Topics in the Foundations of Quantum Theory and Relativityfy.chalmers.se/~hawe/PhD.pdfan apparent...

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Topics in the Foundations of QuantumTheory and Relativity

HANS WESTMAN

Department of Theoretical PhysicsChalmers University of Technology and Goteborg University

Goteborg, Sweden 2004

Topics in the Foundations of Quantum Theory and Relativity

Hans WestmanISBN 91-628-6280-4

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Hans Westman, 2004.

Department of Theoretical PhysicsAstronomy and Astrophysics groupChalmers University of Technology and Goteborg UniversitySE-412 96 GoteborgSwedenTelephone +46(0)31-772 10 00

Front cover: A numerical simulation of a simple quantum system illustrating howquantum probabilities emerges through a relaxation process in the deterministic hid-den variable theory of de Broglie and Bohm.

Back cover: An illustration of the meaning of violations of Bell inequalities in de-terministic hidden variable theories. The amount those inequalities are violated putsa lower bound on the dark region. The four circles represent the nonlocal transitionsets �� , ��

� , ��

� , and � � �

� .

Chalmersbibliotekets reproserviceGoteborg, Sweden, 2004

Abstract

In this thesis we focus on foundational issues in both quantum theory and general rel-ativity. As regards quantum theory, we discuss the measurement problem and howit is solved in the hidden variable theory of de Broglie and Bohm. We discuss howquantum probabilities arises in that theory through a relaxation process �� .This relaxation process is explicitly demonstrated by aid of a numerical simulation.Then we discuss quantum non-locality both from a deterministic hidden variablepoint of view as well as for the standard quantum theory. As regards general rela-tivity, we discuss the status of the equivalence principle in quantum field theory incurved spacetime. Then we discuss the the issue of general covariance, Einstein’s‘hole problem’, its solution, and the nature of observables in general relativity.

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Acknowledgments

My five years as a PhD-student would not have been the same were it not for numer-ous friends, family, mentors, and an open-minded supervisor. The first big thanksgoes to Marek Abramowicz for letting me work freely on subjects that I find interest-ing.

It is difficult for me to find words to express my gratitude to the two mentors thatguided me through the jungle of the foundations of physics. A plant can only growin good soil. Dearest mentors, Antony and Sebastiano, I have learned from you morethan you can possibly imagine.

Nothing would have been the same (briefly) if it were not for my true friend and‘spacetime lover’ Rickard Jonsson. Thanks for helping me out in numerous situa-tions. May the gravitational force be with you!

I would like to thank Nikola Markovic, Ulf Torkelsson, and Andreas Back forintroducing me to the Fortran 90 programming language. Special thanks goes toAndreas Back for numerous coca-cola breaks at 3 o’clock, interesting discussions onstatistical physics, and for our many laughs. May they prolong your life!

Heartfelt thanks goes to my father for helping me out with numerous things dur-ing the stressful time of writing this thesis. I also thank my mother and father fortheir musical genes. Playing the piano is a wonderful way to relax the brain from allthinking of physics. I would also like to thank my brothers Olle and Johan, one for agood jogging company, and the other for inspiring me with interesting music.

I want to thank Behrooz Razaznejad and Kristian Dimitrievski, my friends duringmy student years. Much of my interest in the foundations of physics were fueled byour early discussions. Finally, I would like to thank all the people in the string the-ory group, especially Martin Cederwall and the PhD students for several interestingdiscussions.

October 2004

Hans Westman

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Appended Papers

Paper I

Particle Detectors, Geodesic Motion, and the Equivalence PrincipleS. Sonego, H. Westman - gr-qc/0307040, Class. Quant. Grav. 21 (2004) 433-444

Paper II

Dynamical Origin of Quantum ProbabilitiesA. Valentini and H. Westman - quant-ph/0403034, accepted for publication in Proc.Roy. Soc.

Paper III

Nonlocality, Contextuality and Transition SetsH. Westman - To be submitted to Annals of Physics.

Paper IV

Events as Point-CoincidencesH. Westman and S. Sonego - in preparation for submission to Class. Quant. Grav.

Papers not Appended

Generalizing Optical GeometryR. Jonsson, H. Westman – To be submitted to Class. Quant. Grav.

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CONTENTS

1 Introduction 1

2 The Measurement Problem 32.1 Precise argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Different reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Pilot Wave Theory 113.1 The pilot wave dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Solving the measurement problem . . . . . . . . . . . . . . . . . . . . . 143.3 The infamous ‘impossibility’ proofs . . . . . . . . . . . . . . . . . . . . 16

4 The Origin of Quantum Probabilities 194.1 The subquantum � -theorem . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Proof that � � � �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Understanding quantum equilibrium . . . . . . . . . . . . . . . . . . . 23

4.3.1 The nesting theorem . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.2 The approach of Durr, Goldstein and Zanghı . . . . . . . . . . . 25

5 A Numerical Study 275.1 Relaxation in a two-dimensional box . . . . . . . . . . . . . . . . . . . . 275.2 Numerical technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Two ways of significantly decreasing computation time . . . . . . . . . 315.4 Program listing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Quantum Non-Locality 456.1 Non-locality in deterministic hidden variable theories . . . . . . . . . . 45

6.1.1 The EPRB gedanken experiment . . . . . . . . . . . . . . . . . . 456.1.2 The idea of hidden variables . . . . . . . . . . . . . . . . . . . . 466.1.3 Nonlocal transition sets . . . . . . . . . . . . . . . . . . . . . . . 476.1.4 The meaning of violations of Bell inequalities . . . . . . . . . . . 486.1.5 Signal locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2 The non-locality of quantum theory . . . . . . . . . . . . . . . . . . . . 526.2.1 The EPR argument . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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viii CONTENTS

6.2.2 The issue of counter-factual definiteness . . . . . . . . . . . . . . 536.2.3 Bell’s notion of local causality . . . . . . . . . . . . . . . . . . . . 54

7 Particle Detectors, Geodesic Motion and the Equivalence Principle, 577.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.2 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2.1 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.2.2 The Dewitt monopole detector . . . . . . . . . . . . . . . . . . . 60

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8 On General Covariance 638.1 General invariance as a mathematical symmetry . . . . . . . . . . . . . 64

8.1.1 The Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . 648.1.2 Einstein’s vacuum field equations . . . . . . . . . . . . . . . . . 65

8.2 Einstein’s hole problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.3 Solution to the hole problem . . . . . . . . . . . . . . . . . . . . . . . . . 688.4 General invariance vs Lorentz invariance . . . . . . . . . . . . . . . . . 698.5 The role of the manifold in general relativity . . . . . . . . . . . . . . . 708.6 The manifold structure and quantum non-locality . . . . . . . . . . . . 71

9 Conclusions and Outlook 73

Introduction

This thesis is a study of foundational issues in both quantum theory and relativity.Despite the intense effort of physicists, a quantum theory of gravity is largely miss-ing. As well as technical problems, such as the infinities in perturbative attempts toquantize gravity, there are also conceptual problems like the problem of time and theproblem of observables.

Quantum theory, even taken separately, poses difficult conceptual problems. De-spite almost 80 years since the Schrodinger equations were first published, there isstill no consensus of how to make sense of quantum theory. More acutely, as noted byJohn Bell, standard formulations of quantum theory is intrinsically vague and needundefined notions such as ‘macroscopic’, ‘classical apparatus’, etc.. This is related tothe notorious measurement problem which we will discuss along with several pro-posals of how to solve it, or ignore it.

As a specific answer to the measurement problem we will discuss the hidden vari-able theory of de Broglie and Bohm. This deterministic theory which reproduces allpredictions of quantum theory, has existed ever since 1928, but curiously has mostlybeen ignored. We shall see how it solves the measurement problem, and perhapsmore interestingly, how quantum probabilities can be explained (not postulated) byit. In this thesis a numerical simulation is presented that shows in detail how thequantum probabilities emerge for a simple system.

We also discuss hidden variable theories in general. In particular we providea clear graphical illustration of both the meaning of violations of Bell inequalitiesand how an effective locality emerges as a contingent feature of the quantum statis-tics. We will also discuss the irreducible non-locality inherent in quantum theory, asdemonstrated by Bell.

We will then turn to general relativity. First we discuss the status of the equiva-lence principle in curved spacetime. It is shown that there is in general no correlationbetween the response of a particle detector and its motion being a geodesic.

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2 1. Introduction

Then we proceed to discuss the issue of general covariance. Einstein’s theory isinvariant with respect to all differentiable coordinate transformations. This leads toan apparent problem, which is Einstein’s ‘hole problem’. The solution of the problem,as Einstein came to realize, is that the coordinates � � in a general relativity are verydifferent from those used in, for example, special relativity. While the coordinatesin special relativity can be interpreted as readings of physical objects such as clocksand rulers, the coordinates � � in general relativity are mere parameters devoid ofany such physical meaning. The parameters � � are needed only in order to writedown field equations and to construct solutions. However, after such solutions areobtained, the parameters can, and should be, completely eliminated. One is left withgauge independent quantities that summarizes relations between field values alone.We end by a discussion of the role of the manifold in general relativity.

The Measurement Problem‘They somehow believe that the quantum the-ory provides a description of reality, and evena complete description; this interpretation is,however, refuted most elegantly by your sys-tem of radioactive atom + Geiger counter...’.A. Einstein in a letter to E. Schrodinger

The measurement problem dates back to 1927 (see ref. [35], p. 27) when Einsteinrealized that the superposition principle (the linearity of the Schrodinger equation)generates difficulties for recovering classical behavior of macroscopic systems. Someeight years later Schrodinger, after a congenial correspondence with Einstein, pub-lished his notorious article with the malicious cat experiment. It was conceived, byboth Einstein and Schrodinger, as a reductio ad absurdum type argument aimed atdemonstrating the incompleteness of the quantum mechanical formalism. The argu-ment is very general with basically only two assumptions:

� The Schrodinger equation is universally valid: Everything can be described byit, even macroscopic measuring devices.

� The wavefunction, as given by the linear Schrodinger equation, provides a com-plete description of an individual system.

The completeness assumption amounts to this: There is nothing further to say aboutthe state of affairs of an individual system than what is encoded in the wavefunction.In particular, one should not think of the electron in a hydrogen atom in its groundstate as occupying a precise position in space. For example, pilot wave theory (seesection 3), in which there actually is such a well defined position of the electron, isruled out by the completeness assumption.

The completeness assumption becomes deeply problematic when quantum me-chanics is used to describe the interaction between a measurement apparatus and a

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4 2. The Measurement Problem

quantum system. For it is an immediate mathematical consequence of the linearityof the Schodinger equation that if the microscopic system is in a linear superposi-tion then, after such an interaction, the apparatus will be in a linear superposition ofmacroscopically distinguishable configurations. We see only one configuration (thespecific outcome) but the wavefunction (as calculated from the linear Schrodingerequation) makes no such commitment. This is the measurement problem that Bell(see ref. [1], p. 201) summarized so succinctly

‘Either the wavefunction, as given by the Schrodinger equation, is not everything, or itis not right.’

2.1 Precise argument

We shall now turn to the precise argument in its modern form. We essentially followBallentine’s treatment (see [37] p. 232-7).

In quantum mechanics we are dealing with microscopic objects that cannot beseen by the naked eye. Therefore, since we cannot observe these systems ‘directly’,all we can do is to try to produce a reliable correlation between some ‘property’ ofthe microscopic system and some ‘property’ of a macroscopic object directly visibleto human beings. The result of the measurement is inferred from the configuration ofan appropriate macroscopic indicator variable, belonging to the measurement device(i.e a computer output, a meter needle, etc.).

Definition: Measurement. Let (I) stand for the system we wish to measure and(II) the apparatus that records the result. A measurement in quantum mechanicsis an interaction between a system and the experimental apparatus that producesa unique correlation between some dynamical variable of (I) and an appropriateindicator variable of (II).

In order to make use of this definition we must treat the interaction between thesystem and apparatus in a fully quantum mechanical way. It is here we make use ofthe first assumption, that the Schrodinger equation is universally valid.

The indicator variable of the experimental device may be the center of mass co-ordinate of a meter needle or some other thing. This argument is completely generaland does not depend on such details. Nevertheless the example of the meter needleis good to have in mind. In order to distinguish between the final results, it must beassumed that the final wave-packet of the indicator variable (e.g the center of masscoordinate of the meter needle) is sufficiently localized in configuration space. If itis not sufficiently localized it would be impossible to read off the outcome since itsconfiguration would then not be well-defined.1

1Here we are tacitly assuming that what we see is encoded in the wavefunction.

2.1. Precise argument 5

First, expand the state � � �� of the system (I) in the eigenvectors of the operator�we wish to ‘measure’, i.e � � �� where we have

� � �� . Further,let the system (II) be in a state �� where � is the indicator variable (perhaps thecenter of mass coordinate of a meter needle) and � the other degrees of freedom thatare needed in order to give an exhaustive description of the experimental device (e.gquantum numbers of all the atoms etc.).

Let us first consider the simple case where the system (I) is in an eigenstate � ��of the observable

�. In this case we may apply the definition of a measurement. Ac-

cording to it we should apply an interaction that produces a unique correspondencebetween the value of the system (I) and the indicator variable � of the system (II).How will such a interaction Hamiltonian look like? Well, the argument is so generalthat we do not need to go into specific details.2 The only thing that is important isthat a unique correspondence is produced by the interaction and that the evolutionoperator � is linear. Let the system (II) be prepared in a state �� prior to themeasurement. The state will then evolve to

� � �� ! "�

# " �" �� $ ��&% (2.1)

Here the unitary evolution � has evolved the system (II) such that the value � of theindicator variable is in unique correspondence with the value r. We have made theconvenient restriction that the state � �� of the system (I) is not changed by the inter-action, but this is strictly speaking not necessary and may be dropped [37]. Further,we have allowed for possible changes in the system (II) by introducing a sum over� $ .

