions ϕ 0ð Þ ¼ 0 and h ϕ 1ð Þð Þ ¼ 0. The h; ϕð Þ-entropies can beclassified according to whether they satisfy the additivity relation

Figure 16.2 Set of vectors majorized by a fixed vector y 2 Δ3, x : x≺yf g. The setis given by the points inside the convex hull of the orbit of y under the group ofpermutation matrices.

326 Guido Bellomo and Gustavo M. Bosyk

H h;ϕð Þ x⊗yð Þ ¼ H h;ϕð Þ xð Þ þ H h;ϕð Þ yð Þ (16.6)

or not. Particular instances of Eq. (16.6) that are additive are the well-knownShannon (1948), Rényi (1961) and Burg (1967) entropies, given respectively by

H1 xð Þ ¼ �Xdi¼1

xi log xi, (16.7)

Hα xð Þ ¼ sign α1� α

logXdi¼1

xαi , for all α 2 R∖ 0; 1f g, (16.8)

HBurg xð Þ ¼ 1d

Xdi¼1

log xi: (16.9)

On the other hand, a paradigmatic example of a nonadditive entropy is that byTsallis (1988),

Hq xð Þ ¼ 11� q

1�Xdi¼1

xqi

!, for all q 2 R∖ 1f g, (16.10)

for which the additivity rule is given by

Hq x⊗yð Þ ¼ Hq xð Þ þ Hq yð Þ þ 1� qð ÞHq xð ÞHq yð Þ: (16.11)

We have already pointed out that, from the viewpoint of order theory, majoriza-tion gives a partial order among probability vectors belonging to Δd. This meansthat this binary relation fulfils, for all x, y, z 2 Δd, the following three properties:

• reflexivity: x≺ x,

• symmetry: if x≺ y and y≺ x, then x ¼ y,

• transitivity: if x≺ z and z≺ y, then x≺ y.

Remarkably enough, Cicalese and Vaccaro (2002) have shown that majorizationover Δd defines an even more complex structure: a lattice. This means that therealways exist the infimum (join), x∧y, and the supremum (meet), x∨y. By defin-ition, x∧y means that x∧y≺ x, x∧y≺ y, and z≺ x∧y for all z such that z≺ x andz≺ y. In a similar way, x∨y means that x≺ x∨y, y≺ x∨y, and x∨y≺ z0 for all z0

such that x≺ z0 and y≺ z0 (see Figure 16.3). The algorithms to obtain both theinfimum and the supremum between arbitrary vectors in terms of its elements havealso been given by Cicalese and Vaccaro (2002).

Furthermore, by appealing to the subadditivity and supermodularity of Shannonentropy, the authors introduced a proper distance on the lattice (Cicalese, Gargano,and Vaccaro 2013): given x, y 2 Δd, the distance D : Δd � Δd ↦R is defined as

Majorization, across the (Quantum) Universe 327

D x; yð Þ ¼ H1 xð Þ þ H1 yð Þ � 2H1 x∨yð Þ: (16.12)

It can be shown that this definition provides a proper distance that satisfies, forx, y, z 2 Δd:

• nonnegativity: D x; yð Þ � 0, with D x; yð Þ ¼ 0 if and only if x ¼ y;

• symmetry: D x; yð Þ ¼ D y; xð Þ;• triangle inequality: D x; yð Þ þ D y; zð Þ � D x; zð Þ;• compatibility with the majorization lattice: if x≺ y≺ z, then D x; yð ÞþD y; zð Þ ¼ D x; zð Þ.Before surfing across the quantum universe, we should introduce a couple of

important results involving majorization and matrices, because our study of finitedimensional quantum systems ultimately relies on the density matrix representa-tion of their states. The first theorem is due to Schur (1923), and establishes thatgiven a selfadjoint matrix H 2 CN�N with diagonal d ¼ d1; d2 . . . dN½ � and eigen-values λ ¼ λ1; λ2 . . . λN½ �, then it is always the case that d≺ λ. On the other hand,Horn’s theorem (Horn 1954) tells us that for any two vectors d ¼ d1; d2 . . . dN½ �and λ ¼ λ1; λ2 . . . λN½ � such that d≺ λ, there exists a selfadjoint matrix H 2 CN�N

with diagonal d and eigenvalues λ.

16.3 Majorization in the Quantum Universe

16.3.1 Quantum Basics and Our Roadmap

We are now in position to translate some of the previous discussion into thequantum setting. First, we should review the concepts of quantum states andquantum maps. Now, instead of considering the set Δd of real d-dimensional

Figure 16.3 Hasse diagram for probability vectors x and y, its infimum (join) x∧y,and its supremum (meet) x∨y.

328 Guido Bellomo and Gustavo M. Bosyk

vectors, we are going to work over a set of finite dimensional (positive semidefiniteand trace-class) density matrices or operators over a Hilbert space H ffi Cd,

D ¼ ρ 2 Cd�d : ρ � 0 and Trρ ¼ 1� �

(16.13)

These matrices are also selfadjoint, ρ† ¼ ρ. The set D is convex, meaning thatgiven ρ1, ρ2 2 D, then any mixture of the form p ρ1 þ 1� pð Þρ2 2 D, whenp 2 0; 1½ �. The extreme states of D are the pure states that satisfyρ2 ¼ ρ ¼jψi ψ jh , with ψj i 2 H . States which are not pure are called “mixed.”Notice that there exist infinite ways to decompose a mixed state in terms of convexcombinations of pure states, but they are not arbitrary (we will go more deeply intothis when we discuss the Schrödinger theorem; see Figure 16.4).

Moreover, any state ρ has a spectral decomposition

ρ ¼Xdi¼1

λi eij i eij,h (16.14)

with λi � 0,P

iλi ¼ 1, and eijej� ¼ δij.

Since we plan to later talk about quantum correlations, we should rememberhow to describe composite systems. The states of multipartite systems act overthe tensor product of the individual Hilbert spaces. For example, for a bipartiteA [ B system, where H A and H B are the respective individual Hilbert spaces, thejoint state acts over H AB ¼ H A⊗H B. If ρAB is the state of the bipartite system,we can obtain the (reduced) states of each subsystem by performing the partialtraces: ρA Bð Þ ¼ TrB Að ÞρAB. When the joint state is pure, ρAB ¼jψABi ψAB j�

, we canuse the Schmidt technic to decompose the same in terms of local basis, jiAi� �and jiBf ig, as

ψAB ¼Xn

i¼1

ffiffiffiffiffiψi

piA iB , with ψi � 0,

Xi

ψi ¼ 1 and n ¼ min dA; dB� �

:

(16.15)

Figure 16.4 Illustration of the convex set of quantum states. The pure states arethe extreme points of the set (the border). Notice that a given mixed state can bewritten in infinitely many ways as convex combinations of pure states. In theexample in this figure, ρ can be expressed by mixing either ρ1 and ρ2, or ρ3 and ρ4.

Majorization, across the (Quantum) Universe 329

The Schmidt decomposition offers a natural viewpoint to study the correlations ofbipartite systems. Indeed, if only one Schmidt coefficient is different to zero thenthe state is a product (or noncorrelated) state:

ψAB ¼ ψA

⊗ψB. In other words,

the system is in a pure separable state. When more than one Schmidt coefficient ispositive, the system is in an entangled state. The extreme case is that of amaximally entangled state, characterized by the equality of the Schmidt coeffi-cients, ψi ¼

ffiffiffiffiffiffiffiffi1=n

pfor all i. Hereafter, let σ ψð Þ denote the probability vector

formed by the squared Schmidt coefficients of a bipartite pure state ψAB i.

Mixed joint states demand a more complex hierarchy of correlations. A mixedstate ρAB is a product state whenever it is expressible as a product of individualstates as ρA⊗ρB. In turn, a mixed separable state can be written as a convexcombination of product states,

ρAB ¼Xi

pi ρAi ⊗ρBi , with pi � 0 and

Xi

pi ¼ 1: (16.16)

Mixed product states are particular cases of separable states. Any state that isnonseparable is called “entangled.”

We are also going to consider linear, completely positive, trace-preserving mapsfrom the set of density operators into itself ℰ : D↦D, also called “quantumoperations.” All these operations have a Kraus representation of the form

ℰ ρð Þ ¼Xi

PiρP†i with

Xi

P†i Pi ¼ I (16.17)

in terms of its Kraus operators Pi and I the corresponding identity operator.Quantum operations include a wide variety of possible quantum dynamicalmaps. For example, if we restrict ourselves to the set of bistochastic maps,for which

PiPiP

†i ¼ I, we find the unitary evolutions, ℰ ρð Þ ¼ UρU† with

UU† ¼ U†U ¼ I, as well as the projective measurements,

ℰ ρð Þ ¼Xdi¼1

piPi with pi ¼ Tr ρPið Þ and PiPj ¼ δij, (16.18)

where Pi ¼ jif i ij; ijh i 2 H g is a set of orthogonal rank-one projectors.Finally, we want to introduce some quantum analogues for the classical entro-

pies. The von Neumann entropy (see, e.g., Wehrl 1978),

S ρð Þ ¼ �Trρ log ρ, (16.19)

can be viewed as the quantum version of the classical Shannon entropy, byreplacing the sum operation with a trace. We recall that for a selfadjoint operator,over H , X ¼Pixi j iXi iX jh , with j iXi being its eigenvectors in H and xi being the

330 Guido Bellomo and Gustavo M. Bosyk

corresponding eigenvalues, one has f Xð Þ ¼Pi f xið Þ j iXi iX jh , and the trace oper-ation is the sum of the corresponding eigenvalues.

So far, we have identified two privileged descriptions of the quantum states interms of probability vectors. First, for an arbitrary d-dimensional mixed state, wehave the spectral decomposition with its associated probability vector of eigen-values, λ ρð Þ 2 Δd. Second, for any d-dimensional bipartite pure state jψABi, wehave its Schmidt decomposition with the corresponding coefficients, which afterbeing squared also give a probability vector, σ ψð Þ 2 Δn, with n ¼ min dA; dB

� �.

For a given pure state, both descriptions are connected, because the squaredSchmidt coefficients coincide with the eigenvalues of the reduced states, that is,σ ψð Þ ¼ λ ψAð Þ ¼ λ ψBð Þ, where ψA Bð Þ ¼ TrB Að Þ jψABi ψAB

�. Each of these probabil-

ity vectors entails complementary information about the quantum state. Addition-ally, there is still a third probability vector that one can obtain from the state ρ, ifone considers a preferred observable quantity X, namely, the one defined by thecomponents of the expectation value in this way. Let X be an observable withdiscrete and nondegenerate spectrum, that is, X ¼Pd

i¼1 xi iXj i iX jh . The correspond-ing probability vector, p X; ρð Þ, has i component pi X; ρð Þ ¼ Tr ρjiXð ihiXjÞ. In whatfollows, we are going to present some paradigmatic problems in quantum mech-anics in which the notion of majorization, together with one of those descriptions,gives the right starting point toward their solutions (see Figure 16.5).

16.3.2 Schrödinger’s and Ulhmann’s Theorems

Let us start by attacking the question mentioned in the first line of this chapter,namely: In which ways can one represent a given quantum mixed state as a mixtureof pure states? It seems that the same was answered for the first time bySchrödinger (1936), who proved that a given state ρ ¼Pd

i¼1 λi eij i eijh can bewritten in the form ρ ¼PM

i¼1 pi ψij i ψijh , with pi � 0 andP

i pi ¼ 1, if and onlyif there exists a unitary matrix U 2 CM�M such that

ψij i ¼ 1ffiffiffiffipi

pXdk¼1

Uik

ffiffiffiffiλk

pekj i: (16.20)

It follows that the proposed expression is possible whenever the vector p, withcomponents pi, satisfies the relation p ¼ Bλ ρð Þ, where B is some doubly stochasticmatrix. Hence, recalling Schur’s theorem, we have (Nielsen 2000)

9 pi; ψij if g : ρ ¼XMi¼1

pi ψij i ψih j , p≺ λ ρð Þ: (16.21)

This last equation gives a conclusive requisite that any ensemble of pure statesmust fulfill to represent a given mixed state.

Majorization, across the (Quantum) Universe 331

From the example just shown, we can also have some insight on how tonaturally define a notion of majorization between quantum states by appealing toits spectral decomposition. Given ρ, σ over H , we say that ρ is majorized by σ asfollows:

ρ ≺ σ , λ ρð Þ≺ λ σð Þ: (16.22)

Now, a reasonable question regards the connection between two given states thatsatisfy a majorization relation from the viewpoint of quantum operations. In otherwords, whether there is any operation linking ρ and σ which satisfies ρ≺ σ. WhatUhlmann proved is that ρ≺ σ if and only if the majorized state can be obtainedfrom the latter by means of convex combinations of unitary maps (Uhlmann1970)

ρ ¼Xi

piUiσU†i , with UiU

†i ¼ I ¼ U†

i Ui, pi � 0 andXi

pi ¼ 1: (16.23)

That is, ρ lies in the convex hull of the unitary orbit to which σ belongs. It isinteresting to observe that, contrary to the classical case, convex combinations ofunitary operations give a more general class of bistochastic channels, also knownas random external fields (Alicki and Lendi 1987). Moreover, one can also prove

Figure 16.5 Our roadmap. Given a quantum state, ρ, there are several associatedprobability vectors, which involve different informational aspects of the same.Each probability vector allows complementary descriptions that are related todifferent quantum information problems, which are discussed throughout thischapter in the sections indicated between brackets.

332 Guido Bellomo and Gustavo M. Bosyk

that there exists a completely positive quantum bistochastic map transformingevery possible initial state σ into ρ if and only if ρ≺ σ (Chefles 2002):

σ �������!bistochasticρ , ρ≺ σ: (16.24)

In other words, majorization between quantum states offers an equivalent classifi-cation to that of the action of quantum bistochastic maps, in the same way asmajorization between probability vectors can be rephrased as transformations bydoubly stochastic matrices. Finally, notice that any map of the form Eq. (16.23) is aunital map, that is, a map with the identity map as a fixed point.

16.3.3 Quantum Entropies and Operations

We have already seen that there is a large class of functions that preserve themajorization relation between two probability vectors, commonly referred to asentropies. The same can be settled in the quantum case by appealing to thequantum analogues of the classical entropies. Indeed, the quantum h; ϕð Þ-entropiesare defined as (Bosyk, Zozor, Holik, et al. 2016)

S h;ϕð Þ ρð Þ ¼ h Trϕ ρð Þð Þ, (16.25)

where the entropic functions h and ϕ satisfy the same requirements as that in theclassical case. Notice that the quantum entropy of a given state ρ coincides with theclassical one for the probability vector formed by the eigenvalues of ρ, that is,S h;ϕð Þ ρð Þ ¼ H h;ϕð Þ λ ρð Þð Þ. In this way, we find, in the quantum realm, the same closeconnection between majorization, bistochastic maps, and entropies.

Several properties of Eq. (16.25) have been studied by Bosyk, Zozor, Holik,et al. (2016). In addition to the Schur-concavity of Eq. (16.25), two specific links tomajorization are the following. On the one hand, as a consequence of Schur-concavity and Schrödinger theorem, one has that the quantum h; ϕð Þ-entropy ofan arbitrary statistical mixture of pure states ρ ¼PM

i¼1 pi ψij i ψij,h is upper-bounded by the classical h; ϕð Þ-entropy of the probability vector formed by themixture weights, that is,

S h;ϕð Þ ρð Þ � H h;ϕð Þ pð Þ: (16.26)

On the other hand, when dealing with quantum systems, it is of interest to estimatethe effect of a given quantum operation on them. In particular, one may guess thata measurement can only perturb the state and, thus, that the entropy will increase.This is also true for more general quantum operations. Indeed, as a consequence ofthe Schur-concavity and a result by Chefles (2002), one has that any bistochasticmap ℰ has a nondecreasing effect on the entropy, that is,

Majorization, across the (Quantum) Universe 333

S h;ϕð Þ ℰ ρð Þð Þ � S h;ϕð Þ ρð Þ for all ρ 2 D, (16.27)

where the equality holds if and only if the map is unitary ℰ ρð Þ ¼ UρU†: Notice thatin the case of a nonbistochastic map, the entropy can decrease. For instance, let ρbe the density operator of an arbitrary mixed qubit system, with nonvanishingquantum h; ϕð Þ-entropy, and ε a completely positive and trace-preserving (but notunital) map characterized by Kraus operators P1 ¼ 0j i 0 jh and P2 ¼ 0j i 1 jh . Then,the system, after the action of this map, is on the pure state ℰ ρð Þ ¼ 0j i 0 jh ; thus, ithas zero quantum h; ϕð Þ-entropy.

Projective measurements are particular cases of bistochastic maps (see Eq. (16.18)).Hence, for a given quantum system, the difference of quantum entropies between thepostmeasurement and the premeasurement states works as a signature of the disturb-ance of the state of a system due to the measurement. In particular, consideration oflocal disturbances gives place to a type of quantum correlations measures and a way tocharacterize them (see, e.g., Luo 2008, Horodecki et al. 2005). As the main ingredientneeded to guarantee the validity of these measures is the property expressed inEq. (16.27), these measures can be defined in terms of generalized quantum entropies(see, e.g., Rossignoli, Canosa, and Ciliberti 2010, Bosyk, Bellomo, Zozor, et al. 2016).