If we think of the indicator variable as the center of mass coordinate of a meterneedle, the above implies that it occupies a certain region in space. But of course wemust be careful to design the interaction so that the possible results are macroscopi-cally discernible. Hence the positions of the meter needle corresponding to differentresults of the experiment should be macroscopically distinguishable.

Let us now consider a more general initial state of the system (I). The most generalis � � ��'� � � � �� for some complex coefficients

� . Now we make use of the linearity

of the unitary evolution � , in order to deduce

� � � �� (� � � � �� (2.2)� *) "

�

� � �� # " �" � � ��$ �� (2.3)

2One example of such an interaction Hamiltonian would be +-,/.&02143 in the case where the indi-cator variable is the mass center coordinate of a meter needle. . is a coupling strength and 1 3 is themomentum conjugate to 5 . The interaction is taken to be an impulse measurement, i.e during a veryshort time it is so strong that all other terms in the Hamiltonian may be neglected. This is calleda von Neumann impulse measurement and is frequently used when one wishes to model quantummeasurements theoretically.


1

34

5

2

Figure 2.1: The picture illustrates the ‘ghostly’ superposition of macroscopically dis-tinct configurations, i.e the outcomes labeled � �� %/% % in the picture, of a meter needledepicted as the grey arrows. In reality we see only one outcome (depicted here bythe black arrow) while the wavefunction in the ghostly superposition makes no suchcommitment.

It is now readily seen that the system (II) is in a superposition of macroscopicallydistinct indicator states. If we let the indicator variable � be the center of mass co-ordinate of a meter needle, we have just deduced that the final state is not localizedaround some definite point. Rather, it is a superposition of such well-localized states.But it is a fact of experience that we do not see this in a laboratory experiment. Weget one outcome rather than a superposition of outcomes. It is now clear that there isis a clash between the linearity of the Schrodinger equation and the assumption thatthe wave function provides a complete description of reality. Indeed, there is moreto say about the state of the affairs, namely the particular outcome.3 In Fig. 2.1 theperplexing situation is illustrated.

2.2 Different reactions

The measurement problem is the major crossroad when it comes to interpreting quan-tum theory. Here is a list of reactions that are commonly discussed in the literature.Needless to say, even within these views there are differences between its proponents.

3It is important to realize just how general this argument is. Effectively we use only the linearityof the evolution operator � . One can make a seemingly good objection though. We have assumedthat we could prepare the system (II) in an exact state every time we carry out an experiment. But inpractice it is impossible to prepare a complex system in exactly the same state each time. However,our inability to prepare the system (II) in exactly the same way every time can be taken into accountby ‘describing’ the system (II) as a mixed state. If one carries out the same calculation as above usingdensity matrices (see [37] p. 238-9) we end up with the same contradiction: the indicator will haveevolved into macroscopically distinct configurations.

2.2. Different reactions 7

Isn’t this just a triviality?To the above argument it might be objected that, of course, quantum mechanics can-not make any commitment to any specific outcome. If quantum mechanics woulduniquely single out any of these results it would be a deterministic theory. But quan-tum theory is a probabilistic theory that predicts probabilities of measurement out-comes and nothing more. So, trivially, it is incomplete in this sense. The wavefunc-tion, in the superposition of macroscopically distinguishable configurations, does notencode all states of affairs, especially not the outcome.

The reader may be interested to know that this was precisely the conclusion ofEinstein and Schrodinger. They also thought this was the trivial solution. Quantummechanics is incomplete in precisely this way. However, this is not the orthodoxinterpretation (see below).

But perhaps there is a way to save the completeness assumption. Could it notbe that quantum mechanics provides a complete description only for microscopicsystems and somehow fails to do so when macroscopic systems, like ‘classical’ mea-suring devices, are involved? In order to make sense of this idea it is an absolutenecessity to make clear what macroscopic and microscopic means. After all a macro-scopic system is just a collection of atoms. Thus, one needs to give precise answers toquestions like: why should the wavefunction provide a complete description of oneatom but not a collection of them? How many particles can at most be involved in or-der for the wavefunction to provide a complete description, �&�� particles, or perhapsmore? Perhaps the number of particles is not the relevant thing. Maybe a systemneeds to be heavy. But how heavy? One Planck mass, roughly the weight of a grainof sand?

BohrBohr’s solution(?) to the measurement problem was to censor any analysis of the in-teraction between a ‘classical apparatus’ and ‘quantum system’ (see ref. [9], p. 329).To him, these two formed an indivisible and unanalyzable whole. The interactionbetween macroscopic and microscopic objects is thus, according to Bohr, beyond sci-entific understanding, and in particular not through a purely quantum mechanicaltreatment. This is not a view without problems. Macroscopic objects are composedof atoms. So, put in a different way, this is what is being said: When one atom in-teracts with a collection of atoms one cannot analyze the situation scientifically.4 Bellwrote wittily about the orthodox view (see ref. [1], p. 83)

‘By resisting the impulse to analyze and localize, mental discomfort can be avoided.This is, as far as I understand it, the orthodox view... Many people are quite con-tent with it.’

4Of course, Bohr was aware of this problem and tried to rectify it by somehow defining whata classical system is. One should also keep in mind that Bohr thought that classical concepts areautonomous from, and conceptually prior to quantum theory [54].


In any case, the quantum mechanical treatment of the measurement process hasproven valuable in highlighting technically and conceptually important points, inparticular the emergence of classical physics and the process of decoherence [19].

Decoherence?A not uncommon answer is that decoherence [19] solves the measurement problem.5

It is true that decoherence provides a satisfactory explanation for why interferenceeffects do not show up for macroscopic objects. But it does not deal with the per-plexity that, according to the quantum dynamics, macroscopic objects will in generalbe in superpositions of macroscopically distinct configurations. In fact, the unitaryevolution equation (2.3) is but an instance of a decoherence process that leads to asuppression of interference phenomena. Decoherence does not take away the su-perpositions of macroscopically distinct configurations. It predicts them. Therefore,decoherence cannot solve the measurement problem [54].

Does the wavefunction represent a maximal state of knowledge? Could it not bethat the wavefunction encodes, not the actual state of affairs, but merely a state ofmaximal knowledge? When the observer knows about the result of the measurementhe can safely ‘collapse’ the wavefunction according to his increased knowledge (seee.g ref. [62], p. 70–82). But collapsing the wavefunction is not a passive act. It willdestroy the possibility of interference for future experiments (see e.g ref. [49], p. 84-92). Therefore this view is problematic because we are not dealing with a probabilitybut a probability amplitude. We are interested in matters of principle and thereforeit is of no concern that such interference experiments are (because of decoherence)practically impossible to carry out.

In any case, it is not always clear how one actually should apply the projectionpostulate (i.e the how to collapse the wavefunction) [38].

Furthermore, it is not clear how the word ‘knowledge’ is used. Normally we un-derstand that knowledge is about something. And that something should, for thesake of precision, be subject to an exact mathematical description. But clearly, thisis missing here. We cannot say that the observer has acquired knowledge about themacroscopic settings of the measurement apparatus since the word ‘macroscopic’ isnot well-defined.

Wavefunction collapse as a physical processA different view is that a collapse of the wavefunction is a physical process forwhich a mathematical precise recipe must be provided. The universal validity ofthe Schrodinger equation [47] is the questioned. Perhaps there are corrections thatcome into play when macroscopic systems are involved which brings about a wave-function collapse yielding one of the states in the superposition. Such a proposal is

5For example, in [53] Omnes writes about Schrodinger’s cat ‘Because of internal decoherence, it can besaid that the cat is necessarily either alive or dead. It is not in a quantum superposition.’

2.2. Different reactions 9

given by Ghirardi, Rimini, and Weber [23]. The present mathematically elegant for-mulation, in terms of states in Hilbert spaces and linear unitary evolution operators,surely speaks against any such arbitrary ‘ad hoc’ modification. But if quantum me-chanics and its formalism is phenomenological and just the limit of some other the-ory, such corrections would not be very strange. In Newtonian theory we are sincelong accustomed to post-Newtonian corrections. These corrections are not viewed asarbitrary or ‘ad hoc’ because we know there is a beautifully simple theory underlyingNewtonian mechanics accounting precisely for those corrections.

However, this approach is not without its share of problems. Infinite energy pro-duction is predicted in Lorentz invariant collapse theories.

More worlds?Let us really take seriously the resulting superposition of macroscopically distin-guishable configurations. Let us take the resulting superposition not as a problem,but as a factual description of the situation. This idea, taken to its logical conclusion,implies that there are actual parallel universes ‘out there’. This is the many-worldsinterpretation. Below is a rough description of the idea together with some standardcritique, as I have understood it.

First of all there is not only one interpretation but many. However, there are essen-tially two different types of many-worlds interpretations: complete and incompleteones. The incomplete interpretations [69, 70] postulate extra structure in addition tothe wavefunction. In the complete interpretations [71, 72, 48] one would like to viewquantum mechanics as a complete description without any need for extra ‘baggage’.

As a response to a criticism of Bell (see [58] or [1], p. chapter 11) there has recentlybeen a trend to take the complete many worlds interpretations more seriously. I shalltherefore focus on these types here and the problems associated with them.

If the wavefunction is everything, how do we identify in it the different worlds?To do that one needs to pick out some preferred basis to represent a state in Hilbertspace. This is a technical problem and the decoherence program is believed to be ableto solve it. See however [20] for technical and conceptual difficulties with mathemat-ical attempts.

Suppose however that such a solution to the preferred basis problem exist. Canwe explain the relative frequencies, the statistics that quantum theory predicts, withinthis approach? To highlight this difficulty, consider a billion Stern-Gerlach appara-tuses all measuring spin along a � -axis. To each apparatus there is a correspondingelectron ‘almost’ in the spin �� state. Now let the electrons each pass through itscorresponding apparatus. This results in a measurement with two possible outcomes( � or � ) for each apparatus-electron pair, and a corresponding creation of macro-scopically distinguishable configurations, that the many-worlds people call ‘worlds’.Since there are �&� � apparatuses and two possible outcomes there will be ��

�worlds

after the measurement.It is easily seen that the overwhelming majority of these worlds are compatible


with roughly a � � �� distribution of � and � outcomes, i.e roughly an equal amountof � and � outcomes. However, since the electrons were all prepared in an ‘almost’� � � � state quantum mechanics predicts probabilities �� and �� . Conse-quently, most of the outcomes should be � and only very few should be � .

In order to deal with this situation one must then suppress the ‘bad’ � � �� worldssomehow, and bring forth those that are more compatible with quantum statistics.One then introduces notions like ‘measures of existence’ [48]. All worlds are not tobe on equal footing. I am not sure what is being said here. Does this mean that someworlds exists less that others? But then, what does it mean to say that some worldsexists just a little bit?

In the incomplete many-worlds interpretations, I do not see that there is, in prin-ciple, a problem with relative frequencies. I think any clean solution of the problemof relative frequencies will involve postulating extra structure and thereby acceptingthat the quantum mechanical description is incomplete.

Hidden variables?Finally, perhaps one should accept that the wavefunction cannot provide a completedescription of reality. This is the hidden variable program. The position of the meterneedle depicted as a black arrow in Fig. 2.1 is not encoded in the wavefunction butis something separate. More generally, the world we see is not encoded in the wave-function and therefore one should not be concerned about the wavefunction being insuch strange superpositions. The measurement problem arises only when an imageof the world is sought in the wavefunction. We shall now have a closer look at aparticular hidden variable theory due to de Broglie and Bohm and see how it solvesthe measurement problem.

Pilot Wave Theory

‘For twenty years people were saying thathidden variable theories were impossible. Af-ter Bohm did it, some of the same people saidnow it was trivial. They did a fantastic somer-sault. First they convinced themselves, in allsorts of ways, that it couldn’t be done. Andthen it becomes ‘trivial’.’ J. Bell

We shall now turn to a specific hidden variable theory that goes under the name pilotwave theory, or de Broglie-Bohm theory, or simply Bohmian mechanics. This theorywas first introduced by de Broglie in 1928 [8] and later rediscovered 1952 by Bohm[5] in a slightly different form.

In a sense, pilot wave theory can be viewed as a conceptually unambiguous andmathematically precise implementation of Bohr’s idea of the conceptual priority of‘classical terms’ in quantum mechanics, although it should be seriously doubted thathe would have liked it. In addition to the usual Schrodinger wavefunction �� de-fined on a classical configuration space � , there is also a definite classical configura-tion �� whose evolution is choreographed in a precise mathematical way by thewavefunction. In accordance with Bohr’s view the ‘classical terms’, i.e the classicalconfigurations, are truly fundamental in this theory, and in this respect the theorydisplays a sort of quantum-classical dualism.

Bohm did not merely rediscover de Broglie’s 1928 pilot wave theory, he alsodemonstrated that pilot wave theory was capable of reproducing all of the predic-tions of quantum theory [5].1 By assuming that an ensemble of systems (all preparedwith the same wavefunction) whose respective classical configurations are initiallydistributed according to the Born rule � � ��

� � � � �� , one recovers the quantum1Pilot wave theory has been extended to relativistic field theory [11, 25]. See however [26] for a

possible problem with the Grassmann field theory of fermions.

11

12 3. Pilot Wave Theory

statistics for all possible future measurements on that system. The peculiarity is thatin pilot wave theory the wavefunction has nothing a priori to do with probabilities,its ‘role’ being to guide the configuration. As argued by Pauli [6] and Keller [7], thisparticular assumption can be regarded as physically dubious. Why should the wave-function, whose role is to guide the configuration, also play the role as a probability?

However, there is a similar issue in classical statistical mechanics. There is a well-defined way to compute probabilities in thermal equilibrium using the Gibbs statis-tical postulate

� �� ) �� (3.1)

where�

is a normalization (also called the partition function) and � is the temper-ature of the system. And as in pilot wave theory, the probability distribution � �� involves an object � �� (the Hamiltonian) that at the deeper deterministic level‘guides’ the configuration, which in the case of classical statistical mechanics is theHamilton’s equations.

The statistics (3.1) can be understood in classical mechanics as arising in suffi-ciently complex situations. Therefore the above similarity between � and the Hamil-tonian (both ‘guides’ the configuration) suggests that quantum probabilities couldbe understandable within pilot wave theory without any need for the extra assump-tion that Bohm made in his 1952 papers. This research field, the origin of quantumprobabilities in pilot wave theory, was suggested by Bohm himself (see [5], p. 185)but crucial developments happened only in the early nineties [13, 24]. There are twoconceptually different approaches, the first due to Valentini and the second due toDurr, Goldstein, and Zanghı. My own part in this research field has been to study theemergence of quantum probabilities numerically in the first approach. The resultsare presented in Paper II.

3.1 The pilot wave dynamics

The following is a very brief introduction to pilot wave theory. For a more detailedtreatment see for example [1, 5, 9, 10] and the forthcoming book [17]. Pilot wavetheory, being in a sense just a simple completion of quantum theory2, shares with itthe Schrodinger equation ( �� )

��

� �� (3.2)

2Historically this view is perhaps mistaken [18] since de Broglie’s pilot wave ideas were the seedsof Schrodinger’s wave mechanics. Thus, in one sense, quantum mechanics began as a deterministictheory and onle later were stripped of its corpuscle (i.e the classical configuration) by Schrodinger toyield the standard quantum theory.

3.1. The pilot wave dynamics 13

First let us present the idea in its full generality and then proceed to a concrete exam-ple. Let � be the relevant configuration space (if we are dealing with particles it is thespace of all particle configurations and if we are dealing with fields it is the space ofall field configurations) and � a specific configuration, � � � . Whenever it is possibleto write a continuity equation for � � � � � �

� ��

� �� (3.3)

where� �

is some vector defined on configuration space constructed out of the wave-function � 3, then the de Broglie-Bohm guiding equation is well-defined and reads

�� (3.4)

Together with the Schrodinger equation (3.2), equation (3.4) constitutes a determinis-tic dynamics where a complete specification of initial data fully determines the futurefor all times.

As a concrete example, consider a system of interacting particles with masses� � and configuration � � �� %/% % �� with potential � �� . In this case the Schrodingerequation reads

� � ��

��

��

��

� �� % %/% �� % %/% �� (3.5)

and the corresponding continuity equation is �� where

� � ��"�� Im � �� . Using equation (3.4) this implies the the guiding equation for the th

particle � � � � ��

� �� Im

� � �� % %/% �� (3.6)

From equation (3.6) it is clear that the dynamics is explicitly nonlocal. The motionof the th particle is, for a general wavefunction, non-locally dependent on the po-sitions of all the others. Pilot wave theory played a crucial role for Bell when hedeveloped his critique of the ‘impossibility theorems’ as well as his famous theoremdemonstrating that any hidden variable theory must be nonlocal [73].

If one defines � � �"!$#&%then the guiding equation can be written as� � � �

� � � �$' (3.7)

3There is an infinite amount of possible choices of ($) [22]. However, mathematical simplicity, e.gsymmetries, often impose strong restrictions [24].


One should however exercise some caution since the phase ' is in general multi-valued. This happens, for example, for a hydrogen atom in the first excited state� � � �

� . But locally equation (3.7) is perfectly well-defined.By the definition of the guiding equation, if we consider an ensemble of such sys-

tems, prepared at some time � � � , with the same wavefunction � � and an distribu-tion of particle configurations taken to be

� � � � �� %/% % �� % %/% �� (3.8)

then, for all future times �� , the distribution will be given by

� � � �� %/% % �� % %/% �� % (3.9)

In analogy to classical statistical mechanics this distribution is called an equilibriumdistribution. Any distribution with the above property is called equivariant [24].However, since this is a deterministic theory with well-defined particle positions,there is no a priori reason for why it should have this particular distribution. It iscertainly logically possible to imagine other distributions.

Assuming that the ensemble is initially delivered with an equilibrium distribu-tion, and using the guiding equation, the pilot wave theory yields the quantum statis-tics for all possible measurements on that system. Therefore, given this assumption,no experiment can discriminate between quantum theory and pilot wave theory. Thisresult is a double-edged sword. On the one hand it is very nice to see that one repro-duces all the experimentally verified predictions of quantum theory and that with aconceptually clean and mathematically unambiguous theory, but on the other handone gets into trouble with Popper’s idea of falsification. The details of the dynamicsequation (3.4), one may argue, can not be falsified in equilibrium.4 Since there areinfinitely many possible choices of guiding equations [22] I do think this is a fair ob-jection. Mathematical simplicity may suggest some forms of dynamics over others[24] but in the end no empirical discrimination will be possible. This situation is,however, different if we could have access (somehow) to ensembles not distributedaccording to the equilibrium distribution equation (3.9). In [16] it is suggested thatparticles that decouple soon after the big bang might be characterized by an anoma-lous statistics.

3.2 Solving the measurement problem

‘That way seems to cheap to me’ A. Einstein

4It might be, however, that pilot wave theory can be unambiguously applied to situations whereit is difficult to see what quantum theory predicts. One example of this is quantum tunneling times[56]. Here it is hard to know what quantum theory really predicts (time is not an operator observablein quantum mechanics) while pilot wave theory seems to give a simple answer.

3.2. Solving the measurement problem 15

Pilot wave theory clearly disposes with the idea that the wavefunction provides acomplete description of reality. Along with the wavefunction there is a classical con-figuration. For a ‘believer’ in pilot wave theory it is of no concern that, after a mea-surement, the wavefunction is in a linear superposition of macroscopically distinctconfigurations, since the well-behaved classical configuration is not. And, accordingto pilot wave theory, it is the classical configuration we see, not the wavefunction. InFig. 2.1 the black arrow is, in pilot wave theory, represented by a classical configu-ration guided by the wavefunction. The wavefunction in the ghostly superpositionis not what we see. For a more detailed analysis of the measurement in pilot wavetheory see [9, 11].

Pilot wave theory deals with the measurement problem surgically. It cleanly am-putates the problematic part (the wavefunction evolving into linear superpositions ofmacroscopically distinct states) by postulating that what we see and experience hasnothing to do with the wavefunction, but has everything to do with the position inconfiguration space occupied by the classical configuration. While it is true that thisconstitutes a logically consistent solution to the measurement problem one may havemixed feelings how satisfactory this solution is. What follows is a common ‘meta-physical’ objection to the way pilot wave theory solves the measurement problem.

The wavefunction is a dynamical entity that through decoherence will multiplyinto approximately autonomous branches. Those will presumably evolve in config-uration space in a manner as to dynamically imitate the behavior of real worlds con-taining particles, dust, people, cars, etc. [58]. In pilot wave theory, the configurationwill only end up in one of these branches. In this sense these worlds appear veryreal and it seems strange to me that those worlds should have nothing to to with theworld we see.5 Pilot wave theory might then be criticized in this way. I think thisis a valuable critique that lends credit to the many-worlds interpretation. But ratherthan adopting the many worlds view (that has yet to be formulated consistently inmy opinion) I have instead come to think of this as a serious problem of quantumtheory itself. If I must accept parallel universes, that will per definition never be seenby ‘me’, in order to make sense of a theory, I prefer to take that as a reductio adabsurdum type argument.

Interpretations naturally fall into two classes; (1) those interpreting the wavefunc-tion as representing an ensemble of equally prepared systems and (2) those for whichthe wavefunction describes a property of an individual system (be it completely orincompletely). In the former class we find interpretations such as the statistical in-terpretation [37] or the epistemic interpretations (e.g Peierl’s view [62]) and the latterfinds its supporters among collapse proponents, ‘Everettists’ and ‘Bohmians’. To meit seems that, in the end, neither of these classes of views will be correct. Quantum

5One way to deal with this problem is to hope for a static wavefunction [57]. Such a static wave-function is predicted by the Wheeler-de Witt equation. The time-evolving trajectory will then enter‘channels’ giving rise to time-dependent effective wavefunctions of subsystems. If this is doable (it isa technically difficult problem) that would eliminate the above critique.


theory somehow lies in between these two extremes, ‘flirting’ with both and yet re-fuses commitment. I have therefore come to think that the measurement problemconstitutes a real problem for quantum theory, not to be solved by interpreting thetheory ‘correctly’ but instead by replacing it by something better.

My current view of pilot wave theory is that it is an advanced toy model. Com-pleting quantum theory by adding a well-defined classical configuration constitutesa valuable trick that allows us to ‘peek’ into the subquantum domain. This trick givesvaluable hints as to how any such underlying theory must be constructed. Many fea-tures of pilot wave theory are in fact generic features of any theory underlying thestatistics of quantum theory. By Bell’s theorem ([1], chapter 2) such a theory mustbe nonlocal, and by the Bell-Kochen-Specker theorem ([1, 60, 52]) we know that con-cepts like momentum, energy, spin, etc. cannot be thought of as intrinsic propertiesof the system alone. Pilot wave theory also gives us very interesting hints concerningthe origin of quantum probabilities. We shall turn to this subject in chapter 4.

3.3 The infamous ‘impossibility’ proofs

‘That the apparent indeterminism of quan-tum theory can be simulated deterministicallyis well known to every experimenter. It isnow quite usual, in designing an experiment,to construct a Monte Carlo computer pro-gramme to simulate the expected behaviour.The running of the digital computer is quitedeterministic – even the so-called ‘random’numbers are determined in advance.’ J. Bell

It is often claimed that no deterministic theory can reproduce the quantum statis-tics. The indeterminism of quantum theory is therefore thought of as a fundamentalaspect of nature. To support this view, certain ‘impossibility’ theorems are cited.Without exception, they all make assumptions that is not met by pilot wave hiddenvariable theory. Here we briefly discuss some examples.

In 1932 von Neumann claimed to show that no deterministic theory could re-produce the statistics of quantum theory. von Neumann assumed that expectationvalues must be additive even for dispersion-free states.6 Three years later, in 1935, awoman (Grete Hermann) pointed out a ‘glaring deficiency’ in the argument [60]. Notvery surprisingly she was completely ignored.

Later in the early sixties Bell rediscovered ([1], chapter 1) Grete’s critique of vonNeumann’s ‘impossibiliy’ theorem. Nowadays (informed) scientists agree that the

6A dispersion free state is characterized by having probability one or zero for the outcomes for allpossible types of quantum measurements. In a sense, dispersion-free states are nothing but an extremeform of non-equilibrium �� .

3.3. The infamous ‘impossibility’ proofs 17

von Neumann proof does not even get off the ground. The assumption that expec-tation values must be additive even for dispersion-free states implies, as clarified byBell, a trivial contradiction � ��

��

� ��

� in the case of spin measurements (seee.g [1], p. 31-32). It is by no means a logical necessity that expectation values must beadditive. Pilot wave theory, for example, is simply not like that. It is enough that theadditivity of expectation values is recovered for equilibrium distributions, and thatis what happens in pilot wave theory.

In the same paper Bell also provided another ‘impossibility’ proof (later redis-covered by Kochen and Specker7) that apparently makes a more convincing argu-ment against hidden variables. Then Bell went on to criticize his own ‘impossibility’proof and noted that it can be dismissed on the same ground as von Neumann’s.The Bell-Kochen-Specker ‘impossibility’ proof assumes that two incompatible exper-iments must yield the same individual outcome for dispersion-free states. Clearly,neither that is a logical necessity, and it is violated by pilot wave theory. It is enoughthat one reproduces the quantum statistics (see Paper II for further details) for theequilibrium distribution.

As stated by Bell himself, in evaluating those impossibility proofs, pilot wavetheory was very valuable. It was simply a matter of understanding how that hiddenvariable theory circumvented the impossibility proofs. Remarkably, even today thereare researchers who claim that hidden variable theories are not possible (see e.g [76]).Naturally, they have not cared to check their general reasoning against pilot wavetheory.

Then of course we have Bell’s own non-locality theorem, which is perhaps themost cited. According to Wigner

... the most convincing argument against the theory of hidden variables waspresented by J. S. Bell (1964).

However, Bell himself did not think of his theorem as establishing any argumentagainst the possibility of hidden variable theories. He thought that his theorem es-tablished an incompatibility between local causality and quantum theory ([1], p. 172).The non-locality was not just established for hidden variable theories but also for thestandard quantum theory. Bell stressed this repeatedly (see e.g ref. [3] and [1], p.55, 106, 107, 110, 143). Conveniently, many researches miss (or ignore) this particularpart of Bell’s work when they talk about hidden variable theories and their supposedimpossibility.

7While Bell’s proof makes use of a sort of continuity Kochen-Specker’s theorem does without it.