16.3.4 Entanglement, Separability and Entropy

There is another important application of majorization theory to the problem ofdetermining the separable or entangled character of quantum states. The resultdepends, not surprisingly, on the relation between the global and local spectralproperties of the states.

Let us consider a bipartite scenario: A quantum system in the state ρAB withmarginal states ρA ¼ TrBρAB and ρB ¼ TrAρAB. A remarkable feature of quantumsystems is that there are states that describe situations such that the local disorder(in terms of the entropy of its reduced states) is greater than the global disorder.And that feature implies nonseparability of the state. That is,

S h;ϕð Þ ρA� �

> S h;ϕð Þ ρAB� �

or S h;ϕð Þ ρB� �

> S h;ϕð Þ ρAB� � ) ρAB is entangled:

(16.28)

The most radical example of these bounds is surely given by a Bell state, in whichcase we have minimal (null) global entropy and maximal local entropies. Classic-ally, this is never the case: Given two classical random variables X and Y , withjoint probability distribution pXY , the Shannon entropy always obeys the relationH1 Xð Þ � H1 X; Yð Þ and H1 Yð Þ � H1 X; Yð Þ. From Eq. (16.27), we see that ananalogous result holds in the quantum realm only for separable states (Horodecki,Horodecki, and Horodecki 1996).

334 Guido Bellomo and Gustavo M. Bosyk

Relying on the strong relationship between disorder, entropies, and majoriza-tion, it is natural to expect some connection between separability and the local andglobal spectrums. Indeed, Nielsen and Kempe (2001) proved that

ρAB separable ) ρAB≺ ρA and ρAB≺ ρB: (16.29)

Notice that violation of the r.h.s of Eq. (16.29) is a sufficient condition to haveentanglement. However, we remark that the spectral properties do not determineseparability in general. Indeed, there exist pairs of states such that they are globallyand locally isospectral, but one of them is separable and the other is not (Bengtssonand Życzkowski 2017: 491).

16.3.5 Pure States Interconversion

Evoking the third introductory question that we asked at the very beginning of thischapter, which referred to the possibility of transforming a given pure state intoanother by means of local operations and classical communication, we nowpropose to study entanglement transformations by using local operations andclassical communication (LOCC). So far, we have always considered majorizationbetween quantum states (namely, between the corresponding spectra), but in thiscase, we have to recall majorization relations among Schmidt coefficients.

So, the question is whether an initial bipartite pure state jψABi 2 H AB can betransformed into another bipartite state jϕABi 2 H AB (the target), by using LOCC.Here, by LOCC one refers to product (noncorrelated) operations, acting over H A

and H B independently, assisted by two-way classical communication. That is, onesupposes a channel that allows communicating the results of a given local oper-ation to the other part. The problem has been originally addressed by Nielsen(1999), who identified the necessary and sufficient condition that enables thisentanglement transformation process. An auxiliary result due to Lo and Popescu(1999), used by Nielsen to prove his theorem, regards the possibility to simulatethe two-way classical communication by a unidirectional classical channel andlocal generalized measurements. Interestingly enough, the conditions elucidated byNielsen, under which entangled states can be achieved by this process, can beestablished in terms of a majorization relation. More precisely, if σ ψð Þ and σ ϕð Þ arethe probability vectors formed by the corresponding squared Schmidt coefficients,the LOCC transformation

ψAB! ϕAB is possible if and only if σ ψð Þ≺ σ ϕð Þ:

ψAB ����!LOCC

ϕAB , σ ψð Þ≺ σ ϕð Þ: (16.30)

In other words, since those squared coefficients coincide with the reduced states’eigenvalues

Majorization, across the (Quantum) Universe 335

ψAB ������!LOCC

ϕAB , TrA ψAB

ψAB� ≺ TrA ϕAB

ϕAB� : (16.31)

We stress that the majorization relationship constitutes the necessary and sufficientcondition under which this transformation is allowed, without any reference to thecorresponding Schmidt bases.

As already mentioned, majorization gives a partial order over Δd and, as such,Nielsen’s result does not hold in general, in the sense that there exists a pair ofstates that neither of them majorizes each other: ψj i↮ jϕi. For instance, it is easyto check that jψi↮ jϕi by LOCC when the squared Schmidt coefficients areσ ψð Þ ¼ 0:60; 0:15; 0:15; 0:10½ � and σ ϕð Þ ¼ 0:50; 0:25; 0:20; 0:05½ �, becauseσ ψð Þ⊀ σ ϕð Þ and σ ϕð Þ⊀ σ ψð Þ. With this in mind, the celebrated result discussedearlier, due to Nielsen, has subsequently been extended to the case of nondetermi-nistic LOCC transformations by Vidal (1999). In that case, one looks for themaximal probability of success. On the other hand, it is also possible to providefurther insight if one considers deterministic transformations but with approximatetarget states. Vidal, Jonathan, and Nielsen (2000) have solved this problem byinvoking a criterion of maximal fidelity. The same problem has been tackled froma different perspective, exploiting the lattice structure of majorization and showingthat both proposals are linked via a majorization relation (Bosyk, Sergioli, Freytes,et al. 2017). The latter seems to be the first attempt to exploit the lattice character ofthe majorization partial order in a quantum information context, beyond its well-known partial-order properties.

Another generalization, proposed by Jonathan and Plenio (1999), consists in theextension of the set of initial and final states by appealing to deterministicentangled-assisted LOCC, that is, considering a shared catalytic entangled statebetween both parts. In this protocol, we have a new partial-order relation that it iscalled “trumping majorization” and reads as follows: Given x, y, z 2 Δd, it is saidthat x is trumping majorized by y (and denoted by x≺ Ty) if and only if there existsa catalytic r such that x⊗ z≺ y⊗ z (see Daftuar and Klimesh 2001 for somemathematical properties of trumping, and Müller and Pastena 2016 for an exten-sion of this concept related to Shannon entropy). Although it is an open questionwhether trumping majorization can be endowed with a lattice structure in thegeneral case (Harremoës 2004), it has been recently shown that the structure holdsfor the minimal nontrivial case, namely the case of four-dimensional vectors andtwo-dimensional catalysts (Bosyk, Freytes, Bellomo, et al. 2018).

It is notable that all these questions can be enclosed under the problemof convertibility of one kind of physical resource into another. Lately, thisresource theoretic approach has been extensively applied to attack a bunch ofquantum information-related topics such as, for instance, nonlocal correlations(see, e.g., Barrett et al. 2005, de Vicente 2014), quantum coherence and asymmetry

336 Guido Bellomo and Gustavo M. Bosyk

(see, e.g., Ahmadi, Jennings, and Rudolph 2013, Piani et al. 2016), quantumthermodynamics (see, e.g., Brandao et al. 2013, Gour et al. 2015) and super-selection rules (see, e.g., Gour and Spekkens 2008). Remarkably enough, thisformalism has recently been applied out of the quantum domain, for instance, tothe study of polarization-coherence properties of classical electromagnetic fields(Bosyk, Bellomo, and Luis 2018a, 2018b) as well as to the study of measures ofstatistical complexity (Rudnicki et al. 2016).

16.3.6 Uncertainty Relations

The uncertainty principle is, without any doubt, another of the fundamentalcharacteristics of quantum mechanics. Its relation to majorization theory is theend of our trip across the quantum universe.

Heisenberg (1927), in his seminal article, whose 90th anniversary was cele-brated in 2017, appealed to a heuristic formulation in order to quantify a funda-mental operational limitation imposed by the quantum laws, namely: theimpossibility of preparing states that give well-defined values for complementaryobservables (such as the position and momentum of a particle).

The best-known formulation of this principle is the Robertson (1929) uncer-tainty relation

V X; ρð ÞV Y ; ρð Þ � 14

X; Y½ �h iρ 2, (16.32)

where V O; ρð Þ ¼ O2�

ρ � Oh i2ρ and Oh iρ ¼ Tr ρOð Þ denote the variance andexpectation value of an observable O ¼ X, Y given the state preparation ρ, respect-ively. This preparation uncertainty relation thus describes a tradeoff between thevariance of two incompatible observables for the same quantum state, but separ-ately obtained in different experiments. However, several authors criticized thatformulation because it does not capture the essence of the uncertainty principle ingeneral. The main shortcomings of Eq. (16.32) appear when the observables havediscrete spectrum. On the one hand, the use of variance as degree of uncertainty ofa given observable with discrete spectrum loses its operational meaning, because amere relabeling of the observable outcomes (without changing its probability ofoccurrence) can give a variation of the uncertainty. Indeed, the only justificationfor the use of the variance given by Robertson is because it is “in accordance tostatistical usage” (Robertson 1929). It seems that he had in mind Gaussiandistributions, where the variance is enough to completely describe it. On the otherhand, the measure of incompatibility of the observables is given by the mean valueof the commutator. For observables with discrete spectrum, the commutator isanother observable for which there always exists a quantum state such that its mean

Majorization, across the (Quantum) Universe 337

value vanishes. For example, this happens when the quantum state is an eigenstateof one the observables. In other words, the r.h.s of Eq. (16.32) is state-dependent(that is, it is not universal) and does not fully characterize the incompatibility of theobservables.

For those reasons, several alternative uncertainty relations have been proposedin order to overcome these issues. Among them, geometric (Landau and Pollak1961, Bosyk et al. 2014), entropic (Deutsch 1983, Maassen and Uffink 1988,Zozor, Bosyk, and Portesi 2014) and majorization (Partovi 2011, Friedland,Gheorghiu, and Gour 2013, Puchała, Rudnicki, and Życzkowski 2013) uncertaintyrelations have appeared as the most prominent ones. In general, an uncertaintyrelation is an inequality of the form

U A;B; ρð Þ � ℬ A;Bð Þ, (16.33)

where U A;B; ρð Þ measures the degree of uncertainty in the observables’ outcomeswhen the quantum system is prepared in the state ρ, and ℬ A;Bð Þ is a measure of theobservables’ incompatibility, which is state-independent and strictly greater thanzero except when the observables share at least one eigenstate.

Here, we are interested in a majorization-based formulation of the uncertaintyprinciple. For simplicity, let us consider observables with discrete and nondegene-rate spectrum, that is, X ¼Pd

i¼1 xi iXj i iX jh andPd

i¼1 yi iYj i iY jh . A majorization-based uncertainty relation has the form

p X; ρð Þ⊗p Y ; ρð Þ ≺ ω X;Yð Þ, (16.34)

where pi O; ρð Þ ¼ Tr ρjiOð i iOjh Þ is the i-th component of the d-dimensional prob-ability vector p O; ρð Þ and ω X; Yð Þ is a d2-probability vector that measures theincompatibility between the observables. Clearly, if the observables do not share acommon eigenstate, then ω X; Yð Þ 6¼ 1; 0; . . . ; 0½ �, giving a nontrivial and universal(i.e., state-independent) bound on how the product distributions must be. Theexplicit expression of the optimal bound ω X;Yð Þ is very difficult to calculate ingeneral, because it involves a hard optimization problem. It can be shown that aweaker uncertainty relation is given as follows,

p X; ρð Þ⊗p Y ; ρð Þ≺ω c; c0ð Þ ¼ ω1 cð Þ;ω2 c0ð Þ � ω1 cð Þ; 1� ω2 c0ð Þ; 0; . . . ; 0½ �,(16.35)

whereω1 cð Þ ¼ 1þ c

2

� 2

and c denotes themaximumoverlap between the eigenbasis

of the observables, that is,

c ¼ maxiX ;iYf g

iXh jiYij j 2 1ffiffiffiffiN

p ; 1

� �, (16.36)

338 Guido Bellomo and Gustavo M. Bosyk

and ω1 c0ð Þ ¼ 1þ c0

2

� 2

with

c0 ¼ maxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiXh jiYij j2 þ i0X

� ji0Y 2q

, (16.37)

where the maximum is taken over the all indexes iX ¼ i0X and iY 6¼ i0Y , and over theall indexes iX 6¼ i0X and iY ¼ i0Y .

Finally, let us observe that from a majorization uncertainty relation one can alwaysobtain the corresponding entropic version by using Schur-concave additive entropies.For instance, the Rényi entropic uncertainty relation obtained from Eq. (16.34) is

Hα p X;ρð Þð ÞþHα p Y ;ρð Þð Þ� 11�α

log ω1 cð Þαþ ω2 c0ð Þ�ω1 cð Þð Þαþ 1�ω2 c0ð Þð Þα� �:

(16.38)

It can be shown that for Shannon entropy (α ¼ 1) this entropic uncertainty relationis stronger than the one derived by Deutsch (1983)

H p X; ρð Þð Þ þ H p Y; ρð Þð Þ � �2 log1þ c

2: (16.39)

Therefore, majorization-based uncertainty relations not only give adequate formu-lations of the uncertainty principle, but also allow stronger entropic-based expres-sions to be obtained.

16.4 Concluding Remarks

We began our tour with the aim of explaining how a variety of quantum problemsultimately depend on possible hierarchizations of quantum states based on the notionof majorization. Our task has been accomplished, after heterogeneous discussionsabout classification of quantum mixtures, entropies, and bistochastic operations,correlations, entanglement, conversion by LOCC, and uncertainty relations. As weanticipate, this journey has been by no means comprehensive. However we hope theparadigmatic examples that we have discussed throughout this chapter could drawattention to the ubiquity of majorization as a natural way to compare quantum statesand as a powerful tool to study quantum information problems.

Acknowledgments

We are extremely grateful to the organizers of the workshop Identity, indistinguish-ability and non-locality in quantum physics (Buenos Aires, June 2017). This workwas partially supported by CONICET and UNLP and Grant 57919 from the JohnTempleton Foundation.

Majorization, across the (Quantum) Universe 339

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342 Guido Bellomo and Gustavo M. Bosyk

Part V

The Relationship between the Quantum Ontology andthe Classical World

17

A Closed-System Approach to Decoherence

sebastian fortin and olimpia lombardi

17.1 Introduction

Decoherence is a process that leads to spontaneous suppression of quantuminterference. The orthodox explanation of the phenomenon is given by the environ-ment-induced-decoherence approach (see, e.g., Zurek 1982, 1993, 2003),according to which decoherence is a process resulting from the interaction of anopen quantum system and its environment. By studying different physical models,it was proved that, when the environment has a huge number of degrees of freedomand for certain interactions, the reduced state of the open system rapidly diagona-lizes in a well-defined preferred basis.

The environment-induced approach has been extensively applied to many areasof physics, such as atomic physics, quantum optics, and condensed matter, and hasacquired a great importance in quantum computation, where the loss of coherencerepresents a major difficulty for the implementation of the information processinghardware that takes advantage of superpositions. In the field of the foundations ofphysics, this approach has been conceived as the key ingredient to explainthe emergence of classicality from the quantum world, because the preferredbasis identifies the candidates for classical states (see, e.g., Elby 1994, Healey1995, Paz and Zurek 2002). It has been also considered a relevant element indifferent interpretations or approaches to quantum mechanics (for a survey, seeBacciagaluppi 2016).

The wide success of the environment-induced approach to decoherence over-shadowed any conceptual difficulty: Only a few works were devoted to analyze theassumptions and limitations of the orthodox approach. In resonance with this fact,the different approaches to decoherence that have arisen to face those difficultieswere not taken into account with the care that they deserve. In this chapter we willshow that there is a different perspective to understand decoherence – a closed-system approach – which not only solves or dissolves the problems of the orthodox

345

approach, but also is in agreement with a top-down view of quantum mechanicsthat offers a new perspective about the traditional interpretive problems.

With this purpose, the chapter is organized as follows. In Section 17.2, we willbegin by contrasting a bottom-up view versus a top-down view of quantummechanics. In Section 17.3, the decoherence resulting from the interaction withthe environment will be explained from a closed-system perspective. This willallow us to introduce, in Section 17.4, a general top-down, closed-system approachto decoherence, in the context of which environment-induced decoherence is aparticular case. The chapter closes with some final remarks.

17.2 Bottom-Up View versus Top-Down View of Quantum Mechanics

The idea that nature consists of tiny elemental entities is deeply entrenched in ourway of conceiving reality. It finds its roots in ancient Greece with atomism, andreappears in the early Modern Age with the corpuscularist philosophy of RobertBoyle, which influenced many contemporary thinkers, including Newton. Sincethose days, it has taken different forms in chemistry, as in Dalton’s atomic theory,and in physics, from the kinetic theory of gases to the standard model of particlephysics. An epistemological strategy becomes natural in the light of this onto-logical picture: In order to understand nature, it is necessary to decompose it intosimple systems. The knowledge about the whole is obtained by first studying thesimple systems and then combining them through their interactions. Of course,there are cases in which this analytical strategy leads to descriptions that cannot besolved by formal means. This is the case of the three-body problem in classicalmechanics. Nevertheless, even if there is no general closed-form solution for theequations describing the many-body system, nobody doubts that the behavior ofthe whole system is determined by the components and its interactions; preciselyfor this reason, those problems are commonly solved by numerical methods.