The Origin of Quantum Probabilities

We shall in this section discuss in detail how the quantum probabilities emergeswithin the pilot wave theory. Given that an ensemble is initially delivered with anequilibrium distribution (3.8), the distribution will remain an equilibrium distribu-tion � � � � � � �� for all times. We shall now try to relax that assumption and insteadstart with a more general distribution. It can then be argued, by aid of an � -theorem[13], that almost all such initial distributions will relax to the equilibrium distribu-tion on a coarse-grained level. What is actually demonstrated is that a certain classof distributions (characterized by the absence of microstructure) will initially evolvetowards the equilibrium distribution on a coarse-grained level, while it is not estab-lished that it will actually reach it. Therefore numerical simulations are important.The main result of the numerical simulation reported in Paper II will be presentedhere.

4.1 The subquantum�

-theorem

In classical statistical mechanics the Gibbs entropy is defined by

' �� % (4.1)

where �� is a phase space volume element and � is a phase space distribution. Sinceboth �� and � obey a Liouville relation � � � � �� 1 this quantity is constant intime. However, if one divides phase space into small cells one may define a coarse-grained distribution such that in each cell � �

� �� . Thus, � is taken to beconstant inside each cell but varying from one cell to another. Then it can be shown,

1The boundaries of the volume element �� is evolved according to Hamilton’s equations.

19

20 4. The Origin of Quantum Probabilities

under the assumption that initially at � � � , � � � � � � � � � 2 , that the coarse-grained' �� is indeed increasing in time.

One should carefully distinguish this type of entropy, that is the Gibbs entropy,from Boltzmann’s. They are conceptually very different. We shall comment on thatlater in section 4.3.

In order to define a Gibbs entropy for pilot wave theory let us try to imitate theclassical case as far as possible. Because of the fact that the particle motion is re-stricted to

�x�� %" , we will work in configuration space rather than in phase space.

For the sake of simplicity we will work in a 3-dimensional configuration space. Gen-eralization to arbitrary dimension is straightforward. First consider the expression(4.1). The important feature was that �� and � separately obey the Liouville relation.However, the probability density � in pilot wave theory is subject to the conservationlaw

� ��

� �� '� � � � (4.2)

which in turn implies that��

x� � � � � � �� %" . Since

�� %" is in general not zero,this means that � does not satisfy the Liouville property. However, we know that � � ��satisfies the continuity equation. This implies that the quantity

� � �� is constantalong trajectories since

��

� � �� %" � � � �� %"

� �� (4.3)

Thus we may think of this as a quantum analog of the Liouville relation � � � � �in classical statistical mechanics. To imitate the behavior �� we can not justconsider the configuration space volume �� since �� . However, the quantity�� has this property. This is readily shown by recalling that

�� %" is ameasure of the increase of the volume element (for an infinitesimal lapse of time dt).Thus

� � � � � � � � �� '� � � � �� '� � � � (4.4)

It is easy to see that

� � � � � � � � � � � � � � � � � � � � � � '� � � � � � � � � � � �� '� � � � (4.5)

Taken together equations (4.4) and (4.5 implies)

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� '� � � � � � �

� � '� � � � � � � � � � � � � � (4.6)

2This is an assumption of absence of microstructure of the initial probability distribution.

4.1. The subquantum � -theorem 21

Thus we are led to the pilot wave analog of (4.1)' ��

� �� %

(4.7)

where is a dimensional constant. Just as in the classical case this entropy is constantin time. So now we simply have to introduce a coarse-grained entropy ' , as we didin the classical case. However, because of the presence of a factor � �� we must alsocoarse-grain this quantity. This is alright because in a real situation it is not likely thatwe can gain precise knowledge of the wave function. Thus we introduce

� � �� (4.8)

Furthermore it will be needed to assume that in each cell (just as in the classical case))

� � � � � � � � � � �� (4.9)

Because of the quantum analog of Liouville’s theorem and the assumption of theinitial distribution we have

�� (4.10)

And since�

is constant in each cell we have

�� (4.11)

The coarse-grained entropy ' ��

is now suspected to increase with time. Here is a proof of this:' � �� ' � ��

��


The integrand is now on a very special form: something positive � �� is multipliedby something that looks like

� � � � � � � �

This quantity happens to always be negative or equal to zero which is easy to checkby solving the equation � ��

� � .3 As a consequence we have the ‘subquantum� -theorem’ [13]

�'� � �'�� (4.12)

If it is assumed that the initial velocity field varies significantly within the coarse-graining cells, then one can show that one has a strict inequality [16]

�'� � �'�� (4.13)

4.2 Proof that � � �� Now that we have proved that ' must increase, we are in the position to prove that� � � � � � . First we observe that the subquantum entropy ' has an upper bound equalto zero. This is because' �

� � ��

� � � �� (4.14)

In the first inequality we made use of

� � � � � � � �� (4.15)

and in the last step we have used the fact that � and � � � � are normalized so that� �� . Although we have shown that this entropy must increase itis not proven that it must eventually reach ' � � . That this actually happens, evenfor simple systems, we will see in the next section by aid of a numerical study for thespecific case of a two-dimensional box.4

However, if it actually reaches it, then since

3Divide it by y: �� +�� then put � � � � and rearrange: �� +!�"�� #� . This equation has asolution � � � and must be the only solution since the derivative (being $% +&� ) is always positive when�(')� and always negative when �+*)� . This implies that ,"�� -��, +�./' � and equality occurringonly for � � � � .

4Also in classical statistical physics one can not prove in general that the classical Gibbs entropyactually reaches its upper bound.

4.3. Understanding quantum equilibrium 23

' ��

� �� (4.16)

��

� �� (4.17)

and since by equation (4.15)

� � ��

� � � � � � � � � (4.18)

it follows that the integrand of equation (4.17) must be zero everywhere. Since � � � � �� only has the solution �

� � we end up with the conclusion that for anequilibrium state ' � � we must have � � � �� everywhere.

4.3 Understanding quantum equilibrium

We will now discuss how the quantum probabilities emerge in pilot wave theory ingreater detail. Here we are entering a highly controversial subject: the foundationsof statistical physics. I am not an expert in this field but nevertheless I would like toexpress my view.

The subquantum � -theorem utilizes an ensemble and a corresponding distribu-tion. There are two ways by which I can understand the use of ensembles. Eitherthe distribution describes an actual ensemble (say of one million particles), or oneinterprets it as an ignorance probability relevant for an individual system. The lat-ter is a measure of our knowledge of a system and the increase of its correspondingcoarse-grained Gibbs entropy describes how such knowledge decreases with time.

4.3.1 The nesting theorem

Consider a probability distribution � of a pure ensemble (i.e prepared with the samewavefunction) of complex systems with many degrees of freedom. By a simple argu-ment Valentini [14] managed to show that if � has an equilibrium distribution, thenany subsystem (e.g a particle) extracted from that system will ‘inherit’ an equilibriumdistribution � � � � � � . This has been called the ‘nesting’ theorem.

By extraction is meant a process that singles out one subsystem (e.g a particle) sothat one may, for all practical purposes, describe the system with an effective wave-function �� , where � is the relevant coordinate for the subsystem. This means that,for all practical purposes, the wavefunction is given by a product state5 [14]

�univ

� �� (4.19)5As we have learned from the decoherence program and in particular of the the measurement


where � collectively represent the configuration space coordinates of the rest of theuniverse.

One might then argue as follows. Consider a hypothetical ensemble of universes,all with the same wavefunction. Since the individual system (a universe) is a com-plex system with many degrees of freedom, by the � -theorem the initial distribution� univ � � � � � � univ � � will efficiently relax towards equilibrium. And when equilibriumis reached, if we extract one subsystem (e.g a particle) from each one of the mem-bers of this hypothetical ensemble then, by the nesting theorem, the correspondingensemble of subsystems will have an equilibrium distribution.

A question that comes to mind is how one should interpret such a hypotheticaldistribution. It can hardly describe an actual ensemble of universes since there is perdefinition only one universe. As far as I can see, there is only the ignorance interpre-tation left. But what has our ignorance to do with the actual relaxation process of theuniverse we live in?

Let us illustrate this more carefully. Suppose that we consider an hypotheticalensemble of

#universes and suppose that each consists of particles distributed in

3-space. The whole situation may then be described by a matrix � � whose elementsare three-vectors describing the coordinates for the particles. The upper index refersto what particle we consider and the lower index to what hypothetical universe wemay be in. For example, � ��

� means the spatial position of particle number � �� inhypothetical universe number � .

The � -theorem plus the nesting theorem establish that the ensemble

� � �� %/% %

(4.20)

referring to one and the same particle6 but in different hypothetical universes is dis-tributed as � � � �� . But clearly, this not what we are interested in. We are interestedin the statistics of an ensemble of particles in one and the same universe. That is (the�:th universe happens to be the actual one)

� �# � �# � �# � �# � �# % % %(4.21)

which refers to different particles in one and the same universe. Therefore one maynot infer from Valentini’s result alone, i.e in the absence of further arguments, thatdifferent particles in this one universe is distributed according to � � � �� . We shallnow turn to a different approach that, in my opinion, provides together with the

� -theorem a satisfactory explanation for the emergence of quantum probabilities.

problem, the total wavefunction will generically be in a superposition of macroscopically distinctstates. However, in pilot wave theory one of these states, or branches, will be singled out by theclassical configuration. Then, since it is overwhelmingly unlikely (by decoherence) that the otherbranches will re-interfere with the one occupied by the classical configuration one can then, for allpractical purposes discard the other branches. This means that the total wavefunction can effectivelybe described by a product state.

6We ignore the Pauli principle in this discussion.

4.3. Understanding quantum equilibrium 25

4.3.2 The approach of Durr, Goldstein and Zanghı

A conceptually different approach has been given by D. Durr, S. Goldstein and N.Zanghı [24]. Let

�be the total wavefunction of the whole universe. Then suppose

one prepares an actual ensemble of similarly prepared systems by extracting thosefrom the universe. As before, the wavefunction [24] will, for all practical purposes,look like (we omit complications due to the Pauli principle)

� � � � � � � �� %/% % � � � � � � � � (4.22)

Here the variables ��

� � �% %/%

� are the configuration variables of the similarly pre-pared ensemble and � is the configuration space coordinate for the rest of the uni-verse. Then it can be shown [24] that almost all points in the total configuration space,with respect to the � � � � measure, are compatible with an equilibrium � � � � � � � � � dis-tribution of the extracted subsystems. Thus, we see the quantum � � �� distribution inthe laboratory because it is overwhelmingly likely.

One should stress the difference between this approach and that of Valentini.While Valentini’s approach utilizes hypothetical ensembles Durr’s et al’s approachdeals only with a single universe. To me this feels more satisfactory. However, onemay object as to why a � � � � measure should be used in the first place. In [24] it is alsopointed out that if the measure is taken to be � � � � then most configuration points willbe compatible with a � �� distribution. On the other hand it can be argued that thereis really only one natural measure, which in pilot wave theory is the equivariant � � � �measure.

My view is that further justification of the � � � � measure can be obtained fromValentini’s � -theorem. What the � -theorem effectively shows is that a typical con-figuration, guided by the wavefunction

�, clearly tends to avoid regions where � � ��

is zero. As demonstrated by Durr et al’s result a typical trajectory will then avoidregions for which � � � � � � (i.e those regions characterized in the limit � � by� � � � � � ), and instead spend the overwhelming amount of time inside regions char-acterized by � � � �� for the ensemble of subsystems, that is where � � � � � � .

Modulo the usual controversies and problems that are there even in the founda-tions of classical statistical mechanics, the above provides a satisfactory explanationof the emergence of quantum probabilities. In this respect, pilot wave theory notonly gives a mathematically precise theory of micro-cosmos but it also explains thequantum probabilities. Thus, one has now been able to do to quantum theory, whatBoltzmann did to thermodynamics.

A Numerical Study

In this section we briefly discuss the main result of the numerical study carried out inPaper II. The idea is to study the evolution of an initially non-equilibrium distribution� � � �� . We shall see that, as expected from the � -theorem of the previous secction,the distribution will evolve towards equilibrium on a coarse-grained level.

5.1 Relaxation in a two-dimensional box

As a concrete example we consider a particle in a two dimensional box of size ��in units where � � �� , with an infinite potential barrier

� � � � � � � � � �(� � � � � � �� otherwise (5.1)

Imposing the appropriate boundary conditions, the general solution to the Schrodingerequation

� � ��

��

��

�� (5.2)

is then

�� ) � �� (5.3)

where the amplitudes� � � satisfy

� � ) � � � � � � � � � . In the following we will restrict our-selves to the sixteen first modes ( � � � � � % % % � � ) and also assume that the amplitudes

27

28 5. A Numerical Study

are of equal weight, which implies that � � � � � � � ��

for all �� . This means that we canwrite

� � � � �� and

� � � � � � � � � � � ) � �

�

��

�� (5.4)

The phases� � � were randomly chosen from the interval � �� using Matlab’s random

number generator. Using phases with equal weight may not be necessary for relax-ation � � � �� to occur. But it will increase the number of nodal points (i.e where�� ) and since nodes are characterized by a chaotic velocity field, one expectsan efficient relaxation � � � �� .

The de Broglie-Bohm guiding equations are��

Im� � �� Im

� � �� (5.5)

Consider now an ensemble of such two-dimensional boxes, all prepared with thewavefunction equation (5.4). Since we are free to contemplate any initial distributionof particles we could choose for instance � � � � � � � � �� . This distributionis very different from the quantum equilibrium distribution. In Fig. 5.1 these twodistributions are displayed.

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at

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� �� at � � �

Figure 5.1: Left: The initial non-equilibrium distribution, chosen to be � � � � � � � �� (equal to the ground-state equilibrium distribution). Right: The

squared-amplitude � � � � � � �� at � � � , for the specified superposition of the first16 modes.

Using the guiding law equation (5.5) one can now study the evolution of theprobability distribution. On smaller scales the distribution forms sharp ridges anda highly complex structure but if we look at things at a coarse-grained level the dis-tribution shows a remarkable relaxation towards the equilibrium one over the timescale of � � (in our units). This is illustrated in Fig. 5.2. The coarse-graining is carried

5.2. Numerical technique 29

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atxy

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�� at � � � �

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� � � � at � � � �PSfrag replacements

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Figure 5.2: Smoothed�� , compared with � �� , at times � � � , � � and � � . While � � � �

recurs to its initial state, the smoothed�� shows a remarkable evolution towards equi-

librium.

out by replacing the exact fine-grained distribution � by a smoothed�� obtained by

averaging over the exact � -values of nearby points.

In Paper II it was also shown numerically that the relaxation is exponential, i.ethe � -function (proportional to the negative subquantum entropy) displayed an ex-ponential decrease �� .5.2 Numerical technique

The numerical computation of the relaxation was only made possible because of afew tricks that I will here describe. In the end I have appended one of the programsused. All other programs used in Paper II are variations of it.


Since it was clear that the numerical calculations would be time consuming thecode was written in FORTRAN 90, which due to a long tradition of developing effi-cient compilers, provides high speed calculation. The specific compiler used here wasIntel’s free (but impressive) compiler.1 This compiler is efficient when trigonometricfunctions are involved which makes it particularly suitable for our purposes since ineach time step in the integration of the trajectories, several trigonometric functions(entering the guiding equation) need to be evaluated.

The obvious way would perhaps be to use the continuity equation in order toevolve the probability distribution. However, it seems to me that the singularitiesin the velocity field will make it very time consuming to integrate the continuityequation numerically. We will therefore proceed differently and make use of the factthat the ratio �� is conserved along trajectories.2 Let �� be the position of aparticle that at the time � � � � was in the position � � . Then we can express theprobability density � in a point � � � � � � at time � by

� � ��

� �� (5.6)

Let us then try to compute the probability density by starting with a distribution ofparticles at � � � � and compute the probability density � at later times using equation(5.6). However, by doing this we can only compute the probability density � at pointswhere a trajectory happens to pass. In particular, nodal regions (where � �� issmall) would be very difficult to reach since trajectories tend to avoid such regions.Using many trajectories would perhaps do the trick, and in this way reach the nodalregions by ‘brute force’. But one would then end up calculating an awful amount of� -values in dense regions, where a lot of trajectories would end up. This would leadto poor accuracy in sparse regions and needlessly good accuracy in dense regions.

There is a very simple cure of this malady. Instead of evolving the trajectoriesforward in time, evolve them backwards. If we want to compute the value of � at apoint , we backtrack the particle from its final position to see where it started. Thismeans that we solve the time-reversed guiding equation to end up with an initialposition � �� . The value of � at a point and time � is then computed from

� �� (5.7)

We now have the appropriate control over where we wish to compute the probabilitydensity. Simply start with an evenly spaced grid of final positions, backtrack to findthe initial positions � � � � � , and then compute � � � � � using equation (5.7).

1The free compiler is available at http://www.intel.com/software/products/compilers/flin/noncom.htm

2This strategy was also used in [11, 16] to study the relaxation in a one-dimensional box. Interest-ingly, even if the trajectories cannot cross in such a simple system, one still obtains a limited approachtowards equilibrium of the system. This can be realized as being due to the presence of nodal regions,i.e regions where � � � � is small [77].

5.3. Two ways of significantly decreasing computation time 31

There is one small catch with this idea. We can only compute the probabilitydensity at one time � and rendering all the data of the intermediate particle positionsuseless. This means that if one wants to compute � � � � � � and � �� ( � � � � � � � � )one has to do two separate simulations. This is in contrast to the first idea for whichonly one simulation would have been necessary.

However, this difficulty is a minor one compared to the previous one. For thecase when we evolve the particles forward in time we are probably facing a numer-ical problem which is computationally intractable. Probably that approach wouldrequire too many particles for a decent accuracy in regions where � is small. Evenfor the case when we evolve trajectories backwards in time, � � � � � � particles wereneeded in order to get decent accuracy in one of the simulations in Paper II. There-fore a significant computation time was needed (about a week for the longer simula-tions3) even in this case when one had perfect control over where one computes theprobability density. If the number of particles used would have to be significantlylarger than this, the numerical problem would be computationally very difficult.

The trajectories has been shown to exhibit chaotic behavior [74], meaning thateven the smallest difference in position can yield very different backtracked posi-tions. Together with the large number of trajectories that must be computed in orderto get reasonable accuracy, this necessitates the use of an efficient and high preci-sion integrator. Both speed and precision are an issue. The Runge-Kutta-Fehlbergintegrator [64] turned out to be satisfactory.

5.3 Two ways of significantly decreasing computationtime

While the backtracking technique makes the numerical problem tractable, there areuseful ways to cut down computation time significantly. Here we present two ofthose.

When getting acquainted with the trajectory calculations one can see that the com-putation time for a single trajectory varies significantly with the ‘final’ positions weare backtracking from. Some trajectories require very little computation time whileother trajectories require significantly more time.

Here is how to speed things up, that at the same time also provides us with aglobal error estimate (i.e an estimate of how close a trajectory has come to the correctposition at � � � ).

The Runge-Kutta-Fehlberg method [64] is a 5th order adaptive time step algo-rithm. The method is efficient because of its clever way of calculating the local er-ror estimate, using an embedded 4th order method, without having to do any extraderivative evaluations. If the local error estimate is greater than a specified value

3The computations were carried out on a 2.4 GHz Intel pentium 4 processor.


abstol then the step size is decreased, and if the local error estimate is smaller thenthe step size is increased.

For those nice trajectories requiring only a small computation time, the value ofabstol needs not be very small. The backtracked position will be quite accurateanyway. However, for the nasty trajectories a smaller value of abstol is necessaryto yield a good accuracy.

In order to make sure that we do not use a needlessly small abstol for the nicetrajectories and too large for nasty ones, one can adopt the following strategy. Beginwith abstol

��&� � � , say.4 Calculate the corresponding backtracked position. Then

put abstol�

� � � � and compute the trajectory again. Now we have in our handsa global error estimate which is the distance between the two backtracked positions,calculated using the two different values of abstol. If the global error is less than forinstance � % � � , 5 we keep the backtracked position calculated with the smaller valueof abstol. If the global error estimate is greater than � % � � we do another trajectorycalculation with abstol set to �&� �� . Then we repeat this procedure, decreasing ab-stol by a factor of ten each time, until we have a acceptable global error (i.e less than� % � � ).

Not only do we get a handle on the global error through this procedure, but it alsosignificantly speeds up the computation (by a factor of 5, or perhaps more).

Furthermore, there are special circumstances where the trajectory is so bad thatthe Runge-Kutta-Fehlberg integrator must take more than �&� � time steps. I chose toabort these trajectory calculations. Here is how the program handled the situation.

Some cases required more than �&� � time steps even for � �� &� � � ; each of thesewere aborted and assigned the value of the previous trajectory calculation. In othercases, decreasing abstol resulted in a exceeded the limit of � � � time steps, and weretained the value obtained using the smallest value of abstol that did not exceedthis limit (despite the global error exceeding � % � � ). Finally, some cases reached theminimum setting � �� &� � � � without achieving the desired limit on the globalerror, and for these we assigned the value obtained using � �� &� � � � . In the worstcase one in � � � � trajectories were aborted.

The last trick to speed things up concerns the calculations of sine values in subrou-tine delS. Several of these values will be used many times. Therefore it is beneficialto calculate the values only once and store them (in the cache memory) as variables.

4If one wants to evolve the trajectories for a longer time one could reasonably expect a smallerabstol in order to get a reasonable accuracy. Therefore, for short times choose a bigger abstol andfor longer times choose smaller abstol.

5Keep in mind that the box is �� so that the relative error is � � � �� . This is alrightfor computing the probability density using equation (5.7). However, if a particle happens to be back-tracked to a region where �� is small then the small relative error in the backtracked position willanyway imply a big relative error for the probability density �� . This is so, because in nodal regions,� ��!#"$�%!#"'& (*)*)+ ) !#"$�,!-"'& (*).& (*��) + / � � �*0213�� depends critically on the backtracked position 0 � �*0213�� ). However, trajectoriestend to avoid nodal regions where � is small and therefore few particles are problematic in this sense.In the code we provide an error estimate even for the probability distribution.

5.4. Program listing 33

Accessing the cache memory is faster than recalculating the sine values.A standard modern processor has sufficient memory capacity to store those values

in the cache memory (located physically in the processor). Since one has to calculatean awful amount of sine-values for each trajectory computation time is improved inthis case by a factor �� .

5.4 Program listing

Here is the program used to calculate the evolution of the smoothly coarse-grainedprobability density

�� illustrated in Fig. 5.2. All other programs used in Paper II aremore or less variations of it.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!In the module all the important constants are set. !!!!!! !!!!!!(ncells+1)ˆ2 is the number of cells. !!!!!!(2*ncell+1)ˆ2 is the number of particles in a cell. !!!!!!(nx+1)ˆ2 is the total number of particles. !!!!!! !!!!!!nsafety is just a precausion so that the solver stops in!!!!!!the case it gets stuck at a position. !!!!!!abstol is the total absolute error allowed in the final !!!!!!position. !!!!!! !!!!!!akl is a randomly generated phase. k and l is bound by n!!!!!!which is 4 in this case. !!!!!! !!!!!!The size of the box is by default taken to be pi since !!!!!!that simplifies the wavefunction expression the most !!!!!!which implies faster computation. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

MODULE constINTEGER, PARAMETER :: DP=KIND(1.0D0),n=4,nsafety=100000000,&

ncells=125,ncell=12,nx=399,ns=3!!!If you change the value of ncells here do not forget to!!!change also the print-out format (approximately at line!!!230) accordingly.

REAL(DP), PARAMETER :: pi=3.14159265358979D0REAL(DP), DIMENSION(1:n,1:n) :: aklCOMPLEX(DP), PARAMETER :: i=(0.0D0,1.0D0)


END MODULE

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!! BEFORE DOING A SIMULATION CHECK... !!!!!! ================================== !!!!!! (1) that (ncells+1)*(2*ncell+1)=nx+1 !!!!!! (2) that the format output matches ncells. !!!!!! (3) that lowexp matches the time !!!!!! (4) that maxerror is suitably set !!!!!! (5) that the time is correctly set !!!!!! (6) that nsafety is set appropriately !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

PROGRAM TwoDboxUSE const,ONLY:DP,n,akl,pi,nx,ncell,ncellsLOGICAL :: isstuck,isreallystuck,isstuckinloopINTEGER :: k,l,kbar,lbar,exponent,lowexp,highexpREAL(DP) :: t1,t2,psi20,p0,x0worse,x0better,y0worse,y0better,&

x0cand,y0cand,maxerrorREAL(DP), DIMENSION(0:nx) :: x,y

REAL(DP), DIMENSION(0:nx,0:nx) :: p,pworse,psi2,x0,y0,delta,&deltap

REAL(DP), DIMENSION(0:ncells) :: xbar,ybarREAL(DP), DIMENSION(0:ncells,0:ncells) :: pbar,pworsebar,&

psi2bar

t1=0.0D0 !Initial timet2=4.0D0*pi !Final time !For small/large times use a

!smaller/higher value of lowexp

maxerror=1.0D-2

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Generating the UNIFORM grid. For each particle we shall!!!!!! calculate where it came from, i.e where it was at time!!!!!!t=t1. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

DO k=0,nxx(k)=pi/(2*DBLE(nx+1))+pi*DBLE(k)/DBLE(nx+1)


y(k)=pi/(2*DBLE(nx+1))+pi*DBLE(k)/DBLE(nx+1)END DO

!!!The first grid point begins half a!!!step (i.e pi/(2*DBLE(nx+1)) inside!!!the box. and the last ends similarly.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Importing the phase data akl from file!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

OPEN(UNIT=11,FILE=’fourbyfour.txt’)DO k=1,n

READ(11,*) akl(k,:)END DOCLOSE(11)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Here we calulate the probability distribution p at time!!!!!!t2 by backtracking each particle to its original !!!!!!position at t=t1. The backtracking is done by the !!!!!!subroutine findx0. Given the position at t=t2 and the !!!!!!time t1 it spits out the origial position at t=t1. !!!!!! p0/modspisquare is a subroutine that returns the !!!!!!(probability density p0 at time t=t1)/(the value of !!!!!!|psi|ˆ2 at specified time). !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

lowexp=6 !For small/large times use ahighexp=12 !smaller/higher value of lowexp.

DO k=0,nxDO l=0,nx

isstuck=.FALSE.isreallystuck=.FALSE.isstuckinloop=.FALSE.

CALL findx0y0(t1,t2,x(k),y(l),x0worse,y0worse,&10.0D0**(-lowexp+1),isreallystuck)


IF (isreallystuck) THENPRINT *,’Solver is REALLY stuck! No x0,y0,p and no &

accuracy estimate.’

PRINT *,k,lGOTO 20 !When we go to line 20 x0 and y0

!get the value of xbetter and ybetter!respectively. In this case x0 and y0!then become equal to the former!values of x0,y0.