With the advent of quantum mechanics, this ontological picture went into crisis.The phenomenon of entanglement, which is not a traditional physical interaction,is responsible of correlations that cannot be understood in classical terms. There-fore, in quantum mechanics, the assumption that the best knowledge of the wholeis obtained by studying the simple systems and their interactions breaks down:Here the state of the composite system is not uniquely determined by the states ofthe component subsystems. Nevertheless, in spite of this well-known fact, it isusual to begin with quantum systems, represented by Hilbert spaces, whichbecome subsystems when they constitute a composite system. The implicitassumption is the atomistic assumption that there are certain elemental “particles”out of which everything is composed. This assumption has even been madeexplicit by the atomic modal interpretation of quantum mechanics, according to

346 Sebastian Fortin and Olimpia Lombardi

which there is, in nature, a fixed set of mutually disjoint atomic quantum systemsthat constitute the building blocks of all the other quantum systems (Bacciagaluppiand Dickson 1999). Good candidates for elemental systems are those representedby the irreducible representations of the symmetry group of the theory.

From this viewpoint, when quantum systems interact, their states may becomeentangled: “By the interaction the two representatives [the quantum states] havebecome entangled” (Schrödinger 1935: 555, when he coined the term ‘entangled’).In this case, it is said that the composite system is an entangled state, because itcannot be obtained as the tensor product of the components’ states. Entanglementis, therefore, responsible for the correlations between the values of the observablesof the two subsystems.

This bottom-up ontological view leads us to first consider two particles, say, aproton p and an electron e, represented by the Hilbert spaces H p and H e and instates ψp 2 H p and ψe 2 H e, respectively. Then, the state ψ 2 H p⊗H e of thehydrogen atom as a composite system is said to be entangled when ψ 6¼ ψp⊗ψe forany pair of states ψp and ψe. This suggests that “entangled” is a property thatapplies or not to the state of a composite system. However, the hydrogen atom canalso be represented as constituted by two different subsystems, the center-of-masssystem ψc 2 H c and the relative system ψr 2 H r, such that the state of thehydrogen atom ψ 2 H c⊗H r can be obtained as ψ ¼ ψc⊗ψr: Now the state ofthe composite system is not entangled. Although conceiving the hydrogen atom asbeing composed of a proton and an electron seems more natural, there are groupreasons that may lead to considering that the decomposition in a center-of-masssystem and a relative system is more fundamental (see Ardenghi, Castagnino, andLombardi 2009). This means that it cannot be said that a state of a compositesystem is entangled or not without first deciding which decomposition of thesystem will be considered.

John Earman stresses this fact by saying:

[A] state may be entangled with respect to one decomposition but not another; hence,unless there is some principled way to choose a decomposition, entanglement is a radicallyambiguous notion.

(Earman 2015: 303)

As a consequence, it is necessary to single out the “correct” decomposition, andtwo positions can be distinguished (Earman 2015: 324–327). For the realist, thereare certain subsystems that are ontologically “real” systems, whereas others aremerely fictional. For the pragmatist, by contrast, the legitimate criterion for decom-position is empirical accessibility.

Although in certain passages of his article Earman talks about relativity, thestronger idea is that of the “rampant ambiguity” of the notion of entanglement

A Closed-System Approach to Decoherence 347

(2015: 324, 325, 327). A notion is ambiguous if it has more than one meaning; so,in science and in philosophy ambiguity must be avoided. Therefore, if the notion ofentanglement is ambiguous, the need for a clear-cut decision about how to split thecomposite system into subsystems seems completely reasonable. Nevertheless, adifferent view is possible: The notion of entanglement is not ambiguous; it isrelative to the decomposition. The difference between ambiguity and relativity isnot irrelevant at all. Whereas the first is a conceptual problem to be solved, thesecond is a common feature of physical concepts. In fact, the concept of velocity isnot ambiguous because it is relative to a reference frame. In the same sense,entanglement is a notion that acquires a precise meaning when relativized to acertain partition of the composite system and, as a consequence, no absolutecriterion to select the right decomposition is needed.

The relative conception of entanglement invites us to reverse the generalapproach to quantum mechanics – from the traditional, classically inspiredbottom-up view, to a top-down view that endows the composite system withontological priority. From this perspective, even if two systems exist independ-ently before interaction, after the interaction their existence is only derivative, theybecome components of the composite system on a par with other subsystemsresulting from any different decomposition. This view finds a significant affinitywith the so called quantum structure studies, which deal with the different ways inwhich a quantum system can be decomposed into subsystems according to differ-ent tensor product structures (Harshman and Wickramasekara 2007a,b, Jeknić-Dugić, Arsenijević, and Dugić 2013, Arsenijević, Jeknić-Dugić, and Dugić 2016,Harshman 2016).

But the top-down view can be generalized a step further. Up to this point, therelation between “top” and “down” was described in terms of decomposing thecomposite system into its subsystems: The result of decomposition are subsystems,represented by Hilbert spaces; the tensor product of the Hilbert spaces of thesubsystems is the Hilbert space of the composite system. But the top-downrelationship can also be conceptualized in terms of algebras of observables, inresonance with the algebraic approach to quantum mechanics (Haag 1992). Thewhole system, represented by its algebra of observables, can be partitioned intodifferent parts, identified by the subalgebras, even when these subalgebras do notcorrespond to subsystems represented by Hilbert spaces. This perspective, releasedfrom the subsystem-dependent view anchored in tensor product structures, wasproposed by Howard Barnum and colleagues (2003) as the basis for ageneralization of the notion of entanglement to partitions of algebras. This gener-alized notion becomes the usual notion of entanglement when the partition of thealgebra of the whole system defines a decomposition of the system into subsystems(Barnum et al. 2004, Viola et al. 2005; Viola and Barnum 2010). A further

348 Sebastian Fortin and Olimpia Lombardi

characterization of pure entangled states can be given by appealing to the notion ofrestriction to a subalgebra, which is a natural algebraic generalization of the partialtrace operation (Balachandran et al. 2013a,b). As a consequence, entanglement isnot a relationship between systems or states, but between algebras of observables(Harshman and Ranade 2011).

At present, this subsystem-independent view has been formally studied withgreat detail in many works and is still in development. However, the point that wewant to stress here is that this view suggests a top-down closed-system ontologicalpicture, according to which the whole closed system is the only autonomous entity:The subentities represented by subalgebras of the whole algebra of observables areonly partial perspectives of the closed system without autonomous existence. In thefollowing section we will show that the phenomenon of decoherence can beexplained from this top-down closed-system view, which, in turn, leads to ageneralized approach to decoherence.

17.3 Environment-Induced Decoherence from a Closed-System Perspective

17.3.1 What Are the Systems That Decohere?

The environment-induced-decoherence program quickly became a new orthodoxy inthe physicists’ community (Bub 1997). Despite this, the program is still threatened bya serious conceptual problem, which is precisely derived from its open system.

According to the orthodox view, the first step is to split the universe into thedegrees of freedom that are of direct interest for the observer, “the system ofinterest,” and the remaining degrees of freedom that are usually referred to as “theenvironment.” Inmanymodels, distinguishing between the system of interest and itsenvironment seems to be a simple matter. This is the case in many typical applica-tions of the decoherence formalism to spin-bath models – devoted to study thebehavior of a particle immersed in a large “bath” of many particles (see, e.g., Zurek1982). But the environment can also be internal, such as phonons or other insideexcitations. This is typically the case when the formalism is applied to cosmology:The universe is split into some degrees of freedom representing the system, and theremaining degrees of freedom that are supposed to be nonaccessible and, therefore,play the role of the environment (see, e.g., Calzetta, Hu, and Mazzitelli 2001). Thepossibility of internal environments leads to the need for a general criterion todistinguish between the system and its environment. The problem is that the envir-onment-induced-decoherence program does not provide such a criterion. WojciechZurek recognized this shortcoming of his proposal early on:

one issue which has been often taken for granted is looming big, as a foundation of thewhole decoherence program. It is the question of what are the “systems” which play such a

A Closed-System Approach to Decoherence 349

crucial role in all the discussions of the emergent classicality. This issue was raised earlier,but the progress to date has been slow at best. Moreover, replacing “systems” with, say,“coarse grainings” does not seem to help at all, we have at least tangible evidence of theobjectivity of the existence of systems, while coarse-grainings are completely “in the eyeof the observer.”

(Zurek 2000: 338; see also Zurek 1998).

It is quite clear that the problem can be removed from a top-down closed-systemperspective as that delineated in the previous section.

In order to explain decoherence from a closed-system perspective, let us beginby recalling the definition of the concept of reduced state, because the environ-ment-induced-decoherence program decides to study the time behavior of thereduced state of the system of interest. The reduced state ρr1 of a system S1,subsystem of a system S, is defined as the density operator by means of whichthe expectation values of all the observables of S belonging exclusively to S1 canbe computed. As Maximilian Schlosshauer emphasizes, strictly speaking, areduced density operator is only a “calculational tool” for computing expectationvalues (Schlosshauer 2007: 48). This means that the description of decoherence interms of the reduced state of the open system is conceptually equivalent to thedescription in terms of the expectation values of the observables of the opensystem but viewed from the perspective of the whole closed system. This is thepath we will follow here.

17.3.2 The Perspective of the Closed System

Let us consider a closed system U partitioned as U ¼ S [ E, where S is the opensystem of interest and E is the environment. Let us call OU the space of observ-ables of U, and OS and OE the spaces of observables of S and E, respectively; thenOU ¼ OS⊗OE. If ρU is the state of U, the reduced state of S can be computed bymeans of the partial trace as ρS ¼ TrE ρU . The environment-induced-decoherenceformalism proves that, in many physically relevant models with environments ofmany degrees of freedom, the nondiagonal terms of the reduced state ρS tð Þ rapidlytend to vanish after an extremely short decoherence time tD:

ρS tð Þ���!t�tD ρdS tð Þ (17.1)

where ρdS tð Þ is diagonal in the preferred basis of OS. The evolution just describedexpresses the following evolution in the expectation values of the observablesOS 2 OS of the open system S:

OSh iρS tð Þ ���!t�tD OSh iρdS tð Þ (17.2)

350 Sebastian Fortin and Olimpia Lombardi

But, by definition, ρS is the density operator by means of which the expectationvalues of all the observables OS 2 OS in the state ρS can be computed, that is:

8 OUS ¼ OS⊗IEð Þ 2 OU OUSh iρU ¼ OSh iρS (17.3)

where IE 2 OE is the identity of the space of observables of the environment E.Then, it is clear that even when the task is to describe only S, its reduced state is notindispensable. The physically relevant information about that subsystem can alsobe obtained by studying the state ρU of the whole closed system U and its relevantobservables OUS ¼ OS⊗IE. This means that there is no difference between describ-ing the open system S by means of its reduced state ρS and describing it from aclosed-system perspective by means of the expectation values of the relevantobservables OUS of the closed composite system U in the state ρU . Therefore,the evolution of Eq. (17.3) can be expressed from the viewpoint of the closedsystem U as:

OUSh iρU tð Þ ���!t�tD OUSh iρdU tð Þ (17.4)

where ρdU tð Þ is not completely diagonal, but is diagonal in the preferred basisof OS.

17.3.3 The Emergence of Classicality

The emergence of classicality through decoherence can be explained strictly interms of expectation values. The general idea is that the expectation value of anobservable O when the system is in the certain state ρ can be expressed as:

Oh iρ ¼X

i

Oii ρii þX

i6¼j

Oij ρij (17.5)

where the ρii and the Oii are the diagonal components, and the ρij and the Oij are thenondiagonal components of ρ and O, respectively, in a certain basis. The secondsum of Eq. (17.5) represents the specifically quantum interference terms of theexpectation value. If those terms vanished, the expectation value would adopt thestructure of a classical expectation value, where the Oii might be interpreted aspossible values, and the ρii might play the role of probabilities, since they arepositive numbers that are less than or equal to one and sum to one.

In the light of this idea, the process of decoherence described by the evolution ofEq. (17.2) leads to a classical-like expectation value, since ρdS tð Þ is diagonal in thepreferred basis of OS:

OSh iρS tð Þ ���!t�tD OSh iρdS tð Þ ¼

X

i

OSii ρdSii tð Þ (17.6)

A Closed-System Approach to Decoherence 351

where the ρSii and the OSii are the diagonal components of ρS and OS, respectively,in the preferred basis.

However, the same move cannot be applied to the evolution as expressed inEq. (17.4), because ρdU tð Þ is not completely diagonal: It is diagonal only in thecomponents corresponding to the preferred basis of OS. Nevertheless, decoherencecan be described from the closed-system perspective analogously to Eq. (17.6) if acoarse-grained state ρG tð Þ of the closed system U is defined as the operator suchthat:

8 OUS ¼ OS⊗IEð Þ 2 OU OUSh iρU tð Þ ¼ OUSh iρG tð Þ (17.7)

The density operator ρG represents a coarse-grained state because it can beobtained as ρG ¼ ΠρU ¼ ΠΠρU . The projector Π performs the followingoperation:

ΠρU ¼ TrE ρUð Þ⊗~δE ¼ ρS⊗~δE (17.8)

where ~δE 2 OE is a normalized identity operator with coefficients~δEαβ ¼ δαβ=

Pγδγγ (see Fortin and Lombardi 2014). Now, the process of decoher-

ence can be expressed as

OUSh iρU tð Þ ���!t�tD OUSh iρdG tð Þ (17.9)

where ρdG tð Þ remains completely diagonal for all times t � tD. Now it can be saidthat the expectation value also acquires a classical form from the closed-systemperspective since:

OUSh iρU tð Þ ���!t�tD OUSh iρdG tð Þ ¼

X

i

OUSii ρdGii tð Þ (17.10)

where the ρdGii and the OUSii are the diagonal components of ρdG and OUS,respectively, in the basis of decoherence. It is quite clear that ρG, althoughoperating onto OU , is not the quantum state of U: It is a coarse-grained stateof the closed system that disregards certain information of its quantum state.However, ρG supplies the same information about the open system S as thereduced state ρS, but now from the viewpoint of the composite system S. In fact,if the degrees of freedom of the environment are traced off, the reduced state ρS isobtained:

TrE ρG ¼ ρS (17.11)

Therefore, the reduced density operator ρS can also be conceived of as a kind ofcoarse-grained state of U, which disregards certain degrees of freedom consideredas irrelevant.

352 Sebastian Fortin and Olimpia Lombardi

17.3.4 The Applications of the Closed-System Approach

The closed-system approach was presented from different perspectives, from themore conceptual (Castagnino, Laura, and Lombardi 2007, Lombardi, Fortin, andCastagnino 2012), to the more technical (Castagnino and Lombardi 2005, Castag-nino and Fortin 2011, Fortin, Lombardi, and Castagnino 2014). It was also appliedto a generalization of the spin-bath model (Castagnino, Fortin, and Lombardi2010): A generalized spin-bath model of mþ n spin-1/2 particles, where the mparticles interact with each other and the n particles also interact with each other,but the particles of the m group do not interact with those of the n group. The studyof the model shows that there are definite conditions under which all the particlesdecohere, but neither the system composed of the m group nor the systemcomposed of the n group decoheres.

Once decoherence is understood from this new perspective, the defining-systemproblem, that is, the problem that there is no criterion to distinguish between thesystem and the environment, disappears. In fact, the same closed system can bedecomposed in many different ways. Since there is no privileged or “essential”decomposition, there is no need for an unequivocal criterion to decide where toplace the cut between “the” system and “the” environment. If all the ways ofselecting the system of interest are equally legitimate, decoherence is relative to thedecomposition of the whole system (Lombardi et al. 2012, see also Lychkovskiy2013). In other words, Zurek’s “looming big” problem is not a real threat to theenvironment-induced-decoherence approach: The supposed challenge dissolvesonce it is understood that decoherence is not a yes-or-no process but a relativephenomenon.

17.4 The Top-Down Approach to Decoherence

17.4.1 The Formalism

In the previous section, the closed-system approach to decoherence was stilldiscussed in terms of the possibility of different tensor product structures: Deco-herence is relative to the particular decomposition of the composite system intosubsystems. In this section, the generalization will be taken a step further from thealgebraic viewpoint, by admitting that a closed system may be partitioned intoparts that do not constitute subsystems.

The starting point of the algebraic approach to quantum mechanics (Haag 1992;see also Bratteli and Robinson 1987) is the algebra of observables A Oð Þ, which isthe algebra spanned by a certain set O of observables O represented by self-adjointoperators mapping a suitable Hilbert space H onto itself. When the algebra A Oð Þidentifies a quantum system, the quantum state ω of the system is a prescription of

A Closed-System Approach to Decoherence 353

the expectation values of the observables, and it is formalized as an expectationvalue functional from the observables to the unit interval, ω : A Oð Þ ! 0; 1½ �.A quantum state is said to be normal when there is an associate density operatorρω (with ρω � 0 and Tr ρω ¼ 1) acting on the same Hilbert space H and such thatω Oð Þ ¼ Tr Oρωð Þ. The expectation value ω Oð Þ gives the expected value if onemeasures the observable O when the system is in the state ω, and the equationω Oð Þ ¼ Tr Oρωð Þ is essentially the Born Rule extended to mixed states.

The algebraic notions just stated are sufficient to formulate a top-down approachto decoherence that is independent of the tensor product structures of the Hilbertspaces. Let us consider a closed system U identified by its algebra of observablesA OUð Þ, and its state, represented by the density operator ρU . Now U is notdecomposed into subsystems, but a certain set of relevant observables OR isselected. It is interesting to notice that this move agrees with the approaches ofthe first period in the historical development in the general program of decoherence(see Fortin et al. 2014), when the aim was to understand how classical macroscopicproperties emerge from the quantum microscopic evolution of a closed system. Inthis first period, the approach to equilibrium of quantum systems was studied fromthe behavior of certain observables that supposedly should behave classicallybecause they are accessible from the macroscopic viewpoint, e.g., “gross observ-ables” (van Kampen 1954), “macroscopic observables of the apparatus” (Daneri,Loinger, and Prosperi 1962). In the present case, no restriction is imposed on theselection of the relevant observables: Any set of observables can be selected. Inany case, the algebra of the relevant observables, subalgebra of A OUð Þ, will beconsidered: A ORð Þ � A OUð Þ.

Once the relevant observables are selected, the second step consists in comput-ing the expectation values of the observables of the relevant algebra A ORð Þ:

8OR 2 A ORð Þ ORh iρU tð Þ (17.12)

Then, a coarse-grained state ρG tð Þ is defined, such that:

8OR 2 A ORð Þ ORh iρU tð Þ ¼ ORh iρG tð Þ (17.13)

Now, the nonunitary evolution (governed by a master equation) of this expectationvalue is computed. Decoherence occurs when, after an extremely short decoher-ence time tD, the expectation acquires a particular form:

ORh iρU tð Þ ¼ ORh iρG tð Þ ���!t�tD ORh iρdG tð Þ (17.14)

where ρdG tð Þ remains diagonal in the preferred basis for all times t � tD. Thismeans that, although the off-diagonal terms of ρU tð Þ never vanish through itsunitary evolution, it might be said that the system decoheres relatively to the

354 Sebastian Fortin and Olimpia Lombardi

observational point of view given by any observable belonging to the algebra ofthe relevant observables A ORð Þ.

17.4.2 Classically-Behaving Observables

Let us recall that decoherence has been considered the essential element to explainthe emergence of classicality from the quantum world. But if decoherence is arelative phenomenon, classicality also seems to be relative – the fact that a systembehaves classically or not cannot depend on the way in which the observer decidesto split the original closed system into relevant and irrelevant observables. Thissituation also challenges the orthodox open-system approach: In certain situationsthe fact that classicality emerges in an open system or not depends on whatcomposite system that open subsystem is embedded in. More precisely, giventwo partitions of a closed system U, U ¼ S1 [ E1 and U ¼ S2 [ E2, it may be thecase that S1 and S2 decohere and behave classically, but S1 [ S2 does not decohereand, so, classicality does not emerge in it (see the model in Castagnino et al. 2010).This is a difficulty if one considers that the classical world is objective, independ-ent of any observer’s decision. Recall Zurek’s rejection of any solution of thedefining-system problem that relies on “the eye of the observer” (Zurek2000: 338).

Despite what it seems, the top-down view of decoherence based on the algebraicapproach is not affected by this difficulty. Given the closed system U, saying that itdecoheres from the perspective of the relevant observables OR 2 A ORð Þ amountsto saying that, after a very short decoherence time, the interference terms of theexpectation values of those observables tend to vanish with the unitary time-evolution of the state ρU of U. But the vanishing of the interference terms of theexpectation values of an observable is not a relative fact that depends on theobserver: What depends on the observer is the selection of the relevant observableswith the purpose to see whether the closed system decoheres relative to it or not.

When this fact is understood, it turns out to be clear that all the observables ofthe closed system U can be considered one by one, their trivial algebras can bedefined, and the decoherence of the system U relative to each one of those algebrascan be studied. As a result, one is in a position to know the set of all theobservables of U that behave classically after a certain time, with neither ambiguitynor relativity.

Another difficulty of the orthodox approach that is not usually stressed is thatcertain systems have a classical behavior with respect to certain observables and aquantum behavior with respect to others. For instance, a transistor behaves classic-ally with respect to its center of mass when it falls off the table, but it also has thequantum behavior characteristic of its specific use. When decoherence is conceived

A Closed-System Approach to Decoherence 355

as a phenomenon that occurs or not to a quantum system, these common situationscannot be accounted for. By contrast, the top-down approach that relies on thesubalgebras of observables can easily explain how a single system may combineclassical and quantum behaviors of its different observables.

In summary, according to the explanation of the emergence of the classicalworld given by the top-down algebraic approach just proposed, strictly speaking,classicality is not a property of systems: Thinking of systems that become classicalin their whole leads to the difficulties mentioned earlier. The difficulties can beovercome once it is recognized that classicality is a property of observables. Theemergent classical world is, then, the world described by the observables thatbehave classically with respect to their expectation values.

17.5 Concluding Remarks

In this chapter we have proposed a closed-system approach to decoherence which,at first sight, seems to be a rival of the orthodox open-system approach. However,as we have argued, our proposal is compatible with the environment-induced-decoherence view, but generalizes it by including the treatment of situations thatcould not be studied with that orthodox view.

As already explained, this closed-system approach is in resonance with a top-down view of quantum mechanics, usually based on the algebraic formalism,which is gaining ground in the physics community. It is also interesting to noticethat understanding decoherence from the viewpoint of a closed system representedby its algebra of observables stands in close agreement with the modal-Hamiltonian interpretation of quantum mechanics (Lombardi and Castagnino2008, Ardenghi et al. 2009, Lombardi, Castagnino and Ardenghi 2010; see alsoChapter 2) also developed in our research group. This interpretation, also based onthe algebraic approach, makes the rule that selects the definite-valued observablesto depend on the Hamiltonian of the closed system. Moreover, the definition of thesystem in terms of its algebra of observables leads to an ontological picture wherequantum systems are bundles of properties without individuality (da Costa, Lom-bardi, and Lastiri 2013, da Costa and Lombardi 2014, Lombardi and Dieks 2016).In summary, the general view that endows closed systems with ontological priorityhas different but converging manifestations, in the light of which it deserves to befurther developed.

Acknowledgments

We are grateful to the participants of the workshop Identity, indistinguishabilityand non-locality in quantum physics (Buenos Aires, June 2017) for their

356 Sebastian Fortin and Olimpia Lombardi

interesting comments. This work was made possible through the support of Grant57919 from the John Templeton Foundation and Grant PICT-2014–2812 from theNational Agency of Scientific and Technological Promotion of Argentina.

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A Closed-System Approach to Decoherence 359

18

A Logical Approach to theQuantum-to-Classical Transition

sebastian fortin, manuel gadella, federico holik, andmarcelo losada

18.1 Introduction

The description of the classical limit of a quantum system is one of the mostimportant issues in the foundations of quantum mechanics (see Cohen 1989). Thisproblem has been formulated in different ways and explained by appealing todifferent interpretations (see Schlosshauer 2007). The attempts to explain theclassical limit go back to the correspondence principle, proposed by Niels Bohr.This principle establishes a connection between quantum observables and theirclassical counterparts when Planck’s constant is small enough in comparison withrelevant quantities of the quantum system. In particular, this happens in the limit oflarge quantum numbers.

Nowadays, the most important approach to describe the classical limit is basedon the decoherence process (see Schlosshauer 2007). The general idea of thisapproach is to explain the disappearance of the interference terms of quantumstates by appealing to the decoherence process induced by the environment. In thisway, the coherence needed for most typical quantum phenomena is lost, and theclassical features appear instead.

As is well known, the set of observables associated with a quantum systemforms a noncommutative algebra. This differs from the classical description ofphysical systems, in which observables are represented by functions over a phasespace, which form a commutative algebra. This difference between quantum andclassical systems has a correlate in terms of the elementary properties of physicalsystems. The elementary properties of quantum systems (also known as “yes-notests” or elementary experiments) are represented by orthogonal projectors actingon a Hilbert space. These projectors form a non-Boolean lattice (more specifically,a complete, atomic, atomistic, orthomodular lattice, satisfying the covering law,see Kalmbach 1983). Instead, the elementary properties of a classical system arethe measurable subsets of the phase space, which form a Boolean lattice.

360

The decoherence approach to the quantum-to-classical transition is based on theSchrödinger picture, in which states evolve over time, while observables andphysical properties are taken to be constants. As a result, the structure of quantumproperties remains the same for all times: The quantum logic associated with thesystem does not change (Bub 1997). Therefore, in this approach it is not explainedhow the structure of quantum properties becomes classical. However, as it wasremarked in Fortin and Vanni (2014), a reasonable condition for the existence of aclassical limit is that the lattice of elementary properties becomes Boolean, orequivalently, that the algebra of observables becomes commutative (Fortin andVanni 2014, Fortin, Holik, and Vanni 2016, Losada, Fortin, and Holik 2018).

In this chapter, we present a logical approach to the classical limit, whichdescribes how the logical structure of the elementary properties of a quantumsystem becomes classical when the classical limit is reached. In order to describethe evolution of logical structure, we consider the Heisenberg picture. According tothis picture, observables and physical properties evolve in time, while states remainconstant. In this way, we can consider the algebra of observables and the lattice ofelementary properties as dynamical objects, depending on time or other relevantparameters, such as action, temperature, particle number, or energy.

As we will show later, this offers an interesting perspective for studyingdifferent physical processes. In particular, we discuss the possibility of connectingthe approach of dynamical algebras developed in recent papers (Fortin and Vanni2014, Fortin et al. 2016, Losada et al. 2018) with the description of the classicallimit based on deformation of algebras. We also discuss the case of quantumstatistical mechanics, where intermediate logics are interpreted as phase transitions.

The chapter is organized as follows. In Section 18.2, we review the problem ofthe classical limit as it was traditionally considered in the literature. In Section18.3, we briefly summarize the logical structure of the elementary properties ofclassical and quantum physical systems, and we discuss the main differencebetween both logical structures. In Section 18.4, we introduce the logical approachto the classical limit, and we illustrate this approach with four different examples.Finally, in Section 18.5 we draw our conclusions.

18.2 Different Approaches to the Classical Limit

One of the first explanations was proposed by Niels Bohr, who appealed to thecorrespondence principle. This principle establishes a connection between quantumobservables and their classical counterparts by asserting that, if the ratio between theaction of the system and Planck’s constant is large enough, the classical limit shouldbe recovered. This implies that the quantum-to-classical transition should beattained in the limit of large quantum numbers, such as large orbits, large energies,

A Logical Approach to the Quantum-to-Classical Transition 361

or a large number of particles. A result that goes in line with the correspondenceprinciple is the Ehrenfest theorem. This approach is still important today, in particu-lar for studying quantum phenomena in the semiclassical level.

Paul Dirac proposed another explanation of the classical limit, appealing to thedestructive interference among all the possible paths of the physical system (Dirac1933). In this way, he showed that the classical action path has the dominantcontribution. This idea was subsequently elaborated by Richard Feynman (1942)in his thesis, opening the door to the celebrated path-integral formulation ofquantum mechanics.

All these approaches presented problems, which where extensively discussed inthe literature. In particular, it is important to remark that Bohr himself did notconsidered the classical limit as an explanation of the emergence of classicalreality. Quite on the contrary, Bohr believed that the classical realm exists inde-pendently of quantum theory and cannot be derived from it. As is well known, thediscussion about the classical limit is subtle and problematic, and there is no realagreement on a solution.

Nowadays, the most important approach for describing the classical limit isbased on the environment-induced decoherence. In this approach, it is consideredthat the quantum-to-classical transition is the result of the loss of coherence of thesystem due to the interaction with its environment (Schlosshauer 2007). Manyphysicists considered this proposal as the correct explanation of the classical limit(and also of the measurement process); however, some objections were raised,because the decoherence process would not explain how the logical structure of theelementary properties becomes a classical logic.

Another important approach to the study of the classical limit is based onalgebras’ deformation (see Landsman 1993). In this formalism, quantum commu-tators (or equivalently, Moyal brackets) reduce to Poisson brackets, deforming thealgebra involved.

In what follows, we present an alternative approach to describe the classicallimit. This is a logical approach, based on the evolution of the quantum observ-ables, and it allows describing the quantum-to-classical transition of the logicalstructure of the quantum systems. In the next section, we review some basicfeatures about the lattice of the elementary properties of classical and quantumsystems, which are relevant to our logical approach of the classical limit.

18.3 Logical Structure of Quantum Mechanics

In classical and quantum mechanics, the physical properties of a system areendowed with a lattice structure. These structures are different in the classicaland quantum case, and they determine the logical structure of the physical system.

362 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

In classical mechanics, physical systems are represented by the phase space, andtheir properties are represented by subsets of the phase space (see Kalmbach 1983).In quantum mechanics, the physical systems are represented by Hilbert spaces andthe properties are represented by closed vector subspaces or by their correspondingorthogonal projectors (von Neumann 1932; for a recent discussion about the logicapproach to quantum mechanics, see Domenech, Holik, and Massri 2010, Holik,Massri, and Ciancaglini 2012, Holik, Massri, Plastino, and Zuberman 2013, Holik,Plastino, and Sáenz 2014, Holik and Plastino 2015; for applications to quantumhistories see Omnès 1994, Losada, Vanni, and Laura 2013, 2016, Griffiths 2014,Losada and Laura 2014a, b). In both cases, the set of all the properties of a systemhas an orthocomplemented lattice structure. This implies that there is an orderrelation (�) such that for each pair of properties there are an infimum (∧) and asupremum (∨), and each property p has a complement p⊥ with adequate proper-ties. All orthocomplemented lattices satisfy certain inequalities, called distributiveinequalities (see Kalmbach 1983):

a∧ b∨cð Þ � a∧bð Þ∨ a∧cð Þa∨ b∧cð Þ � a∨bð Þ∧ a∨cð Þ (18.1)

When the equalities hold, the lattice is distributive. An orthocomplemented anddistributive lattice is called a Boolean lattice. The distributive property is anessential feature that differentiates classical and quantum lattices of properties.

In the classical case, the properties of the system are represented by the subsetsof its phase space. The partial-order relation is given by the inclusion (�) of sets.The infimum and the supremum are the intersection (\) and the union ([) of sets,respectively, and the complement of a property p is the complement of sets pc. Theset of classical properties is not only an orthocomplemented lattice, but also adistributive one, i.e., classical properties satisfy the distributive equalities. There-fore, the logical structure of a classical system is Boolean. This structure is usuallycalled classical logic.

The quantum case is very different. The properties are represented by closed-vector subspaces (or by their corresponding orthogonal projectors; von Neumann1932). Thus, the logical structure of quantum systems is given by the algebraicstructure of closed subspaces. The set of all quantum properties is also anorthocomplemented lattice, and, as in the classical case, the partial order relationis given by the inclusion of subspaces, and the infimum is given by the intersectionof subspaces. However, the supremum and the complement of properties aredifferent from the classical ones. The supremum is given by the sum of subspacesand the complement of a property is its orthogonal subspace. The resulting latticeis nondistributive (see Kalmbach 1983), and therefore, it is not Boolean. Thisstructure is called quantum logic (Birkhoff and von Neumann 1936).

A Logical Approach to the Quantum-to-Classical Transition 363

The distributive inequalities are the main difference between classical andquantum logic. In the classical lattice, all properties satisfy the distributive equal-ities, but in the quantum lattice, only distributive inequalities hold, in general.However, for some subsets of quantum properties the equalities hold. When asubset of properties satisfies the distributive equalities, they are called compatibleproperties. It can be proved that a sufficient and necessary condition for a set ofproperties to be compatible is that the projectors associated with the propertiescommute. Moreover, it can be shown that properties associated with differentobservables are compatible if the observables commute. If, on the contrary, twoobservables do not commute, some of the properties associated with them are notcompatible. Therefore, by extension, commuting observables are called compatibleobservables.

The differences between classical and quantum logic are of fundamental import-ance for the classical limit problem. If a quantum system undergoes a physicalprocess such that its behavior becomes classical, then its logical structure ofproperties should undergo a transition from quantum logic to classical logic, i.e.,its lattice structure should become distributive. However, the description of theclassical limit of a quantum system usually focuses on the state of the system.The mathematical description of this process does not explain how the logicalstructure changes on time. Therefore, in these approaches it is not possible todescribe how the structure of quantum properties becomes classical. In order togive an adequate approach to the classical limit, we need a description in whichobservables and physical properties evolve over time, changing the logical struc-ture of the system.

18.4 Logical Quantum-to-Classical Transition

A complete description of the quantum-to-classical transition should explain howthe logical structure of the system changes from a quantum logic to a classicallogic. In order to adequately describe this transition, we consider a quantum systemwith a general time evolution and a time-dependent set of relevant observablesO tð Þ ¼ Oi tð ÞgiϵI

�(where I is a set of indexes).

Each set of relevant observables O tð Þ generates an algebra of observables V tð Þ,and each algebra has associated an orthocomplemented lattice of properties LV tð Þ.We assume that in the initial set of observables O 0ð Þ there are some incompatibleobservables, and therefore its corresponding algebra of observables V 0ð Þ is non-commutative and the associated lattice of properties LV 0ð Þ is nondistributive.

For a quantum system with a decoherence time tD, the quantum-to-classicaltransition is characterized by a process that transforms noncommutative observ-ables into commutative ones

364 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

Oi 0ð Þ; Oj 0ð Þh i

6¼ 0⟶ Oi tDð Þ; Oj tDð Þh i

¼ 0, 8i, j: (18.2)

After time tD, the algebra of observables V tð Þ becomes commutative, and thecorresponding orthomodular lattice LV tð Þ becomes nondistributive. The logicalclassical limit is expressed by the fact that, while LV 0ð Þ is a nondistributive lattice,LV tDð Þ is a Boolean one. In this way, we obtain an adequate description of thelogical evolution of a quantum system.