ENDIF

CALL findx0y0(t1,t2,x(k),y(l),x0cand,y0cand,&10.0D0**(-lowexp),isstuck)

IF (isstuck) THENPRINT *,’Solver is stuck! No accuracy estimate.’PRINT *,k,lx0better=x0worsey0better=y0worseGOTO 10

ENDIF

x0better=x0candy0better=y0cand

delta(k,l)=sqrt((x0worse-x0better)**2+&(y0worse-y0better)**2)

IF (delta(k,l)<maxerror) GOTO 10

DO exponent=lowexp+1,highexpCALL findx0y0(t1,t2,x(k),y(l),x0cand,y0cand,&

10.0**(-exponent),isstuckinloop)IF (isstuckinloop) THEN

PRINT *,’Solver is stuck! Got accuracy estimate.’PRINT *,k,lGOTO 10

ENDIFx0worse=x0bettery0worse=y0betterx0better=x0candy0better=y0cand


delta(k,l)=sqrt((x0worse-x0better)**2+&(y0worse-y0better)**2)

IF (delta(k,l)<maxerror) GOTO 10ENDDO

PRINT *,’High exponent!’,k,l

!!!Since it has not jumped to line 10!!!delta cannot be smaller than maxerror!!!which means that the solver failed to!!!pass the required accuracy test even!!!for the highest allowed exponent.!!!Thus the warning for the questionable!!!accuracy of x0 and y0.

!!!!!!!!!!!!!!!!!!!!Computing p!!!!!!!!!!!!!!!!!!!!

10 CONTINUE

CALL modpsisquare(t2,x(k),y(l),psi2(k,l))CALL modpsisquare(t1,x0better,y0better,psi20)CALL p0dist(x0better,y0better,p0)p(k,l)=psi2(k,l)*p0/psi20

IF (isstuck) GOTO 20

!If isstuck is true then there is no less!accurate x0,y0 to compute pworse in order!to get a estimate for p. Therefore we!go directly to the end. delta and deltap!get the values 0.0D0 which is the default!value of any object of real numbers.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Computing pworse for the less accurate x0,y0 !!!!!!in order to get an estimate of the error in p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!


CALL modpsisquare(t1,x0worse,y0worse,psi20)CALL p0dist(x0worse,y0worse,p0)pworse(k,l)=psi2(k,l)*p0/psi20

deltap(k,l)=abs(p(k,l)-pworse(k,l))

x0(k,l)=x0bettery0(k,l)=y0better

20 CONTINUEx0(k,l)=x0bettery0(k,l)=y0better

!PRINT *,k,l,deltap(k,l),delta(k,l)

ENDDOPRINT *,k

END DO

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!The subroutine coarsegrain takes a (nx+1)*(nx+1)-matrix !!!!!! and returns a smaller smoothly coarse grained !!!!!!(ncells+1)*(ncells+1)-matrix. Note that the coarse !!!!!!graining cells are over lapping so we cannot depend !!!!!!on this to give us an accurate Hbar. This is because !!!!!!Hbar is defined for non-over-lapping cells. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

CALL smoothgrain(x,y,p,xbar,ybar,pbar)CALL smoothgrain(x,y,pworse,xbar,ybar,pworsebar)CALL smoothgrain(x,y,psi2,xbar,ybar,psi2bar)

!!!It is not very nice!!!programing that I generate!!!xbar and ybar trice, but!!!since it does notchange!!!computation time much I!!!am too lazy to fix it!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Everything is now done and we can write to files. Here !!!!!!I could make the formating better. I now have to insert!!!


!!! how many cells I have in the formating string. This !!!!!!process I should automatize. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

OPEN(UNIT=12,FILE=’xybar.txt’)DO kbar=0,ncellsWRITE(12,*) xbar(kbar),ybar(kbar)

ENDDOCLOSE(12)

OPEN(UNIT=13,FILE=’pbar.txt’)DO kbar=0,ncellsWRITE(13,"(126f25.16)") pbar(kbar,:) !Do not forget to put

ENDDO !the formating toCLOSE(13) !mf25.16 where m is

!ncells+1.OPEN(UNIT=14,FILE=’pworsebar.txt’)DO kbar=0,ncellsWRITE(14,"(126f25.16)") pworsebar(kbar,:) !Same here!

ENDDOCLOSE(14)

OPEN(UNIT=15,FILE=’psi2bar.txt’)DO kbar=0,ncellsWRITE(15,"(126f25.16)") psi2bar(kbar,:) !And also here!

ENDDOCLOSE(15)

OPEN(UNIT=16,FILE=’x0y0data.txt’)DO l=0,nx

DO k=0,nxWRITE(16,"(4f25.16)") x0(k,l),y0(k,l),REAL(k,DP),&

REAL(l,DP)ENDDO

ENDDOCLOSE(16)

OPEN(UNIT=17,FILE=’accuracydata.txt’)DO l=0,nx

DO k=0,nxWRITE(17,"(4f25.16)") delta(k,l),deltap(k,l),&

REAL(k,DP),REAL(l,DP)


ENDDOENDDOCLOSE(17)

!!!!!!!!!!!!!!!!!!!!!!!!That’s all fox!!!!!!!!!!!!!!!!!!!!!!!!!

END PROGRAM

SUBROUTINE smoothgrain(x,y,psi2,xbar,ybar,psi2bar)USE const,ONLY:DP,ncells,ncell,nx,nsINTEGER :: k,l,kbar,lbarREAL(DP), DIMENSION(0:nx,0:nx), INTENT(IN) :: psi2REAL(DP), DIMENSION(0:nx), INTENT(IN) :: x,yREAL(DP), DIMENSION(0:ncells,0:ncells),INTENT(OUT) :: psi2barREAL(DP), DIMENSION(0:ncells), INTENT(OUT) :: xbar,ybar

DO kbar=0,ncellsDO lbar=0,ncells

psi2bar(kbar,lbar)=0.0D0DO k=-ncell,ncellDO l=-ncell,ncell

psi2bar(kbar,lbar)=psi2bar(kbar,lbar)+psi2(ncell+ns*kbar+k,&ncell+ns*lbar+l)

END DOEND DO

psi2bar(kbar,lbar)=psi2bar(kbar,lbar)/((DBLE(2*ncell+1))**2)END DO

END DO

DO kbar=0,ncellsxbar(kbar)=x(ncell+ns*kbar)ybar(kbar)=y(ncell+ns*kbar)

END DORETURNEND SUBROUTINE

SUBROUTINE findx0y0(t1,t2,x,y,x0,y0,abstol,flag)USE const,ONLY:DP,nsafetyREAL(DP),INTENT(IN) :: t1,t2,x,y,abstolREAL(DP),INTENT(OUT) :: x0,y0


LOGICAL, INTENT(OUT) :: flagINTEGER :: lREAL(DP) :: xyerr,xout,yout,h,t

l=0t=t2h=1.0D-4x0=xy0=y

!First we integrate backwards as long as t>t1.

DO WHILE(l<nsafety)CALL rkfstep(t,x0,y0,-h,xout,yout,xyerr)IF (xyerr>h*abstol) THEN

h=0.9D0*h*(abstol*h/xyerr)**(0.25D0)l=l+1

ELSEIF (t-h<t1) EXITt=t-hx0=xouty0=youtl=l+1IF (xyerr<0.5D0*h*abstol) THENh=0.9D0*h*(abstol*h/xyerr)**(0.2D0)

END IFEND IF

END DO

IF (l>nsafety-2) flag=.TRUE.

h=t-t1

!Now we are just above t=t1 and need only to do the last!step. Since the routine above were about to do this step!for a larger h than h=t-t1 (it must be larger or equal!)!the estimate of the error will be LESS than abstol*h.

CALL rkfstep(t,x0,y0,-h,xout,yout,xyerr)x0=xouty0=youtRETURN


END SUBROUTINE

SUBROUTINE rkfstep(t,x,y,h,xout,yout,xyerror)USE const,ONLY:DPREAL(DP), INTENT(IN) :: t,x,y,hREAL(DP), INTENT(OUT) :: xout,yout,xyerrorREAL(DP), PARAMETER :: a2=0.2D0,a3=0.3D0,a4=0.6D0,a5=1.0D0,&

a6=0.875D0,b21=0.2D0,b31=3.0D0/40.0D0,b32=9.0D0/40.0D0,&b41=0.3D0,b42=-0.9D0,b43=1.2D0,b51=-11.0D0/54.0D0,&b52=2.5D0,b53=-70.0D0/27.0D0,b54=35.0D0/27.0D0,&b61=1631.0D0/55296.0D0,b62=175.0D0/512.0D0,&b63=575.0D0/13824.0D0,b64=44275.0D0/110592.0D0,&b65=253.0D0/4096.0D0,c1=37.0D0/378.0D0,&c3=250.0D0/621.0D0,c4=125.0D0/594.0D0,&c6=512.0D0/1771.0D0,dc1=c1-2825.0D0/27648.0D0,&dc3=c3-18575.0D0/48384.0D0,dc4=c4-13525.0D0/55296.0D0,&dc5=-277.0D0/14336.0D0,dc6=c6-0.25D0

REAL(DP) :: kx1,kx2,kx3,kx4,kx5,kx6,&ky1,ky2,ky3,ky4,ky5,ky6

REAL(DP) :: xerror,yerrorCALL delS(t,x,y,kx1,ky1)CALL delS(t+a2*h,x+b21*h,y+b21*h,kx2,ky2)CALL delS(t+a3*h,x+h*(b31*kx1+b32*kx2),y+h*(b31*ky1+b32*ky2),&

kx3,ky3)CALL delS(t+a4*h,x+h*(b41*kx1+b42*kx2+b43*kx3),y+&

h*(b41*ky1+b42*ky2+b43*ky3),kx4,ky4)CALL delS(t+a5*h,x+h*(b51*kx1+b52*kx2+b53*kx3+b54*kx4),&

y+h*(b51*ky1+b52*ky2+b53*ky3+b54*ky4),kx5,ky5)CALL delS(t+a6*h,x+h*(b61*kx1+b62*kx2+b63*kx3+b64*kx4+&

b65*kx5),y+h*(b61*ky1+b62*ky2+b63*ky3+&b64*ky4+b65*ky5),kx6,ky6)

xout=x+h*(c1*kx1+c3*kx3+c4*kx4+c6*kx6)xerror=h*ABS(dc1*kx1+dc3*kx3+dc4*kx4+dc5*kx5+dc6*kx6)yout=y+h*(c1*ky1+c3*ky3+c4*ky4+c6*ky6)yerror=h*ABS(dc1*ky1+dc3*ky3+dc4*ky4+dc5*ky5+dc6*ky6)xyerror=ABS(MAX(xerror,yerror))RETURNEND SUBROUTINE

SUBROUTINE delS(t,x,y,vx,vy)USE const, ONLY:DP,n,akl,i


INTEGER :: k,lREAL(DP), INTENT(IN) :: t,x,yREAL(DP), INTENT(OUT) :: vx,vyREAL(DP), DIMENSION(1,1:n) :: sinkx,kcoskxREAL(DP), DIMENSION(1:n,1) :: sinly,lcoslyCOMPLEX(DP), DIMENSION(1:n,1:n) :: expdataCOMPLEX(DP), DIMENSION(1,1) :: psi,Dxpsi,Dypsi

DO k=1,nsinkx(1,k)=SIN(k*x)sinly(k,1)=SIN(k*y)kcoskx(1,k)=k*COS(k*x)lcosly(k,1)=k*COS(k*y)DO l=1,n

expdata(k,l)=EXP(-i*((k**2+l**2)*t/2.0D0+akl(k,l)))END DO

END DO

psi=MATMUL(MATMUL(sinkx,expdata),sinly)Dxpsi=MATMUL(MATMUL(kcoskx,expdata),sinly)Dypsi=MATMUL(MATMUL(sinkx,expdata),lcosly)

vx=AIMAG(Dxpsi(1,1)/psi(1,1))vy=AIMAG(Dypsi(1,1)/psi(1,1))

END SUBROUTINE

SUBROUTINE modpsisquare(t,x,y,psi2)USE const,ONLY:DP,akl,n,pi,iREAL(DP), INTENT(IN) :: t,x,yREAL(DP), INTENT(OUT) :: psi2COMPLEX(DP) :: psiINTEGER :: k,lpsi=(0.0D0,0.0D0)DO k=1,nDO l=1,n

psi=psi+SIN(k*x)*SIN(l*y)*EXP(-i*((k**2+l**2)*t/2.0D0+&akl(k,l)))

END DOEND DOpsi=psi*2.0D0/(pi)psi2=psi*CONJG(psi)


END SUBROUTINE

SUBROUTINE p0dist(x,y,p0)USE const,ONLY:DP,piREAL(DP), INTENT(IN) :: x,yREAL(DP), INTENT(OUT) :: p0p0=(2.0D0/pi)**2*(SIN(x))**2*(SIN(y))**2END SUBROUTINE

Quantum Non-Locality

Here we will discuss a feature of quantum theory that has triggered a lot of debate:quantum non-locality. First we will discuss non-locality in deterministic hidden vari-able theories. Then we will see that this non-locality is not merely a feature of deter-ministic hidden variable theories but also of quantum theory itself as was repeatedlystressed by Bell. This chapter is an introduction to Paper III.

6.1 Non-locality in deterministic hidden variable theo-ries

Until now, we have only discussed the particular hidden variable theory of de Broglieand Bohm. In this section we will consider any hidden variable theory capable of re-producing the quantum statistics. Such a theory must violate the Bell’s inequalitiesand therefore be nonlocal. Here we shall provide a graphical illustration of whatviolations of Bell inequalities actually means and exactly how it is connected to non-locality. We shall also graphically illustrate how it is possible for this non-locality tobe hidden in the quantum uncertainty noise.