In what follows, we discuss the dynamics of the quantum algebra of observablesand the logic structure of properties in some physical models.

18.4.1 Quantum Operations

A quantum operation is a linear and completely positive map from the set ofdensity operators into itself (Nielsen and Chuang 2000). For each time t, weconsider a quantum operation ℰt, which maps the initial state of the system ρ0 tothe state at time t, i.e.,

ℰt ρ0ð Þ ¼ ρ tð Þ: (18.3)

If we use the sum representation, we can express the quantum operation ℰt asfollows (Nielsen and Chuang 2000):

ℰt ρ0ð Þ ¼X

μEμ tð Þρ0bE†

μ tð Þ, (18.4)

where Eμ tð Þ are the Kraus operators of ℰt.We define the Heisenberg representation of ℰt as the operator ~ℰt that evolves the

observables from an initial time up to time t, i.e., ~ℰt O� � ¼ O tð Þ. The operator ~ℰt

must preserve the mean values of the observables for all times,

Tr ρ tð ÞO� � ¼ TrX

μEμ tð Þρ0bE†

μ tð ÞO� �

¼

¼ Tr ρ0X

μbE†μ tð ÞOEμ tð Þ

� �¼ Tr ρ0O tð Þ� �

: (18.5)

Hence, we can express the operator ~ℰt as follows:

~ℰt O� � ¼ O tð Þ ¼

XμbE†μ tð ÞOEμ tð Þ: (18.6)

Since O is self-adjoint, then ~ℰt O� �

is also a self-adjoint operator. Therefore, ~ℰt

maps observables to observables.Once we have defined the temporal evolution of quantum operations, we can

describe the logical classical limit of a quantum system as it was explained before.

A Logical Approach to the Quantum-to-Classical Transition 365

We illustrate the logical approach with a simple example: the amplitude dampingchannel.

The amplitude damping channel is useful for describing the energy dissipationdue to the environment effects. It is relevant for quantum information processing,because it is an adequate model for quantum noise. In particular, this model can beapplied to the decay of an excited state of a two-level atom due to spontaneousemission of photons. If the atom is in the ground state, no photon is emitted, andthe atom continues in the same state. But, if the atom is in the excited state, after aninterval of time τ, there is a probability p that the state has decayed to the groundstate and a photon has been emitted (see Nielsen and Chuang 2000).

The quantum operation of the amplitude damping channel can be expressed asfollows:

ℰτ ρ0ð Þ ¼ E0ρ0E†0 þ E1ρ0E

†1, (18.7)

where the Kraus operators are

E0 ¼1 0

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pð Þp

!, E1 ¼

0ffiffiffip

p

0 0

!: (18.8)

The associated quantum map ~ℰτ, acting on the space of observables is given by~ℰτ O� � ¼ bE†

0OE0 þ bE†1OE1. In matrix form, we have

~ℰτ O� � ¼ O00

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pð Þp

O01ffiffiffiffiffiffiffiffiffiffiffi1� p

pO10 pO00 þ 1� pð ÞO11

!: (18.9)

Applying the amplitude damping channel n times, we obtain the map ~ℰnτ O� �

,which has the form

~ℰnτ O� � ¼ O00

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pð Þn

pO01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� pð Þnq

O10 1� pð ÞnO11 þ O00�1� 1� pð Þn

0@

1A: (18.10)

If n⟶∞, all the observables become proportional to the identity. This implies thatthe algebra of observables becomes trivially commutative, and its correspondinglattice of properties becomes a classical logic.

18.4.2 Rigged Hilbert Space

One of the most investigated fields in quantum foundations is the quantizationproblem, which consists in obtaining quantum observables from their classicalcounterpart. Much less considered has been the problem of dequantization: thetransition from quantum to classical observables. It was shown (Castagnino and

366 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

Gadella 2006) that dequantization may require two steps, one is a type of deco-herence and the other is the notion of macroscopicity, which is implemented by thelimit ℏ ! 0. In the present section, we intend to give a brief account of anothernotion of dequantization in the presence of unstable quantum systems orresonances.

In a previous paper (Fortin et al. 2016), we argued that an essential characteristicof the quantum-to-classical transition should be the transition from a noncommu-tative algebra of observables to a commutative one, when t⟶þ ∞. This can berigorously formulated for unstable quantum systems, provided we considered thelinear space spanned by the resonance state vectors, also called Gamow vectors.

Resonances are usually characterized as poles of some analytic continuations ofa reduced resolvent or a scattering matrix. Both formulations are not alwaysequivalent – one may construct models for which the poles in one of these twoformulations are not poles in the other. In the energy representation, these polesappear in complex conjugate pairs and have the form ER � iΓ=2, where ER is theresonance energy and Γ is related to the inverse of the mean lifetime (see Bohm1993). Notice that Γ must be always positive.

From the observational point of view, single resonances show an exponentialdecay, provided that the time intervals are not too short and not very large either(Fonda, Ghirardi, and Rimini 1978). However, these deviations are very difficult toobserve. Therefore, most experiments with resonances show exponential decaysfor practically all values of time (Fischer, Gutierrez-Medina, and Raizen 2001,Rothe, Hintschich, and Monkman 2006).

Now, pure stable states have a mathematical representation in terms of vectorstates. The difference between a stable state and a resonance state is just thatthe value of the parameter Γ is equal to zero for stable states. Then, one istempted to introduce a definition of resonance states in such a way that, if theresonance poles are ER � iΓ=2, we have either H ψDj i ¼ ER � iΓ=2ð Þ ψDj i orH ψGj i ¼ ER þ iΓ=2ð Þ ψGj i (Nakanishi 1958). Here H ¼ H0 þ V is the totalHamiltonian which produces the resonance phenomenon. Note that, in the firstcase, formal time evolution gives e�itH ψDj i ¼ e�iERte�tΓ=2 ψDj i, which is an expo-nential decay for t⟶þ ∞. On the other hand, a similar formal time evolutiongives e�itH ψGj i ¼ e�iERtetΓ=2 ψGj i, which decays exponentially as t⟶� ∞. Vectorstates ψDj i and ψGj i are known as the decaying and growing Gamow vectors,respectively.

As a matter of fact, both vectors ψDj i and ψGj i (where the D stands for decayand the G stands for growing) are equally suitable for a vector state for theconsidered resonance. Nevertheless, the choice ψDj i seems more natural as thetime flows in the positive direction. The point is that both are time reversal of eachother and represent the same physical phenomenon. Note that both vector states

A Logical Approach to the Quantum-to-Classical Transition 367

describe the part of the resonance that behaves exponentially with time. Deviationsadd a background term (Bohm and Gadella 1989), but here we can consider it asnegligible.

The previous considerations have an important mathematical flaw, however.Gamow vectors are eigenvectors of the total Hamiltonian H with complex eigen-values. This is not compatible with the assumption that H is self-adjoint. However,this property is essential if we want the Gamow vectors to have an exponentialbehavior with time. There are two possible remedies for this problem:

18.4.2.1 Non-Hermitian Hamiltonian

This is the approach known as dilation analytic potentials (Balslev and Combes1971). It gives normalizable Gamow vectors belonging to the Hilbert space onwhich the total Hamiltonian H is defined as a self-adjoint operator. However, theseGamow vectors depend on a nonphysical parameter, precisely the parameter thatprovides the dilation, which is arbitrary at some extent (Reed and Simon 1978).There are other possibilities for using non-Hermitian Hamiltonians (see forinstance Eleuch and Rotter 2017).

18.4.2.2 Rigged Hilbert Space

The second possibility is the extension of the Hilbert space to a rigged Hilbert space(RHS). A RHS is a triplet of three spaces Φ � H � Φ�, with the following proper-ties: (i) H is an infinite dimensional separable Hilbert space (a Hilbert space isseparable if any orthonormal basis is countable); (ii) Φ is a dense subspace (asubspace of H is dense if any neighborhood of any vector in H contains vectorsin Φ) with a topology such that the Φ has less convergent sequences than what itwould havewith the topology inherited fromH ; and (iii)Φ� is the vector space of allcontinuous linear mappings from Φ to the space C of complex numbers. RHS alsoserves for a rigorous presentation of the Dirac formulation of quantum mechanics(Roberts 1966, Antoine 1969, Melsheimer 1974, Bohm 1978, Gadella and Gómez2002, 2003), and it has some other applications concerning group representationsand special functions (Celeghini, Gadella, and del Olmo 2016, 2017, 2018).

Then, if H is the Hilbert space on which the total Hamiltonian H ¼ H0 þ Vacts, we may construct two RHS Φ� � H � Φ�

�, with the property that

ψDj

E2 Φ�

þ and ψGj

E2 Φ�

�, where the index j stands for the number of reson-

ances in the system with resonance complex energies ERj � iΓj=2. Decaying andgrowing Gamow vectors have the desired time behavior, a fact that can berigorously proven (Bohm and Gadella 1989, Civitarese and Gadella 2004).

A nonrelativistic quantum system may have infinitely many resonances. Thismeans that only a finite number of resonances may be considered. We recall that

368 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

resonances are determined by the poles of a complex analytic function which arealways isolated points in the complex plane. For large values of ER, the energies goto the relativistic regime, so that we have to discard this possibility. But then,resonances with large imaginary part are not observable, because their meanlifetimes are extremely small. This means that only a finite number of resonancesmay be considered within the nonrelativistic regime for a given unstable quantumsystem.

In addition, if we only focus our attention on the resonance behavior, we mayconsider the space spanned by the Gamow vectors. For decaying (growing)Gamow vectors, this is a finite dimensional subspace of Φ�

þ Φ��

� �. Let us assume

that our system has N resonances with zj≔ERj þ iΓj=2 and z∗j being its complexconjugate. We may consider the 2N dimensional space H G spanned by all Gamowvectors n

ψD1

�; ψG

1

�; ψD

2

�; ψG

2

�; : . . . ; ψD

N

�; ψG

N

�o:::: (18.11)

We define on H Ga pseudometric that on the vectors of the basis (2) is

ψDi

ψDj

� �¼ ψG

i

ψGj

� �¼ 0, ψD

i

ψGj

� �¼ ψG

i

ψDj

� �¼ δij, (18.12)

where δij is the Kronecker delta. We extend this pseudometric to the whole of H G

by linearity.We may write the restriction of the total Hamiltonian H to H G as (Losada,

Fortin, Gadella, and Holik 2018)

H ¼XN

j¼1zjψD

i

�ψGj

þXN

j¼1z∗j

ψGi

� �ψDj

:�(18.13)

Note that H in Eq. (18.13) is formally Hermitian. Using the pseudometric Eq.(18.12), we find that

Hn ¼XN

j¼1znj

ψDi

�ψGj

þXN

j¼1z∗j

� �nψGi

� �ψDj

:�(18.14)

This suggests a possible choice for the time evolution operator on H G as

U tð Þ≔e�itH ¼XN

j¼1e�itzj

ψDi

n �ψGj

þ e�itz∗jψG

i

� �ψDj

� o: (18.15)

The identity I on H G in this representation is given by

I≔XN

j¼1ψDj

n �ψGj

þ ψGj

� �ψDj

� o: (18.16)

Using the pseudometric, we obtain that IψD

j Þ ¼ψD

j Þ and IψG

j Þ ¼ψG

j Þ,j ¼ 1, . . . ,N, so that this is indeed the identity. With this identity, one possiblechoice of the inverse of U tð Þ is

A Logical Approach to the Quantum-to-Classical Transition 369

U�1 tð Þ ¼ U �tð Þ ¼XN

j¼1eitzjψD

i

n �ψGj

þ eitz∗j

ψGi

� �ψDj

� o: (18.17)

Then, we obtain U tð ÞU�1 tð Þ ¼ I as we would expect.The time evolution for any observable O ¼ O 0ð Þ on H G should be defined as

O tð Þ≔U �tð ÞOU tð Þ: (18.18)

This O tð Þ is well defined for all values of time t. However, with definitions Eq.(18.15) and Eq. (18.16), O tð Þ diverges as t⟶� ∞. This result is not satisfactory.This is the reason why we have chosen instead, as evolution operator on H G:

U tð Þ≔XN

j¼1e�itzj

ψDi

n �ψGj

þ eitz∗j

ψGi

� �ψDj

� o, (18.19)

which is Hermitian. In this case, we have that U tð ÞU† tð Þ ¼ e�tΓI. Nevertheless, weshould keep the definition of O at time t as O tð Þ≔U† tð ÞOU tð Þ. In this case, wehave the following relationship for the commutator of two observables at time t:

O1 tð Þ;O2 tð Þ½ ¼XN

j¼1e�2tΓj αj tð Þ ψD

j

n �ψGj

þ βj tð Þ ψGj

� �ψDj

� o, (18.20)

where αj tð Þ and βj tð Þ, i ¼ 1, 2, . . . ,N, are constants for which the dependence on tis just a phase of the form e�2itERj . Since all Γ j > 0, one concludes that, in the limitt ! þ∞, the commutator Eq. (18.20) vanishes.

In conclusion, for quantum decaying systems and with a correct choice of theform of our operators, commutators vanish for long values of time.

18.4.3 Decoherence and Irreversible Processes

Many attempts have been made to recover the laws of classical mechanics throughsome classical limit. The more relevant approaches include the quantum decoher-ence process, which is responsible for the disappearance of the interference termsof quantum states, which are inadmissible for a classical description. In addition,decoherence provides a rule for choosing the candidates for classical states.

As it is indicated in Castagnino, Fortin, Laura, and Lombardi (2008), threeperiods can be identified in the development of the general program of deco-herence (see also Omnès 2005). In the first period, the arrival to the equilibriumof irreversible systems was studied. During this period, authors such as vanKampen, van Hove, Daneri, et al. developed a formalism that was not success-ful for explaining the decoherence phenomenon, but it established the basisfor its future development. The main problem of this period was that thedecoherence times that were found were too long in comparison with theexperimental ones.

370 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

In the second period, the decoherence in open systems was studied. The maincharacters of this period were Zeh (1970, 1973) and Zurek (1982, 1991). Thedecoherence process is described as an interaction process between an openquantum system and its environment. This process, called environment-induceddecoherence (EID), determines a privileged basis (usually called pointer basis ormoving decoherence basis), which defines the observables that acquire classicalfeatures. Nowadays, this is the orthodox position on the subject (Bub 1997). Thedecoherence times in this period were much smaller, solving the problem of thefirst period.

In the third period, the arrival to equilibrium of closed systems was studied (Casatiand Chirikov 1995a, b, Ford and O’Connel 2001, Frasca 2003, Casati and Prosen2005, Gambini, Porto, and Pulin 2007, Gambini and Pulin 2007, 2010). Within thisperiod, a new approach to the decoherence was presented by Castagnino et al.According to this approach, the decoherence process can occur in closed systems,and it depends on the choice of some observables with some particular physicalrelevance (for example, the van Hove observables). This process, called self-induceddecoherence (SID), also determines which is the privileged basis, called the finaldecoherence basis, that defines which observables acquire classical features.

In some works (Castagnino and Lombardi 2004, Castagnino et al. 2008, Cas-tagnino and Fortin 2013), the common characteristics of the different approachesto decoherence were summarized, and a general framework for decoherence wasproposed. According to the general framework, decoherence is just a particularcase of the general problem of irreversibility in quantum mechanics. Since thequantum state follows a unitary evolution, it cannot reach a final equilibrium statewhen time goes to infinity. Therefore, another element must be considered in sucha way that a nonunitary evolution is obtained. The way to introduce this nonunitaryevolution has to include the splitting of the whole space of observables O into arelevant subspace OR � O and an irrelevant subspace. Once the essential roleplayed by the selection of the relevant observables is clearly understood, thephenomenon of decoherence can be explained in four general steps (reproducedfrom Castagnino and Fortin 2013):

• First step:The space of relevant observables OR is defined. For example, in the EIDapproach the relevant observables are OR ¼ OS⊗IE, where OS is an arbitraryobservable of the system S, and IE is the unit operator of the environment E.SID-relevant observables were defined in Castagnino and Fortin (2013).

• Second step:The expectation value ORh iρ tð Þ, for any OR 2 OR, is obtained. This step can beformulated in two different but equivalent ways:

A Logical Approach to the Quantum-to-Classical Transition 371

i. ORh iρ tð Þ is obtained as the expectation value of OR in the unitarily evolvingstate ρ tð Þ (this way is typical of SID) and its evolution is studied.

ii. A coarse-grained state ρR tð Þ is defined as

ORh iρ tð Þ ¼ ORh iρR tð Þ, (18.21)

for any OR 2 O and its nonunitary evolution, governed by a master equation, isobtained (this way is typical of EID).