6.1.1 The EPRB gedanken experiment

1935 was the year when Einstein, Podolsky, and Rosen published their famous ‘EPR’argument [50] aimed at establishing the incompleteness of the quantum mechanicalformalism. While the original argument involved momentum and position measure-ments of a specific entangled state, Bohm developed a more transparent version ofit in terms of particles with spin [12]. The following gedanken experiment therefore

45

46 6. Quantum Non-Locality

BA

C

time

space

electrons

PSfrag replacements

aa’ bb’

Figure 6.1: A spacetime diagram of the standard EPRB scenario. Two electrons fly offto the spatially separated regions � and � that each contain a Stern-Gerlach apparatusmeasuring spin. The apparatus at � can be aligned along either � or � $ , and

�or

� $ forthe apparatus at � . The angle at � is assumed to be chosen only outside the past lightcone of � (depicted as the � �� lines) and vice versa for the angle at � .

goes under the name the EPRB gedanken experiment.Consider two electrons prepared in a singlet state ��

� � �� '� � � � � � � �'� � � � � at

a spacetime region � (see Fig. 6.1). The two electrons are assumed to pass throughtwo spatially separated spacetime regions � and � each containing a Stern-Gerlachapparatus measuring the spin along some direction (see Fig. 6.1). At � we can chooseeither to measure the spin along the direction � or � $ and similarly at � one can choosebetween � or � $ . We will confine ourself to situations for which � , � $ , � , and � $ areperpendicular to an axis joining the two apparatuses. Therefore an angle � , � $ , �

, or� $ is sufficient to specify the direction � , � $ , � , or � $ respectively. It shall be assumedthat the angles of the Stern-Gerlach apparatuses in their respective spacetime regionswere chosen outside of the past light cones emanating from the regions � and � (SeeFig. 6.1).

6.1.2 The idea of hidden variables

The only feature of a deterministic hidden variable theory we shall use here, is theexistence of a function that determines the outcome ( � or � ) with probability one,given the specific angles of the Stern-Gerlach apparatuses plus some extra parame-ters � . Such function must exist since we are dealing with a deterministic theory forwhich the outcome is in principle predictable. It shall prove unnecessary to providemore information about the nature of these parameters � . However, as a concreteexample one may think of the de Broglie-Bohm hidden variable theory. In the EPRBscenario the parameters � collectively represent the (spinor) wavefunction and thepositions of the electrons (see ref. [1], p. 130-132). The wavefunction and the particle

6.1. Non-locality in deterministic hidden variable theories 47

positions are the hidden parameters � .1

Hence, we shall assume that the outcomes are of the form� �3� � � � � � � � �� 3� � � � � � � � � (6.1)

Quantum probabilities in hidden variable theories represent our ignorance of thedetailed microscopic state of affairs. Since one cannot (at least not for the moment)prepare an ensemble with a definite hidden variable configuration (i.e an ensemblewith a definite � ) one must consider ensembles of such hidden variable configura-tions with a certain probability distribution � � � � . For example, in pilot wave the-ory for the EPRB scenario one can prepare a ensemble of systems with the same(spinor) wavefunction (e.g the singlet state), but the particle positions �� will bedistributed according to the Born rule � � � �� because of the reasons discussed insection 4.3.

In order to reproduce the quantum probabilities one cannot choose any distribu-tion � � � � . Rather there will be a special distribution that is suitably called the equilib-rium distribution � eq � � � that yields the quantum statistics for all possible measure-ments. It seems natural to think that such a distribution arises in the same way as forpilot wave theory. For the following discussion the equilibrium distribution � eq � � �will be assumed.

One might object to expressions like (6.1). Why is the outcome at � dependenton the angle at � , and vice versa? By construction, there is no signal with a speedless or equal to that of light, which can carry information about the choice of angleat the other spatially separated region. However, as proven first by Bell (the articlereprinted in [1], chapter 2), any hidden variable theory must be nonlocal. If oneassumes a local functional dependence like

� �3� � � � � � � and� �� , one can

in fact derive a direct logical contradiction with quantum theory as is shown in PaperIII.

6.1.3 Nonlocal transition sets

Let us proceed to the notion of transition sets [15] in the EPRB scenario. Let� ��

and� �3� � � � � � be functions such that the outcome ( � � ) of an individual experiment at

� and � is completely determined by specifying the angles � and�

and the hiddenvariable � . A nonlocal transition set for this system is then defined as (see Fig. 6.2)

� �� 3� � � � � � � � �3� $ � � � � �� (6.2)

where � is the space of all possible hidden variable configurations. If � � � �� thenthe outcome at B will depend non-locally on the choice of angle ( � or � $ ) at A. In a

1Contrary to our notation, it is also common in pilot wave theory to call only the particle positionsthe hidden variables.


PSfrag replacements

��

��

�-space

Figure 6.2: The figure illustrates the intersection of two subsets of the space � ofall possible hidden variable configurations. In one set the measurement outcome inregion � is � � , and in the other set the outcome is � � . The intersection of these twosets is a nonlocal transition set � �� represented in the picture by the dark region.

similar manner, we define three more transition sets

� � � ��

� � � � � �3� $ � � � � � � � �3� $ � � $ � � � � (6.3)� ��

��

� � � � � �3� � � $ � � � � � �3� $ � � $ � � �� (6.4)� � �

��

� � � � � �3� � � � � � � � �3� � � $ � � � � (6.5)

Notice the convention of putting the � -angles upstairs and the�-angles downstairs.

Of course, in a local hidden variable theory these nonlocal transition sets must beempty � ��

� � � ��

��

��

. This implies a local functional depen-dence of the type

� �3� � � � and� � � � � � and this, as mentioned above, yields a logical

contradiction with quantum theory.One might ask how probable it is for a � to belong to the transition set (6.2). This

is given by the measure [15]

� �� ! � �#"$� �� &% � � ��

� �3� $ � � � � � � � � � � � � (6.6)

where � � � � is the probability distribution of hidden variables. The other measuresare defined analogously.

6.1.4 The meaning of violations of Bell inequalities

The famous Bell non-locality theorem involves inequalities that must be satisfied byany locally causal theory. In the literature there are nowadays a plethora of simi-lar inequalities involving a variety of different number of possible angular settings


[75], also involving particles of higher spin. For the situation where there is onlytwo possible angles at the two spatially separated regions, there are also a multitudeof inequalities. In Paper III a universal inequality is derived containing all other in-equalities (restricted to two possible angular settings) as special cases. The universalinequality is

� � � � � � ��

� � � ��

� � � � � � � ��

� � � ��

��

� ��

� � � � � � � � � ��

� �� (6.7)

where

��

� �� 3� ��

��

� �� 3� $ � � ��

��

� � �� 3� $ � � $ ��

� ��

� �� 3� �� $ �

�(6.8)

are the quantum probabilities for getting either a correlated outcome (� � � � � � ) or

an anti-correlated one (� � � �

� � ). Quantum theory violates this inequality.In Paper III it is shown that in any hidden variable theory there exists a set � �

such that, if � happens to be in that set, one can derive a direct logical contradictionwith locality, i.e the assumption that all the transition sets are empty � ��

� � � ��

� ��

��

��

.To establish that the set � � is non empty a lower bound on the measure � �� was

also derived in Paper III:

� ��

� ��

� ��

� ��

� ��

� ��

� � � ��

��

� � � ��

�� (6.9)

where the right hand side can be shown to be related to how much the universalinequality (6.7) is violated, i.e how much bigger than � that the left hand side ofequation (6.7) must be according to quantum theory.

This set � � can then be shown to be related to the transition sets in equations (6.2)–(6.5). In fact, as is shown in detail in Paper III, if � �� then it must also belong toprecisely one or to precisely three transition sets. This is illustrated in Fig. 6.3. If wechoose our possible angular settings as � �

� � �� $ � � $ � � $ � �

� and � �� $ � � �� then

� � � � �� % � � � � . It is interesting to note that the violation of Bell’s inequalities

does not reveal the measures of the regions where � belongs to precisely two or tofour transition sets. This means that it is possible to come up with nonlocal hiddenvariable theories that satisfy the Bell inequalities for a fixed set of possible angles. It


PSfrag replacements

�-space

��

��

Figure 6.3: The picture depicts the four (in general intersecting) nonlocal transitionsets � �� , � ��

�� , � ��

� , and � � �

� . Violations of Bell inequalities put a lower bound

only on the dark region which is the set � � . This is the region where � belongs toprecisely one or to precisely three nonlocal transition sets.

might be that this will be ruled out if one requires that the hidden variable theoryshould reproduce the quantum statistics for all possible measurements. See Paper IIIfor more detailes.

In summary, the non-locality of hidden variable theories is very explicit. The out-come at the region � is in general non-locally dependent on the choice of angle at thespatially separated region � , and vice versa.

6.1.5 Signal locality

We have now seen that any hidden variable theory, capable of reproducing the quan-tum statistics, must be nonlocal in the sense that the individual outcome at one regionin spacetime is dependent on what is done in a spatially separated region. But if na-ture is nonlocal in this very explicit way, why can we not see this explicitly? Accord-ing to quantum theory, the marginal statistics at region � is insensitive to the choiceof angles at � . Therefore one cannot make use of this non-locality to send messages.

But why should the statistics of outcomes at � not be dependent on the anglechosen at � when it is clear that an individual outcome in general exhibits such de-pendence? In the book [62] Bell comments

‘It is as if there is some kind of conspiracy, that something is going on behind thescenes which is not allowed to appear on the scenes.’

We shall now see that this masking of non-locality comes about only in quantum


��

��

��

��

PSfrag replacements

BB��

��

Figure 6.4: The pictures illustrate a Stern-Gerlach apparatus in region � . The appara-tus in region � (not in the picture) can be aligned either along � or � $ . The solid linesrefer to the outcome at � if the angle � had been chosen and the dashed if instead � $had been chosen. The left picture illustrate the insensitivity of the marginal statisticsin region � to the choice of angle at A that occurs for the quantum equilibrium dis-tribution. The right picture illustrates that the marginal statistics at � is sensitive tochanges of angle at � when the distribution is not the quantum equilibrium one.

equilibrium. It is the equilibrium distribution, presumably emerging through a re-laxation process, that hides the underlying non-locality.

Following Valentini [15], any transition set � �� may be partitioned into twodisjoint subsets � �� and � �� . � �� is the set for which theoutcome at � is � � if angles � � � were chosen and � � if instead the angles � $ � � werechosen; and vice versa for � �� . In equilibrium the measures of these setshappen to be equal. This means that the amount of outcomes that changes from �

to � is equal to the amount that change from � to � , when the angle is changed�� $ at � . This yields an insensitivity to the angular setting at � of the marginaldistribution at � although the individual outcomes depend on it.

However, for a non-equilibrium distribution one might very well have

� � � �� (6.10)

and

� � � �� (6.11)

for example. That would imply a marginal statistics at � that is sensitive to whatmeasurement ( � or � $ ) is done at � . The situation is illustrated in Fig. 6.4.

Summarizing, according to the hidden variables point of view, the statistical lo-cality we have in nature is not fundamental. It is a contingent feature of the quantumequilibrium distribution. But deeper down we have a truly non-local world.


6.2 The non-locality of quantum theory

In the previous section we discussed quantum non-locality from the perspective ofhidden variable theories. It is a common view that it is only hidden variable theories,and not quantum theory, that are non-local. This situation is however more seriousthan that. As also demonstrated by Bell, quantum theory is not locally causal in avery precise mathematical way.

6.2.1 The EPR argument

‘For me, it is so reasonable to assume that the photonsin those experiments carry programs, which havebeen correlated in advance, telling them how to be-have. This is so rational that I think that when Ein-stein saw that, and the others refused to see it, he wasthe rational man. The other people, although historyhas justified them, were burying their heads in thesand.’ J. Bell

Einstein’s worry about quantum mechanics began early. Already in 1927 he ex-pressed worries that a many-electron system in configuration space involves corre-lations between the electrons that violate the principle of action-by-contact [35]. Atthe same time he also believed that quantum mechanics was essentially a statisticaltheory in the sense that a more complete description was possible. He later cameto believe that the nonlocal feature of quantum theory was inseparably intertwinedwith the idea that quantum theory provides a complete description of reality. Ein-stein therefore saw a way of arguing that quantum mechanics cannot be consideredcomplete if locality is assumed. And this led to the famous Einstein-Podolsky-Rosenargument [50], the ‘EPR argument’ for short.

However, Einstein was never satisfied with the presentation of the argument whichwas, in fact, written by Podolsky. It was, according to Einstein, too formal and tooinvolved. 2 Einstein later published his own version of the EPR argument (see e.gref. [51]) which is much clearer and less involved. But the damage was already doneand trying to resurrect the argument after it had been ‘heroically defeated’ by Bohrwas difficult.

There was, however, one person who saw the relevance and importance of theEPR argument: John Bell. He was not satisfied with Bohr’s response (see ref. [1], p.155–156 for his negative view) and commented slyly

‘Is Bohr just rejecting the premise – ‘no action at a distance’ – rather than refutingthe argument?’

2Apparently, Einstein was very annoyed with Podolsky for publicly announcing in New YorkTimes that they had showed quantum theory to be faulty, and refused to talk to him after that [59].

6.2. The non-locality of quantum theory 53

Bell, in fact, developed his own EPR argument (see e.g ref. [1], p. 54–55) and repeat-edly stressed that quantum mechanics is not locally causal. His argument is a veryclean one and not needlessly entangled with metaphysics in the sense that it doesnot assume the reality of the micro-cosmos (as also commented in ref. [1], p. 150).It is logically very straightforward. The only things involved are the results of ex-periments (the readings of macroscopic measurement devices) and a mathematicallyprecise notion of local causality.

What is missed in many contemporary presentations of Bell’s theorem is thatBell’s work was not a refutation of the EPR argument. It was a logical continua-tion of it. While Bell agreed with Einstein that locality implies the incompleteness ofquantum theory, he went further to ask if any completion of it could be local, andanswered this question in the negative.