• Third step:It is proved that ORh iρ tð Þ ¼ ORh iρR tð Þ reaches a final equilibrium value ORh iρ∗:

lim t!∞ ORh iρ tð Þ ¼ lim t!∞ ORh iρR tð Þ ¼ ORh iρ∗, 8OR 2 OR : (18.22)

This also means that the coarse-grained state ρR tð Þ evolves, with a nonunitaryevolution, toward a final equilibrium state:

lim t!∞ ORh iρR tð Þ ¼ ORh iρ∗, 8OR 2 OR (18.23)

• Fourth step:The moving preferred basis ~j j tð Þi

n ois defined. This basis is the eigenbasis of a

state ρP tð Þ such that

lim t!∞ ORh i ρR tð Þ�ρP tð Þð Þ ¼ 0, 8OR 2 OR : (18.24)

The characteristic time for this limit is the decoherence time tD.

The approaches to decoherence all have one thing in common: They need tointroduce a nonunitary evolution. From a general point of view, it is possible toapproximate the evolution of the system through an effective non-HermitianHamiltonian Heff . It can be proved that the evolution of the mean value is givenby (Castagnino and Fortin 2012)

ORh iρ tð Þ ffi ORh iρ∗ þX

iCie

�γit, (18.25)

where γ�1i are the characteristic times of the system, which are associated with the

complex eigenvalues of the effective Hamiltonian. Then, it is easy to see that thecommutator between two relevant observables is (Fortin and Vanni 2014)

OR;O0R

� � �ρ tð Þ⟶0: (18.26)

This means that, when t⟶þ ∞, the expectation value of the commutator betweenOR and O0

R becomes zero. Therefore, the Heisenberg uncertainty relation becomesundetectable from the experimental viewpoint.

372 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

18.4.4 Quantum Statistics and the Classical Limit

In this section, we show that the dynamics of logics can be related to otherparameters, different from time. In quantum statistics, the mean number of par-ticles occupying a quantum state is given by the formula

�ns ¼ 1exp αþ βϵsð Þ � 1

(18.27)

in which the “+” sign corresponds to Fermi-Dirac statistics and the “�” to Bose-Einstein. The parameter α is related to the particle number according to thecondition X

s�ns ¼

Xs

1exp αþ βϵsð Þ � 1

¼ N : (18.28)

being N the total particle number. The partition function reads

ln Zð Þ ¼ αN �X

sln 1� exp �α� βϵsð Þ½ : (18.29)

When the concentration of the gas is made sufficiently low, quantum effectsshould be important. This limit corresponds to small N. Equivalently, we shouldhave �ns � 1 (or exp αþ βϵsð Þ � 1).

If we now assume that the particle number is fixed, and we increase thetemperature (this is equivalent to β⟶0), we obtain that the most important termsare those satisfying βϵs � α. Under these conditions, we obtain thatexp αþ βϵsð Þ � 1. Or equivalently, that �ns � 1. This is the condition for theclassical limit. In other words, the condition under which quantum effects arenegligible. In this limit, and for both cases, Fermi-Dirac and Bose-Einstein, weobtain

�ns ¼ exp �α� βϵsð Þ: (18.30)

This constraint reduces to Xsexp �α� βϵsð Þ ¼ N; (18.31)

then, we can express

�ns ¼ Nexp �α� βϵsð ÞP

s exp βϵsð Þ : (18.32)

Thus, at sufficiently low density or high temperature, we obtain the Maxwell–Boltzmann distribution, which is a signature of classicality. But the fact that we canattain the classical limit by adjusting the temperature suggests that time is not the

A Logical Approach to the Quantum-to-Classical Transition 373

only parameter that allows us to observe a logic transition. The algebraic aspects ofthis transition will be discussed in a future work, but we can advance somepoints here.

First, some interpretations of quantum mechanics suggest that, when the clas-sical limit is obtained, an irreversible process should be observed. Under thisperspective, this can be related to the mathematical formalism of Gamow vectors.

Second, the approach of dynamical logics can be useful to interpret the quant-ization deformation formalism under a new light. Indeed, some authors (see, e.g.,Landsman 1993) have proposed to study the classical limit and the quantization ofa given theory by appealing to the formalism of deformation quantization. In thisapproach, one starts with a classical (commutative) algebra of observables A0,endowed with a pointwise product ∙ and a Poisson bracket ;f g. Then, a family ofalgebras Ah is introduced, indexed with a parameter h � 0. The parameter h isintended to represent a dimensionless combination of some characteristic param-eters associated with the system and Planck’s constant. An associative product ?his introduced in the indexed algebras, and it is required that (see Landsman 1993for details)

lim h!0i

hf ; g½ h� ¼ f ; gf g (18.33)

and

12lim h!0 f ; g½ hþ ¼ f g: (18.34)

The examples shown in this section suggest that the parameters involved in theclassical limit process could be time, temperature, particle number, or others. Thus,our dynamical logics approach could be connected in a natural way with theformalism of deformation of algebras. We will discuss this possibility elsewhere.

18.5 Conclusions

In this chapter, we have presented a logical approach for the description of thequantum-to-classical transition of physical systems. This approach consists indescribing the system as a collection of observables that evolve over time,according to the Heisenberg picture, but with a nonunitary evolution.

In turn, the algebra of observables determines a lattice of elementary physicalproperties with a logical structure. In the classical case, the properties have aclassical logic structure, and in the quantum case, they have a quantum logicstructure. The time evolution of the algebra induces a time evolution of the latticeof properties. Therefore, in this approach, the classical limit is attained when the

374 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

final structure of properties becomes a classical logic, or equivalently, when theresulting algebra of observables becomes commutative.

We have shown some examples in which this logical transition occurs, amongthem, quantum channels, unstable physical processes and models of self-induced decoherence. We have also shown that our formulation has a naturalapplication in quantum statistical mechanics, where the temperature parameteror the particle number can play the role of the time in reaching the classicallimit. In other words, the classical limit of quantum statistical systems indicatesthat time is not the only parameter that may show a transition from quantum toclassical logic. Furthermore, we have connected our approach with the formal-ism of quantization deformation. In future works, we will develop these ideas inmore detail.

Acknowledgments

We are grateful to the participants of the workshop Identity, indistinguishabilityand non-locality in quantum physics (Buenos Aires, June 2017) for their usefulcomments. This work was made possible through the support of Grant 57919 fromthe John Templeton Foundation and Grant PICT-2014–2812 from the NationalAgency of Scientific and Technological Promotion of Argentina.

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378 Sebastian Fortin, Manuel Gadella, Federico Holik, and Marcelo Losada

19

Quantum Mechanics and Molecular Structure:The Case of Optical Isomers

juan camilo martınez gonzalez, jesus jaimes arriaga, andsebastian fortin

19.1 Introduction

Since its birth, quantum mechanics has enjoyed high prestige thanks to its successin the explanation and prediction of phenomena at the atomic and molecular scales.Indeed, this theory began by explaining the emission lines observed in the hydro-gen atom and, after a few years after its first formulation, it could explain theenergy spectrum of simple molecules. This type of success quickly leads scientiststo suppose that all chemistry can be explained by physics. The famous claim byPaul Dirac is an example of such an assumption:

The underlying physical laws necessary for the mathematical theory of a large part ofphysics and the whole of chemistry are thus completely known, and the difficulty is onlythat the exact application of these equations leads to equations much too complicated to besoluble. It therefore becomes desirable that approximate practical methods of applyingquantum mechanics should be developed, which can lead to an explanation of the mainfeatures of complex atomic systems without too much computation.”

(Dirac 1929: 714)

However, as time went by, it turned out to be clear that the attempt to explainchemistry from physics leads to complications that allow us to question Dirac’sclaim. One of these problems is to explain molecular structure from quantummechanics. There are several ways to approach this problem, but in this work wewill do it by means of a particular case: optical isomers and the Hund paradox.

When young Kant meditated upon the distinction between his right hand andhis left hand, he could not foresee that the problem of incongruent counterpartswould be reborn in the twentieth century under a new form. The so-called Hundparadox points to the difficulty of giving a quantum explanation to chirality, thatis, to the difference between the members of a pair of optical isomers orenantiomers. The question about whether the quantum formalism can accountfor chirality concerns philosophy of science for (at least) three reasons. First, it

379

introduces an interesting case for the debate about the relation between physics(quantum mechanics) and chemistry (molecular chemistry), which has been thefocus of many philosophical works in recent years. Second, and related to theprevious point, the analysis of the paradox can enrich the discussion aboutwhether quantum mechanics can provide an explanation of molecular structure.Third, since some approaches attribute the origins of the paradox to a focus onisolated molecules, the solution is believed to be found in considering moleculesin interaction; these views pose a relevant question to the ontology of chemistry:Is chirality an intrinsic property of a molecule? These three problematic pointsmake the resolution of the Hund paradox an issue of the utmost importance forthe philosophy of science.

On this basis, in this chapter we will analyze the problem of optical isomerismby proceeding in the following steps. In Section 19.2, the Hund paradox will bepresented in formal terms. Section 19.3 will be devoted to showing the relevanceof the paradox to the relation between physics and chemistry, to the explanation ofmolecular structure, and to the ontology of chemistry. In Section 19.4 the paradoxwill be conceptualized as a case of quantum measurement, stressing that decoher-ence does not offer a way out for this problem. Finally, in Section 19.5 we willargue for the need of adopting a clear interpretation of quantum mechanics; inparticular, we will claim that the modal-Hamiltonian interpretation, which con-ceives measurement as a breaking symmetry process, supplies the tools required tosolve the Hund paradox.

19.2 The Hund Paradox

As it is well known, a chemical formula such as H2O indicates the elements in acompound and their relative proportions, but it does not offer information about thegeometric structure of the molecule. Molecules with the same chemical formulabut differing in the spatial disposition of their atoms are called isomers. The classof isomers includes the subclass of optical isomers or enantiomers: the members ofa pair of enantiomers are mirror images of each other; the property that distin-guishes them is called chirality. The peculiarity of enantiomers of a same com-pound is that they share almost all their chemical and physical properties: Theydiffer in how they rotate the plane of polarization of plane-polarized light.Depending on the direction of the rotation, dextro-rotation or levo-rotation, opticalisomers are called D or L.

The problem of the enantiomers was first formulated by Friedrich Hund (1927),a pioneer in the development of quantum chemistry. From a structural point ofview, the two members of a pair of enantiomers have the same bonds, i.e., the“distance between atoms” is the same. Since the quantum Coulombic Hamiltonian

380 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

depends only on the internuclear distances, the Hamiltonian is exactly the same forthe two members of the pair. Consequently, quantum mechanics provides the samedescription for two chemical species that can effectively be differentiated inpractice by their optical activity (Harris and Stodolsky 1981, Wolley 1982, Berlin,Burin, and Goldanskii 1996, Quack and Stohner 2005, Schlosshauer 2007).

In the quantum domain, the parity operator P is associated with spatial reflec-tion: if Dj i and Lj i are the states of isomers D and L, respectively, P transforms Dj iinto Lj i and vice versa: P Lj i ¼ Dj i, P Dj i ¼ Lj i. Let us suppose that the moleculeconsists of A atomic nuclei and N electrons. Then, the Coulombic Hamiltonian ofthe complete molecule reads

H ¼XAg

Pg2

2mgþ e2

XAg<h

ZgZh

2mgþXiN

Pi2

2me� e2

XAg

Zg

r ig

!þ e2

XNi<j

1r ij

(19.1)

where Pg, Zg, and mg are the momentum operator, the atomic number, and themass of the nucleus g, respectively, with g = 1, 2, . . ., A; e and me are the chargeand the mass of the electrons, respectively; Pi is the momentum operator of theelectron i, with i = 1, 2, . . ., N; r ij is the operator “distance” between the electron iand the electron j, and r ig is the operator “distance” between the electron i and thenucleus g. Since the Coulombic interaction only depends on the distance betweenthe interacting particles, it is symmetric under spatial reflection; therefore, theHamiltonian commutes with the parity operator P:

P; H� � ¼ 0 (19.2)

This means that the eigenstates jωni of the Hamiltonian have definite parity. Inparticular, the ground state jω0i is invariant under space reflection:P jω0i ¼ jω0i. As a consequence, jω0i cannot be a chiral state jDi or jLi, butit is a superposition of them:

jω0i ¼ 1ffiffiffi2

p jDð i þ jLiÞ (19.3)

The question is, then, why chiral molecules are never found in this superpositionstate. The states obtained in the laboratory are jDi and jLi, which are noteigenstates of the Hamiltonian and do not correspond to the ground state. So,why do certain chiral molecules display an optical activity that is stable over time,associated with a well-defined chiral state? Why do chiral molecules have adefinite chirality? (Berlin et al. 1996). The Hund paradox points to the core ofcertain traditional problems of the philosophy of chemistry. Let us consider thembriefly.

Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 381

19.3 Intertheoretic Relationships and Molecular Structure

The relationship between chemistry and physics is one of the hottest topics in thephilosophy of chemistry. In this context, the links between theories coming fromthe two disciplines have been explored in great detail from different perspectives.However, despite this effort, there is no agreement yet with respect to the bestmodel of intertheoretic relationships to describe those links. Although the idea ofreducing different disciplines to physics is much older, the success of the applica-tions of quantum mechanics to chemical systems turned reduction into a regulativeidea in the accounts of the relationship between chemistry and physics. FollowingDirac’s famous dictum (1929), the idea that chemistry can be reduced to quantummechanics pervaded both the communities of physicists and chemists. The neces-sary approximations for such a reduction led to the constitution of quantumchemistry as a new area of scientific research (Gavroglu and Simões 2012).However, the approximate methods frequently distort the principles of quantummechanics in such a way that the interpretation of the intertheoretical links asreductive is seriously disputed (Woolley 1978, 1982; Primas 1983). In fact, thestrategies that make possible the description of chemical phenomena in quantumterms, such as the Born-Oppenheimer approximation or the models of valencebond and molecular orbital, do not strictly satisfy the conditions of Nagelianreduction – not only do they establish loose and noncontinuous connectionsbetween chemistry and physics (Lombardi 2014), but they also introduce assump-tions that stand in conflict with quantum mechanics itself (Lombardi and Castag-nino 2010).

The case of enantiomers would provide a new insight in the discussion about thereduction of chemistry, because it involves a difficulty that does not depend onapproximations. In this case the challenge is more fundamental, because it standsbeyond the Born-Oppenheimer approximation. In fact, even if we cannot writedown the exact Hamiltonian due to its complexity, we know that it only dependson the distance of the component particles and, therefore, it cannot account for thedifference between the members of a pair of enantiomers.

19.3.1 The Concept of Molecular Structure

Molecular structure, given by the spatial arrangement of the nuclei in a molecule, isa main character of molecular chemistry: It is “the central dogma of molecularscience” (Woolley 1978: 1074), as it plays a key role in the explanation ofreactivity. However, the concept of molecular structure finds no comfortable placein the theoretical framework of quantum mechanics, inasmuch as it appeals to theclassical notion of individual nuclei in fixed positions. This problem can be viewed

382 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

as a particular manifestation of the general problem of the intertheoretical linksbetween molecular chemistry and quantum mechanics. Following the traditionalreductionist perspective, some authors consider that the difficulties are only due toour partial knowledge of molecular systems in the theoretical framework ofquantum mechanics (Sutcliffe and Woolley 2011, 2012).

From the opposite perspective, Robin Hendry (2004, 2008, 2010), who rejectsthe epistemological reduction of chemistry to physics, claims that the problem ofmolecular structure must be addressed within the ontological domain: The inter-esting philosophical question is how the entities and processes studied by differentdisciplines are related to each other. In particular, the author considers that thelinks between quantum mechanics and molecular chemistry, embodied in theproblem of molecular structure, must be conceived nonreductively, in terms ofemergence.

Optical isomerism introduces a new perspective to the discussion. In fact, thedifference between two enantiomers lies in their structure. Again, in this case noarguments concerning how to interpret approximations in quantum mechanics areinvolved – the Hund paradox arises in terms of the exact Hamiltonian of themolecules. Due to this particularity, a clarification of the paradox would certainlyenrich the debate about whether quantum mechanics can provide an explanation ofmolecular structure.

19.3.2 The Ontology of Chemistry

Although not as extensively treated as the relationship between chemistry andphysics, a central topic in the philosophy of chemistry is that related to theontology of chemistry, that is, to the object of study of the discipline (see Hendry2018). The central question in this field of inquiry concerns what items inhabit therealm of chemistry and to which ontological categories they belong. For instance,certain works analyze the nature of the chemical bond (Vemulapalli 2008, Hendry2012), or the reference of the concept of chemical orbital (Labarca and Lombardi2010, Llored 2010), or even the ontological status of electronegativity (Leach2013). The common ground in those discussions is the effort to identify and tocategorize chemical properties. Does the term ‘chemical bond’ have an onticreference? Is the shape of the orbitals a real property or a mere theoretical tool?Is electronegativity an intrinsic property of atoms or a loosely defined property ofelements?

From a traditional perspective, chirality was usually conceived as an intrinsicproperty of molecules, defined by the molecular structure. However, more recently,some approaches consider that the Hund paradox is the consequence of focusingon isolated molecules. From this viewpoint the solution must be based on studying

Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 383

the interactions in which the molecule is involved. Therefore, chirality would nolonger be a property of a particular molecule, but it would rather turn out to expressthe result of a relation linking the molecules with its environment. This view, thatmodifies the traditional ontological picture regarding chirality, began to prevailwith the development of the theory of decoherence.

19.3.3 The Reductionist Program: Quantum Mechanics Reloaded

Hinne Hettema (2009, 2012) claims that molecular structure can be understood interm of reduction, in light of certain recent developments in quantum chemistry,such as the quantum theory of atoms in molecules (QTAiM; Bader 1994).

The QTAiM was introduced by Richard Bader in the nineties, and it is based onthe assumption that the distribution of the electron density associated with atoms andmolecules represents the physical manifestation of matter in space (Bader 2010b).Therefore, the topology of the electron density encodes and reflects the notions ofatom, bond, structure, and structural stability (Bader 2011). In this way, Bader arguesthat a theory based on electron density allows linking the language of chemistry withthat of physics. According to Bader, atoms are bounded by zero-flux surfaces in thegradient vector field, which is a result of the dominant morphology of the electrondensity distribution, which leads to the natural partition of the molecular space intomononuclear regions associated with the atoms. The zero-flux surface is not crossedby any trajectory of the gradient vector field. This fact is interpreted in principle asfollows: no electron crosses such a surface in such a way that the electron density ofeach atom in the molecule remains unchanged over time.

Likewise, it is possible to definemolecular structure bymean of the critical pointsat which the gradient vector cancels. A local maximum is associated with nuclearpositions and a saddle point is linked to a chemical bond (which is called bond pathin the theory). Moreover, with the help of these critical points, the classical view ofmolecular structure with “balls and sticks” is recovered through molecular graphs,which are constructed by employing “balls” to represent nuclear positions and“sticks” to represent bond paths (Bader 2010a). Under this context, QTAiM claimsto offer a reductionist scheme for molecular structure. In the author’s words:

The reductionist approach afforded by QTAiM offers a clear solution to the myriad ofpersonal views and models of bonding. As has been amply demonstrated by appeal tophysics, the presence of a bond path linking a pair of atoms fulfills the sufficient andnecessary conditions that the atoms are bonded to one another. This definition, whichnecessarily applies to quantum mechanical densities, transcends all bonding schemes andcategories and provides a unified physical understanding of atomic interactions. Oneassumes such unification to be a primary goal of any physical theory.

(Bader 2011: 20)

384 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

QTAiM was very successful to describe many chemical molecules. However, itis doubtful that it offers a complete reduction of chemistry to quantum mechanics:

– Bader discards the wave function and adopts the density of electrons as afundamental entity.

– QTAiM considers that the maximum of electron density are the positions ofthe atoms. Why? Is this a new postulate of quantum mechanics?

– The theory finds zero flux surfaces and claims that these surfaces encloseatoms. However, it does not consider the holism of quantum systems and thepeculiar problem of individuality in quantum mechanics.

Bader himself presents his theory as an extension of quantum mechanics forchemical systems. Then, he has to modify quantum mechanics to obtain his results.The idea of modifying quantum mechanics just to explain the results of chemistryis not acceptable in a reductionist program with ontological pretensions. In fact, themost accepted proposal to describe molecular structure (in general, and isomers inparticular) from quantum mechanics is based on quantum decoherence (Scerri2011), a phenomenon that can be accounted for in standard quantum-mechanical terms.

19.4 Decoherence, Enantiomers, and Quantum Measurement

As explained in Section 19.2, the ground state of the molecule is not one of thechiral states jDi or jLi, but a superposition of them (see Eq. (19.3)). Then why dowe always find the molecule either in the state jDi or in the state jLi? It is not hardto see that the question is the same as that underlying the quantum measurementproblem: Following Schrödinger’s famous example, if the cat is in a superpositionof “alive” and “dead,” we have to explain why we always see the cat dead or alive.In technical terms, the problem is to explain why we measure definite values of anobservable when the system is in state of superposition of the eigenstates of thatobservable. In the particular case of chirality, the problem is to account for the factthat, although the molecule is in a superposition of the chiral states, it alwaysmanifests a definite chirality.

The orthodox answer to the measurement problem is the collapse hypothesis (orvon Neumann’s projection postulate), according to which, when we measure thesystem, the state collapses to one of the states of the superposition. Then, if theresult of a single measurement is, say, dextro-rotation, then the system is actuallyin the state jDi:

jω0i ¼ 1ffiffiffi2

p jDð i þ jLiÞ ! jDi (19.4)

Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 385

If many measurements are performed on identical systems with the same initialconditions, it is possible to define an ensemble, whose state is represented by adensity operator:

ρcollapsed ¼12

Dj i Dh j þ Lj i Lh jð Þ (19.5)

This state is interpreted as stating that there is a probability 0.5 of finding thesystem in the state jDi and a probability 0.5 of finding the system in the statejLi:the state is a mixture of an equal number of definite chiral states. The collapsehypothesis is very successful in reproducing the experimental results, but it has noexplanatory power, to the extent that it is an ad hoc hypothesis specificallydesigned to account for the quantum measurement problem. Moreover, collapseis a nonunitary process that breaks the Schrödinger evolution; however, thehypothesis does not explain why or when the process happens. For this reason,during the last decades, quantum measurements have been approached fromdifferent perspectives; one of them is that given by the theory of decoherence.

According to the orthodox approach – the so-called environment-induced deco-herence (Zurek 1981, 1993, 2003) – decoherence is a phenomenon resulting fromthe interaction between an open quantum system and its environment. Let usconsider a closed system U with two subsystems: the open system S in the initialstate ρS, and the environment E in the initial state ρE. Then, the initial state ofthe total system is ρU ¼ ρS⊗ρE. This state evolves in a unitary way according tothe Schrödinger equation. But, the theory of decoherence studies the behaviorof the reduced state of the open system, ρreduced ¼ TrE ρUð Þ, obtained by applyingthe partial trace on the state of the whole closed system; the partial trace is anoperation that removes the degrees of freedom of the environment from ρU . Asa consequence, the reduced state of the open system is no longer governedby the Schrödinger equation, but is instead ruled by a master equation: ρreducedmay evolve in a nonunitary way. Moreover, when the number of degreesof freedom of the environment is very high, the reduced state may becomediagonal and mimic the ρcollapsed, obtained by means of the collapse hypothesis(see Eq. (19.5)).

In his “Editorial 37” in Foundations of Chemistry, Eric Scerri (2011) explicitlyrelates the problem of optical isomerism to the quantum measurement problem.According to Scerri, the Hund paradox would be dissolved if the interaction of themolecule with its environment were taken into account:

The study of decoherence has shown that it is not just observations that serve to collapsethe superpositions in the quantum mechanics. The collapse can also be brought about bymolecules interacting with their environment.

(Scerri 2011: 4; see Scerri 2013 for a similar claim)

386 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

The idea is that the enantiomer molecule is in interaction with the environment (air,particles, other molecules, etc.). If the initial states of the molecule and theenvironment are jω0i and jε0i, respectively, the initial state of the whole systemis 1=

ffiffiffi2

p� � jDð i þ jLiÞ ⊗ jε0i. The interaction between the molecule and its envir-onment define the evolution of the total system, which, in some cases, produces acorrelation between the possible states of the system and the environment:

1ffiffiffi2

p jDð i þ jLiÞ ⊗ jε0i ! 1ffiffiffi2

p jDi ⊗ jεDi þ 1ffiffiffi2

p jLi ⊗ jεLi (19.6)

Decoherence occurs when, as the result of the evolution, the states of the environ-ment become rapidly orthogonal: εLjεLh i ! 0. As a consequence, after anextremely short decoherence time, the reduced state of the molecule acquires thesame structure as that of the mixed state after collapse (see Eq. (19.5)):

ρdecohered ¼12

Dj i Dh j þ Lj i Lh jð Þ (19.7)

As in the case of quantum measurement, this state is interpreted as stating that themolecule is in one of the states jLi or jDi, and that probabilities measure ourignorance about which state it is. In this way, the theory of decoherence wouldsolve the problem underlying the Hund paradox.

Although there was a time when, as stressed by Anthony Leggett (1987) andJeffrey Bub (1997), decoherence was considered the “new orthodoxy” in thephysics community to explain quantum measurements, at present it is quite clearthat decoherence does not solve the measurement problem. In fact, collapse is thechange of the state of the system, from a superposition to a definite state; on thisbasis, ρdecohered can be interpreted as a legitimate mixture. On the contrary, in thecase of decoherence, the state of the whole system never collapses, but alwaysevolves according to the Schrödinger equation – the superposition never vanishesthrough the unitary evolution. Therefore, it cannot be supposed that what isobserved at the end of the decoherence process is one of two definite events: eitherthat associated with jLi or that associated with jDi (see Adler 2003). Bub (1997)even claims that the assumption of a definite event at the end of the process is notonly unjustified, but also contradicts the eigenstate-eigenvalue link. These conclu-sions about decoherence can also be drawn from the traditional distinction betweena proper mixture – the mixed state of a closed system – and an improper mixture –the reduced state of an open system. As Bernard d’Espagnat (1966, 1976) repeat-edly stressed, improper mixtures cannot be interpreted in terms of ignorance (foradditional arguments, see Fortin and Lombardi 2014).

Summing up, at present some authors still consider that decoherence, by itself,solves many conceptual problems in quantum physics (e.g., Crull 2015).

Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 387

Nevertheless, in the community of the philosophy of physics it is well known that,although decoherence is a powerful tool to deal with conceptual problems, it doesnot allow us to dispense with interpreting the formalism (Vassallo and Esfeld2015). In the next section we will follow precisely an interpretive path to deal withthe Hund paradox.

19.5 Symmetry Breaking, Enantiomers, and theModal-Hamiltonian Interpretation

As it is well-known, the contextuality of quantum mechanics, derived from theKochen-Specker theorem, implies that all the observables of a quantum systemcannot acquire definite actual values simultaneously. Therefore, any realist inter-pretation of quantum mechanics is forced to select a preferred context, that is, theset of the definite-valued observables of the system. The modal-Hamiltonianinterpretation (MHI; Lombardi and Castagnino 2008, Lombardi, Castagnino, andArdenghi 2010) is a realist, noncollapse interpretation that places the Hamiltonianof the system in the center of the stage. According to the modal-Hamiltonianactualization rule, the observables that acquire actual definite values are theHamiltonian H and all the observables that commute with H and have, at least,the same symmetries (that is, that do not break the symmetries of H).

The justification for selecting the Hamiltonian as the preferred observableultimately lies in the physical relevance of the MHI and in its ability to solveinterpretive difficulties. In fact, the MHI actualization rule can be applied to severalwell-known physical situations, leading to results consistent with empirical evi-dence (Lombardi and Castagnino 2008: section 5), and to the account of quantummeasurements both in the ideal an in the nonideal cases (Lombardi and Castagnino2008: section 6, Lombardi, Fortin, and López 2015). In the present discussionabout the Hund paradox, the relevant point is that the MHI describes measurementas a symmetry-breaking process – measurement breaks the symmetry of theHamiltonian and then turns an otherwise nonactualized observable into an actuallydefinite-valued observable, which thus becomes empirically accessible.

As a simple example, let us consider the case of a free particle. In this case theHamiltonian is symmetric under space-displacements in all space directions: Allthe directions are equivalent with respect to the linear motion of the particle. Thethree components of the momentum, Px, Py, Pz, are the generators of this sym-metry, but they cannot acquire definite values simultaneously because they do notcommute with each other. According to the MHI, none of these three observablesbelongs to the preferred context: They are not definite-valued observables becausethey have less symmetries than H , and this means that the actualization of any ofthem would distinguish a direction of space in a completely arbitrary way. Let us

388 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

now suppose that we want to measure one of those observables, say, the compon-ent Py in direction y. For this purpose, we have to place a wall normal to thedirection y, in such a way that the new Hamiltonian is the original one plus a termthat represents the asymmetric potential barrier. It is precisely this term that breaksthe symmetry of the original Hamiltonian and renders the observable Py actual anddefinite-valued and, as a consequence, accessible to measurement. But the point tostress here is that now the system is no longer the free particle; it is a new system,whose Hamiltonian is not symmetric with respect to displacements in direction y.

In light of these interpretive ideas, the Hund paradox can now be rephrased inMHI’s language. As stressed in Section 19.2, the exact Hamiltonian H of theenantiomeric molecule (see Eq. (19.1)) is symmetric under spatial reflection – itcommutes with the parity operator P (see Eq. (19.2)). Now, let us consider theobservable chirality C , whose eigenstates are jDi and jLi. The eigenvalues d and lof C represent the properties dextro-rotation or levo-rotation, respectively. It iseasy to see that C does not commute with P: P; C

� � 6¼ 0. In this case, as in theexample of the free particle, the actualization of the observable C, would determinethe chirality of the molecule in a completely arbitrary way: It would introduce inthe molecule an asymmetry not contained in its Hamiltonian. As a consequence,from the MHI viewpoint, C has no actual value: Chirality is indefinite in theisolated molecule.

If the observable C is to be measured, the parity symmetry of the molecule hasto be broken. For this purpose, the molecule must interact with another system M,which plays the role of the apparatus, in such a way that the Hamiltonian HT of thenew composite system is no longer parity invariant. For instance, this is obtainedwhen

HT ¼ H þ HM , (19.8)

where the Hamiltonian HM of the new system breaks the original parityinvariance: HM ; P

� � 6¼ 0 ) HT ; P� � 6¼ 0. A good candidate for HM is the Hamil-

tonian usually introduced in quantum chemistry to describe the interaction betweenmolecules and polarized light (see Shao and Hänggi 1997), which is a function ofthe electric field �E and the magnetic field �B of the polarized light. Additionally, Cmust commute with the total Hamiltonian HT in order to obtain a stable reading ofchirality. Under these conditions, according to the MHI the observable C acquiresa definite actual value: We measure dextro-rotation or levo-rotation, but now thesystem is no longer the isolated molecule, but the molecule in interaction with thepolarized light. In a certain sense, this answer to the Hund paradox agrees withthe view according to which the solution must be looked for in the interaction ofthe molecule with its surroundings: Chirality is not an intrinsic property of themolecule, but of the composite system molecule plus polarized light. However, our

Quantum Mechanics and Molecular Structure: The Case of Optical Isomers 389

view does not appeal to decoherence, but rather to an interpretation of quantummechanics that explicitly accounts for measurement from the perspective of thesymmetries of the system.

19.6 Final Remarks

In this chapter we have argued that the problem of enantiomers cannot be solvedby appealing to decoherence, but it requires a precise interpretation of quantummechanics capable of dealing with quantum measurement. In particular, we haveshown that the MHI provides us with the adequate tools, since it conceptualizesmeasurement as a symmetry breaking process. It is important to stress that,nevertheless, this result does not supply an indisputable answer to the problemof molecular structure. In fact, the proposed interpretive approach only accountsfor the different behavior of the members of a pair of enantiomers in theirinteraction with polarized light, but it does not take a stand about molecularstructure understood as a spatial geometric property. This is an issue that deservesa further discussion, even in the context of the present solution of the Hundparadox.

Acknowledgments

This work was made possible by the support of Grant 57919 from the JohnTempleton Foundation and Grant PICT-2014–2812 from the National Agency ofScientific and Technological Promotion of Argentina.

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392 Juan Camilo Martínez González, Jesús Jaimes Arriaga, and Sebastian Fortin

Index

Action at a distance, 45, 96, 99, 127, 172, 176,178–179

Aharanov, Yakir, 129Aharonov-Bohm effect, 47, 96Albert, David, 13, 37, 95, 166, 287AlgebraGalilean, 273, 299, 310Heisenberg, 298non-commutative, 74, 155, 367of observables, 153, 298, 307–310, 348, 353–354,

356, 361, 364, 366, 374von Neumann, 22

Algebraic approach to quantum mechanics, 34, 43, 47,348, 353, 355–356

Allori, Valia, 15, 166, 178, 228Araújo, Mateus, 59Arenhart, Jonas, 3, 185Arnold, Barry, 323Arrow of time, 282Auyang, Sunny, 279

Bacciagaluppi, Guido, 39, 279Bader, Richard, 384–385Baker, David, 279Ballentine, Leslie, 247, 283, 288, 290Barnum, Howard, 348Bauer, Edmond, 52–57Beables, 9–12, 14, 22–25, 131, 172Bedingham, Daniel, 25, 28Bell, John, 74, 77, 79–80, 86, 126, 133, 165, 167,

171–172, 175, 181, 222, 229, 247Bell’sexperiment, 247, 249, 251, 262inequalities, 247–250, 252, 255–256, 260theorem, 75, 124, 127, 129

Bellomo, Guido, 4, 323Benatti, Fabio, 13, 15, 25Big Bell Test, 260Bohm, David, 77, 81, 83, 124, 126, 129, 133, 135,

159, 161, 231

Bohmianmechanics, 71theory, 78, 90–91, 99, 241

Bohr, Niels, 52–53, 71, 73, 75–76, 83, 93, 121–123,130, 133, 135, 154–160, 245, 249, 360–361

Born Rule, 65, 97, 104, 107–108, 111–112, 170, 174,297, 354

Born, Max, 83, 279Born-Oppenheimer approximation, 38, 382Born-Vaidman rule, 101–103Bosyk, Gustavo, 4, 323, 333Boyle, Robert, 346Brown, Harvey, 39, 126–127, 279–281Brun, Todd, 21Bub, Jeffrey, 33, 60, 62–63, 136, 281, 387Bueno, Otávio, 3, 185Buonomano, Vincent, 248Butterfield, Jeremy, 23

Callender, Craig, 136, 287Cantor, Georg, 199Carnap, Rudolf, 135, 154, 157–159Carroll, Sean, 103, 122, 134–135, 142–143Cartwright, Nancy, 123Casimir operators, 40–42, 274, 281, 291Cassirer, Ernst, 155–158Cauchy surfaces, 16–17, 19–22, 26–27Causation, 169, 175Chakravartty, Anjan, 150–151Chaos, 256, 317–318Chen, Eddy K., 158Chirality, 379–381, 383–385, 389Church, Alonzo, 198Clarke, Samuel, 179Classical

configuration space, 164, 166, 170–171limit, 42, 68, 82, 205, 214, 216–220, 360–362,

364–365, 370, 373–375logic, 72, 188, 196, 199, 202–203, 235, 362–364,

366, 374–375

393

Classical (cont.)mechanics, 71, 76–77, 81–82, 89, 93, 108, 130, 191,

226, 236–237, 239, 242, 275, 278, 280, 283, 346,363

physics, 11, 67, 73, 76, 81, 94, 96, 137–139, 158,177, 205, 235–237

statistical mechanics, 89, 236–237statistics, 209, 214universe, 237variables, 76–77, 83

Clifton, Rob, 14Collapsehypothesis, 385–386of the wave function, 87–88, 97, 225–227, 233relativistic, 9, 16, 20, 22, 24, 26theories, 9–10, 13, 15–16, 20, 22, 25, 95, 100, 104,

125, 165–167, 173–174, 224, 228Collier, John, 179Contextuality, 32, 44, 75–76, 279, 388Correlations, 36–37, 45, 53, 63, 94, 168, 175,

237–239, 245, 247, 262, 315–316, 329, 334, 336,339, 346–347

Costa de Beauregard, Oliver, 286Cramer, John, 220Cushing, James, 124, 126, 128

d’Espagnat, Bernard, 387Dalton, John, 346de Broglie, Louis, 80, 124, 126, 229, 231Decision theory, 103Decoherenceenvironment-induced, 345–346relativity of, 353self-induced, 375top-down, closed-system approach to, 346

Democritus, 261Deutsch, David, 103, 121, 129Dieks, Dennis, 2, 51Diosi, Lajos, 95Dirac, Paul, 124, 133, 135, 157, 362, 379, 382Discernibility, 187, 192Distinguishability, 205, 209, 215, 219–220, 228,

296DistributionBose-Einstein, 186, 209, 213, 373equilibrium, 83, 85Fermi-Dirac, 186, 209, 213, 217, 373flash, 238Maxwell–Boltzmann, 208–210, 212, 216, 219, 373Planck, 210Poisson, 226Rayleigh-Jeans, 209

Dürr, Detlef, 166Dynamicscollapse, 21, 167, 227, 230, 232deterministic, 242Markovian, 16, 18nonlinear, 16

Earman, John, 271, 347Egg, Matthias, 229Eigenstate-eigenvalue link, 9, 13–15, 23, 153, 280,

387Einstein, Albert, 45, 77, 90, 121, 123, 150, 164, 168,

176–178, 219–220, 229, 245–247, 249, 263Electron density, 384–385Ensemble

canonical, 211, 214grand canonical, 212, 214

Entanglementas a relationship between algebras of observables,349

generalization of, 348relativity of, 348

EntropyBurg, 327quantum, 333Rényi, 327Shannon, 327, 330, 334, 336, 339Tsallis, 327von Neumann, 330

Ergodicity, 245, 248–249, 255–256, 259–261Esfeld, Michael, 3, 136, 168, 222Euclid, 294Everett, Hugh, 54, 126, 133, 135, 161, 239Experiment

Aspect, 127double-slit, 114–115EPR, 125, 219loophole-free, 245, 248–249, 255, 257noninteracting, 44Stern–Gerlach, 38, 84, 86, 125, 170, 222

Faye, Jan, 154Feynman, Richard, 72, 77, 180, 362Finkelstein, Jerry, 21Fleming, Gordon, 23Folse, Henry, 121Forman, Paul, 124, 126Fortin, Sebastian, 4, 345, 360, 379Frauchiger, Daniela, 58–60, 63, 65French, Steven, 193–194, 196, 206, 208, 220Frigg, Roman, 145Fuchs, Christopher, 136

Gadella, Manuel, 4, 360Galilean

covariance, 277, 279invariance, 291transformations, 272, 278, 280, 282, 289, 300

Galileo Galilei, 71, 245Galois, Évariste, 270Gamow vectors, 367–369, 374Gasiorowicz, Stephen, 285Gell-Mann, Murray, 107Generalized contexts, 107, 112Ghirardi, GianCarlo, 9, 13, 15, 22–23, 25, 95, 226

394 Index

Girardeau, Marvin, 316Gisin, Nicolas, 16, 18Gleason, Andrew, 75Goethe, Johann, 155Goldstein, Sheldon, 15, 136, 166, 247Grassi, Renata, 9, 13, 15, 22–23, 25Griffiths, Robert, 55, 107GroupGalilean, 34, 39–42, 269, 272–274, 276–279,

281–282, 284, 286, 288, 291, 295, 310,312

Lie, 272maximal kinematic, 305, 314Poincaré, 42, 282, 291, 295Schrödinger, 35

GRWflash theory (GRWf ), 230, 233, 235, 238, 240–241matter density theory (GRWm), 227, 229, 232–233,

235, 238, 240theory, 9, 53, 161, 165, 173, 220, 226–230

Guiding equation, 78–80, 82–84, 86, 231–232, 234,238, 241

Haag, Rudolf, 297Hacking, Ian, 123Haecceity, 193–197, 200Halvorson, Hans, 3, 133, 145Hamilton, William, 247Hamiltonianeffective, 372non-Hermitian, 368, 372

Harshman, Nathan, 4, 294Hartle, James, 107Healey, Richard, 136, 140–141, 159–160Hegel, Georg W. Friedrich, 155Heisenberg, Werner, 57, 133, 135, 157, 185, 337Hendry, Robin, 383Hettema, Hinne, 384Hidden variables, 75, 78, 87, 123–124, 165–166, 171,

174, 195, 200, 251, 253, 255, 260Hnilo, Alejandro, 4, 245Holik, Federico, 4, 360Howard, Don, 164, 168Huggett, Nick, 205, 207–208, 211, 214Humeanism, 169, 195, 233, 236Hund paradox, 379, 381, 383, 386–390Hund, Friedrich, 380

Improper mixture, 56, 387Indiscernibility, 187–188, 197–198, 203Indistinguishability, 44–46, 198, 203, 214, 312,

315Individualitynon-, 186, 188, 191, 193–196transcendental, 193, 208

InequalityClauser-Horne (CH), 251Clauser-Horne inequality (CH), 250

Clauser-Horne-Shimony-Holt (CHSH), 250Eberhardt’s, 250Mermin’s, 255

Instrumentalism, 122, 129, 233Interpretation of quantum mechanics

atomic modal, 346Bohr’s, 85consistent-histories, 55Copenhagen, 72, 77, 83, 121–122, 133, 245,

247de Broglie-Bohm pilot-wave, 122, 281Everett, 54, 122, 134, 173–174many worlds (MWI), 59, 68, 98–99, 101–104, 126,

165–167, 224modal, 32–33, 36, 54, 57, 234,

280–281modal-Casimir, 41modal-Hamiltonian (MHI), 33–34, 39–40, 46, 54,

281, 356, 388perspectival, 59QBist (quantum Bayesian), 141, 160relational, 55, 57statistical, 247–248, 262transactional, 220

Invariancedynamical, 304, 308, 313, 317Galilei, 41, 269Parity, 389Poincaré, 42time-reversal, 278, 282, 286–288,

290–291Irreducible representations (irreps), 40, 295,

347Ismael, Jenann, 175

Jaimes Arriaga, Jesús, 4, 379Jaynes, Edwin, 245

Kant, Immanuel, 124, 155, 379Kastner, Ruth, 3, 205Kelvin, Lord, 93–94, 96Kent, Adrian, 17Kepler, Johannes, 294Klein, Felix, 279Kochen, Simon, 75, 234Krause, Décio, 3, 185, 193–194, 196

Ladyman, James, 3, 121, 179Lakatos, Imre, 126, 130Landau, Lev, 76Lange, Marc, 173Laplace, Pierre-Simon, 93, 135Laplace’s demon, 237Lattice

Boolean, 108, 360, 363distributive, 108, 111–112, 363, 365orthocomplemented, 108, 111–113, 116,

363–364

Index 395

Laue, Hans, 288Laura, Roberto, 2, 107Leegwater, Gijs, 65Leggett, Anthony, 387Leibniz law (principle), 198, 200, 203, 205Leibniz, Wilhelm, 179Lewis, David, 100Lewis, Davis, 205Lewis, Peter, 167, 176Lieb, Elliott, 316Lifshitz, Evgeny, 76Liniger, Werner, 316Lo, Hoi-Kwong, 335Loewer, Barry, 13, 37, 169Lombardi, Olimpia, 2, 4, 32, 269, 345London, Fritz, 52–57López, Cristian, 4, 269Lorenz, Max, 323Losada, Marcelo, 2, 4, 107, 360

Mach-Zehnder interferometer, 116, 309Majorization, 323Marshall, Albert, 323Martínez González, Juan Camilo, 4, 379Mass density, 13, 15, 22, 25Matter densityfield, 226–229, 232, 235, 238ontology, 25, 167, 229

Maudlin, Tim, 21, 95, 121, 130, 220, 223–224, 230,234, 239

Maxwell equations, 94, 128, 156, 158, 261, 277Measurementas a symmetry breaking process, 33, 39, 390determinate outcomes of, 223frequency, 36reliable, 37single, 36–37, 246, 385von Neumann model of, 36–38, 55

Mermin, David, 21, 180, 254Mill, John Stuart, 123Minkowski spacetime, 10, 16, 28, 64, 66Molecularchemistry, 38, 380, 382structure, 379–380, 382–385, 390

Møller, Christian, 155Monton, Bradley, 14Motion reversal, 290–291Mott, Nevill Francis, 124Muller, Fred, 192Musgrave, Alan, 126Myrvold, Wayne, 2, 9, 65, 67, 168, 173, 176

Newton laws, 94Newton, Isaac, 71, 178–179, 346Ney, Alyssa, 3, 136, 164–165Nguyen, James, 145Nielsen, Michael, 323, 335No-go theorems, 43, 65–66, 123

Noncommutativity, 73Nonlocality, 45, 74, 80, 90, 222, 229, 232, 238–239Nonseparability, 214North, Jill, 282, 287Nozick, Robert, 279

Ohanian, Hans, 270Olkin, Ingram, 323Omnès, Roland, 107Ontology

distributional, 9–10of properties, 43, 46particle, 159, 222, 226, 231, 233, 240primitive, 15, 91, 166–167, 178, 233, 235–236

Optical isomerism, 38, 380, 386

Parmenides, 261Particles

Bohmian, 97–98, 234Bohmian Dirac sea of, 241Dirac sea of, 241–242identical, 44–45, 78, 304–305, 313, 316, 319indistinguishable, 311, 314–317individual, 167, 222, 226numerical difference of, 191–192point, 186, 225–226, 231–232, 239, 241

Pauli, Wolfgang, 122Pearle, Philip, 9, 14, 22, 24, 28, 95Peirce, Charles Sanders, 124Penrose, Roger, 95, 133Peres, Asher, 136Perspectivalism, 51, 57–59, 61, 64, 67–68Picture/representation

Heisenberg, 90, 108, 110–113, 127, 361, 365, 374interaction, 17Schrödinger, 111Tomonaga–Schwinger, 17, 20

Planck constant, 79, 360–361, 374Plato, 294Popescu, Sandu, 335Popper, Karl, 126, 261Positive-operator-valued measures, 86Principle

of compositionality, 202of correspondence, 246of identity of indiscernibles, 191of impenetrability, 191of individuality, 45, 189, 191–192of local causality, 172of metaphysical continuity, 10–11, 14of superposition, 246, 261, 263of uncertainty, 78, 323–324, 337–339Pauli exclusion, 213

Projection postulate, 161, 219, 385Properties

atomic, 108–109, 111–112bundles of, 43–44, 46, 356case-, 43, 45

396 Index

contrary, 107, 113, 116definite, 65elementary, 360–362natural physical, 148–149possible, 44, 46relational, 57–58, 280type-, 43, 45

Quantumchemistry, 128, 380, 382, 384, 389computation, 72, 129, 345information, 72, 86, 297, 324, 332, 336, 339,

366noise, 366potential, 81–82statistics, 45, 78, 205, 207, 209–211, 214, 373

Quantum field theory, 11, 21, 42, 47, 73, 78, 90, 128,225, 239, 242, 279, 282, 291, 295

Quantum Theory of Atoms in Molecules (QTAiM),384–385

Quasi-set theory, 200Quine, Willard V. O., 157

Racah, Giulio, 286Randomness, 78, 83–84, 87, 100, 124, 249, 256, 259,

318Realismabout the quantum state, 135, 159entity, 123, 129local, 247, 251metaphysical, 129ontic structural, 168scientific, 72, 121–122, 124–125, 127–130, 143,

149spacetime state, 151, 168

Redhead, Michael, 206, 208, 220Reichenbach, Hans, 145Relativitygeneral, 150, 185, 201, 228special, 64, 66, 96, 176

Renner, Renato, 58–60, 63, 65Rigged Hilbert space, 301, 366, 368Rimini, Alberto, 226Roberts, Bryan, 283Ross, Don, 179Rovelli, Carlo, 55, 57, 135, 141, 160Ruffini, Remo, 270

Sachs, Robert, 285Sakurai, Jun, 285, 287, 290Saunders, Simon, 126, 134, 136, 142, 192Scerri, Eric, 386Schaffer, Jonathan, 175Schlosshauer, Maximilian, 350Schmidt decomposition, 330–331Schrödinger equationcovariance of the, 274–277invariance of the, 41, 269–270, 275–276

Schrödinger, Erwin, 72, 80, 83, 186–191, 193–196,201, 261, 331

Schrödinger’s cat, 72, 88, 223, 232Schur, Issai, 323, 326, 328Schur-concavity, 326, 333Sebens, Charles, 103Self-induced decoherence, 371Semi-group, 19Shimony, Abner, 13, 171Signalling, 16, 18–19Spacetime

Galilean, 16, 127globally hyperbolic, 16relativistic, 10, 16–17, 20

Specker, Ernst, 75, 234Spontaneous localization, 9, 227, 229–230Spurrett, Don, 179Standard model, 11, 128, 201, 240–241, 297, 346State

actual, 54, 137Bell, 254, 334entangled, 16, 18, 45, 51, 53–54, 62, 89, 99, 170, 187,

223, 251, 256–257, 261, 330, 335–336, 347, 349extrinsic, 20–22GHZ, 253–255perspectival, 56potential, 137reduced, 12, 20–22, 331, 334–335, 345, 350–352,

386–387relational, 56

Suárez, Mauricio, 39, 279Sudbery, Anthony, 59Superselection rules, 304, 337Symmetrization, 46, 214, 219, 317Symmetry

abelian, 302dynamic, 300, 305Galilean, 79, 281, 299gauge, 42group of, 40, 295, 299kinematic, 300, 304–305, 313–315, 317of particle permutations, 304of the Hamiltonian, 35, 38, 296, 388permutation, 103, 188, 192Poincaré, 299rotational, 35, 299, 304space-time, 299SU(2), 295

Tails problem, 13Tappenden, Paul, 103Tarski, Alfred, 44Teller, Paul, 168Tensor product structure, 34, 306–311, 313–316, 348,

353Theorem

Birkhoff’s, 326Ehrenfest, 362

Index 397

Theorem (cont.)equipartition, 215–217Horn’s, 328Kochen-Specker, 41, 45, 139, 279, 388Schrödinger, 329, 333Schur’s, 331Uhlmann’s, 324Zanardi’s, 309

Timpson, Christopher, 142–143, 149, 151–153, 158, 168Tumulka, Roderich, 25, 166, 230Typicality measure, 231, 236–237, 241–242

Uncertainty relations, 52, 123, 130, 215–216, 226,237, 240, 324, 337–339, 372

Urelemente, 199

Vaidman, Lev, 2, 93, 135, 253van Fraassen, Bas, 32, 129, 149, 151Vanni, Leonardo, 2, 107von Helmholtz, Hermann, 155von Neumann, John, 52–54, 56, 75, 95, 122

Wallace, David, 103, 134, 136, 142–143, 148–149,151–153, 158, 161, 168

Watanabe, Satosi, 286

Wave functionas a field on configuration space, 136as a mere calculational device, 140as a multi-field on physical space, 136as an abstract mathematical object, 143conditional, 85, 88realism, 136, 151, 158representational status of the, 157

Weber, Tullio, 226Weihs, Gregor, 259Weyl, Hermann, 187–188, 191–192, 200, 279Wheeler, John, 121Wigner, Eugene, 287–290, 295, 299Wigner’s friend, 55, 57–58, 65, 67Wilson, Jessica, 175Wiseman, Howard, 171–172

Yang, Chen Ning, 316

Zanardi, Paolo, 297, 308, 310, 315Zanghí, Nino, 2, 71, 136, 166Zeeman effect, 38Zeh, Heinz-Dieter, 371Zeilinger, Anton, 121Zurek, Wojciech, 349, 353, 355, 371

398 Index