The following is the structure of Bell’s paper “The theory of local beables” (seeref. [1], p. 52–62)

� ��

�� " � � � ��

� �� "��

� � � �� # � � ��

� � " � � � � �# � # �� " �� # � � � � # � � �� # � � � � � # � # � � �� " � � � � � # �# * � � � � # � � � � � � � �

Logically this is an argument of the type ‘reductio ad absurdum’. We assume localcausality and arrive at its logical negation (non-locality), expressed through the vi-olation of Bell inequalities. The assumption of locality must then be dropped andquantum theory is therefore irreducibly nonlocal.

The mainstream conclusion of Mermin [60]

‘To those for whom non-locality is anathema, Bell’s Theorem finally spells thedeath to the hidden-variable program’

does not square well with Bell’s own view simply because Mermin somehow forgets,misses, or ignores the first half of Bell’s argument, the EPR argument.

6.2.2 The issue of counter-factual definiteness

‘Ist mir wurst.’ A. Einstein

Not uncommonly it is claimed that non-locality is not a feature of quantum theorybut only of hidden variable theories, and other mutations of quantum theory. Andany argument attempting to establish the non-locality of quantum mechanics itselfmust, it is claimed, involve an illegitimate use of counter-factual reasoning ([52], p.168). While that is true for some arguments (see e.g ref. [52], p. 164), Bell’s argumentin “The theory of local beables” does not make use of counter-factual reasoning.3

Loosely speaking counter-factual reasoning amounts to a speculation about whatwould have been the outcome if another experiment had been carried out instead the

3I am grateful to Tim Maudlin for pointing out this to me.


one that was actually carried out. However, all present and past theories make claimsabout the possible outcomes of experiments that have not, are not, and will not beperformed. That is just the nature of such human constructs. Newtonian mechanicsmakes predictions about what would happen if Saturn should orbit Jupiter even ifsuch experiment will most likely never be performed. Quantum theory, however,does not in general make predictions about individual outcomes but only about thestatistics of the outcomes.

The predictive power of our theories allow us to ask what would have happened ifa different experiment had been performed instead of the one that was actually per-formed. This property is called counter-factual definiteness, or CFD for short. When-ever scientists construct theories of cause and effect relationships they are guilty ofassuming some form of CFD.

Let us now carefully distinguish between two forms of CFD: individual CFD (ICFD)and statistical CFD (SCFD). Assuming ICFD allow scientists to speculate about indi-vidual outcomes of unperformed experiments and assuming SCFD allows specula-tion only about the long term statistics of outcomes of unperformed experiments.Quantum theory presupposes SCFD, because if it did not, it would not predict any-thing at all. However, ICFD in quantum theory is problematic. Quantum theory byno means allows (in general) speculation about what would have been the result inan individual (unperformed) experiment. Only for a restricted class of preparationsand experiments does quantum theory give an answer to such questions. Therefore,it makes little sense to ask quantum theory in a general situation what would havehappened if an other (incompatible) experiment had been performed instead of theone that was actually performed.

6.2.3 Bell’s notion of local causality

When we discussed deterministic hidden variable theories it was natural to requirethat the nonlocal transition sets were empty � ��

� � � ��

��

��

.However, this way of defining locality involves a speculation about what would havehappened if a different experiment had been carried out instead of the actual one. Forexample, requiring � � �

��

means that if the angular settings, in an EPRB experi-ment, were chosen as � and

�instead of � and

� $ (or vice versa), that would not haveaffected the result at � . However, as we have seen in Paper III, these transition setscannot be empty if the hidden variable theory is supposed to reproduce the quantumstatistics. Therefore such a hidden variable theory must be nonlocal.

But one can then correctly point out that this argument involves ICFD and there-fore establishes the non-locality only for deterministic theories (i.e hidden variabletheories). More acutely, it involves extra hidden parameters that are alien to quan-tum theory. Therefore, one can correctly point out that quantum theory escapes thistype of argument.

In order to establish the non-locality of quantum theory Bell proceeded in a dif-

6.2. The non-locality of quantum theory 55

PSfrag replacements

A B

D

Figure 6.5: Bell’s notion of local causality requires the probabilities attached to val-ues of local beables in a space-time region � to be unaltered by the specification ofvalues of local beables in a space-like separated region � , when what happens inthe backward light cone of � is already sufficiently specified, for example by a fullspecification of local beables in a space-time region � . Quantum theory violates thisprinciple of local causality.

ferent way. He used a notion of locality, different from the one above, which does notinvolve ICFD (nor SCFD for that matter4) and in particular not extra hidden param-eters.

Here is Bell’s notion of local causality (see Fig. 6.5):

Local Causality: A theory will be said to be locally causal if the probabilitiesattached to values of local beables in a space-time region � (see Fig. 6.5) areunaltered by specification of values of local beables in a space-like separated region� , when what happens in the backward light cone of � is already sufficientlyspecified, for example by a full specification of local beables in a space-time region

� .

A few comments about terminology. Bell realized that any argument that assumesthe reality of micro-cosmos could easily be called into question by simply denyingsuch reality.5 Therefore Bell introduces the notion of local beables. These are the‘classical terms’ that Bohr so much insisted play a crucial role in any appropriateunderstanding of quantum theory. This is a clever move by Bell because few woulddeny the reality of such ‘classical’ objects.6

Let us discuss Bell’s notion of local causality in the context of the EPRB gedankenexperiment. Let both Stern-Gerlach apparatuses measure the spin along the � -direction.

4SCFD is assumed only in the derivation of the Bell inequalities. See below.5This metaphysical position was advocated by, for example, Heisenberg ‘They would prefer to come

back to the idea of an objective real world whose smallest parts exist objectively in the same sense as stones ortrees exist, independently of whether or not we observe them. This is, however, impossible...’ [55]. Heisenberg’smetaphysical position is problematic because it does not answer questions like: How is it possible forsomething that does not really exist to build up something that does?

6In [1], p. 52 Bell writes ‘So it could be hoped that some increase in precision might be possible by concen-trating on the beables, which can be described in ‘classical terms’, because they are there.’


If the result of the spin measurement in region � is not specified, the probability ofgetting � or � in region � is �

�� for both outcomes. However, if the result at � is

specified, e.g the result � , then the probability for a � outcome at � is one. Sincequantum theory, according to the standard view, provides a full specification of localbeables in region � , we have a violation of local causality.

Violation of Bell’s notion of local causality implies the existence of superluminalcausal influences, as pointed out by Bell ([1], p. 110) and elaborated on by Maudlin[78].7 This places us in an uneasy position. But could it be that quantum theory isnot capable of giving a complete description and that there was more to specify inthe spacetime region � ? And by doing so, could one escape the above conclusion ofnon-locality?

In the second part of the paper “The theory of local beables”, Bell demonstratesthat any theory, satisfying the above notion of local causality, cannot reproduce thestatistics of quantum theory. Therefore they must be nonlocal in the same way quan-tum theory is nonlocal. He arrives at that conclusion, by deriving a pair of inequal-ities that any locally causal theory (deterministic or indeterministic) must satisfy. Itis only in the derivation of these inequalities that SCFD must be assumed. By con-struction, these inequalities involve a comparison of statistics of incompatible mea-surements.

7Strictly speaking, this conclusion rests on that quantum theory will not fail when more efficientdetectors can be constructed, and that the angles can be treated as free variables ([1], p. chapter 12).

Particle Detectors, Geodesic Motion andthe Equivalence Principle,

We will now discuss the status of the equivalence principle in quantum field theoryin curved spacetime and show that a not uncommon application of the equivalenceprinciple is erroneous. This section will serve as an introduction to Paper I.

7.1 Introduction

When discussing different features of quantum field theory, the concept of a particledetector is useful. A particle detector in quantum field theory could be, for example,a hydrogen atom. When a photon is absorbed the hydrogen atom gets excited. In amanner of speaking, one then says that the particle detector (the hydrogen atom) hasdetected one quanta (a photon in this case). In practice, however, a hydrogen atomis very cumbersome to do calculations with, and therefore one has introduced moreabstract models of particle detectors, like the DeWitt monopole detector describedbelow.

One might expect that if a quantum field is prepared in a vacuum state, which ischaracterized by the absence of real quanta, then a particle detector would not getexcited, and thereby not detect any particles. However, as shown by Unruh [45], if aparticle detector in flat spacetime is in a state of constant proper acceleration � , thensuch a particle detector will detect a thermal spectrum of quanta with temperature

��

� � � � (7.1)

The equivalence principle states that for regions of spacetime that are very small withrespect to the typical scales associated with spacetime curvature, any experiment per-

57

58 7. Particle Detectors, Geodesic Motion and the Equivalence Principle,

formed in such region will have the same outcome as in flat spacetime. Now, since aparticle detector (e.g a hydrogen atom) can in normal situations be considered verysmall compared to the typical scales associated with the spacetime curvature, it isthen natural to ask if Unruh’s result (i.e that a detector does not detect particles if andonly the motion is geodesic) can be generalized to an arbitrary curved spacetime. Forexample, in [83, p. 516] Renteln writes (see also [81, 82])

‘Since the equivalence principle states that acceleration and gravitation are locallyequivalent, virtual particles should be observable in a gravitational field.’

We shall in fact show that contrary to these claims, the equivalence principle can ingeneral not be applied in this way. More precisely, in Paper I we show that there isin general no correlation between the motion being geodesic or not and the detectionof quanta. To that effect two examples are exhibited, one in which the motion ofthe detector is geodesic and detects particles, and one in which the motion is non-geodesic and the particle detector remains unexcited.

7.2 Mathematical preliminaries

We now proceed to give a brief introduction to quantum field theory in curved space-time. In order to quantize a classical system the dynamical variables are turned intooperators required to satisfy specific commutation relations. For example, in nonrel-ativistic quantum mechanics of a single particle, the dynamical variables of momenta� and positions � are turned into operators satisfying � � � � � � �

( �� ).Likewise in field theory the corresponding field variable � � � � together with its

canonical momenta � � � � are turned into operators. It is convenient to expand thefield operators in terms of creation and annihilation operators [46].

7.2.1 Normal modes

In order to define creation and annihilation operators one introduces an orthonormalbasis that one can expand the field in. Therefore we must have a notion of a scalarproduct. For the Klein-Gordon equation it is defined by

�� (7.2)

where �� and� � is a future directed time-like unit vector that is orthogonal

to a hypersurface � on which we are performing the integration. �� is a volumeelement in � . We can now define orthonormal modes as those satisfying

�� # �� # � �� # �� # � �� # �� (7.3)

7.2. Mathematical preliminaries 59

where � # are called positive frequency modes and � �# negative frequency modes andthe indices

� � � are allowed to be continuous. Once a set of modes is chosen, one mayexpand the scalar field in terms of these

� � # � # � # � � � � ��# � �# � � � (7.4)

Then we quantize by implementing the commutation relations

� � # �� # � (7.5)

The vacuum �� is then defined by the relation � # � � �� . This taken to correspond toa state of no field excitations. However, there are of course many sets of orthonormalmodes. In Minkowski spacetime one chooses the modes that are Lorentz invariant(i.e

! � # �� ). In a general spacetime, though, there will be no such nice symmetries inorder to guide us in choosing our modes. One may think that nothing should dependon what normal modes we use for expanding the field. This is true for classical fieldtheory but in quantum field theory it is not as trivial. If we change the normal modesaccording to (this is called a Bogolubov transformation)

�� # � � # � # �� # � �# (7.6)

�� # � �� # � �# �� # � # (7.7)

will induce the change

�� # � �� # � # � � �� # � �� (7.8)

of the annihilation operator. As a consequence the vacuum �� will not be annihilatedby �� since it contains a creation operator. One can also define a new vacuum by�� # � �� '� � . This vacuum would not be the same as the first one since according to thefirst set of normal modes the ’vacuum’ � �� contains particles. So there is an intrinsicambiguity of what to call a vacuum state.

In order to sort out such questions it will be helpful to give an operational definitionof vacuum. As we propose in Paper I, one may define a state � � � to be a vacuum stateif there exists a congruence so that a particle detector following any of the congruencelines, will be unexcited, i.e not detect any particles. If that is possible one may saythat the field is in a vacuum state. In order to bring out this more clearly one mustprovide a model for a particle detector which we shall do now.


7.2.2 The Dewitt monopole detector

The Dewitt monopole particle detector is an idealized point particle with internalenergy levels labeled by the energy � , coupled to the scalar field � via a monopoleinteraction

� � �� where � is the detectors monopole moment operator.Suppose we prepare the detector in its ground state � � �

�. Then we let it move

on some spacetime trajectory. In the generic case, the detector will not remain in itsground state but will undergo the transition ��

� � � � � . At the same time the fieldalso undergoes the transition � � � � � � � from the vacuum state �� to an excited state� � � . To the first order the transition amplitude will be given by [46]

� ��

�(7.9)

or using � �� ! # �� ! � # �� and � � ��

� � � � � � � � � � � ��

�� ! # � � � � � � � �� (7.10)

Now this is the transition amplitude for a specific transition �� . The

probability we obtain by squaring the modulus of this transition amplitude. If we donot care about the details of the specific transition we sum over E and a complete setof states � � � [46] yielding

� � ��

� � � ��

��

��$ ! � # � � � � � � � � � � � � � � � � �� $ � � �� (7.11)

Since the last part is independent of the details of the detector it represents in someway the ’bath’ of particles that the detector experiences. The first term determineshow easily the detector gets excited by a certain energy. The latter part is appropri-ately called the response function.

� ��

��

�� $ ! � # � � � � � � � � � � � � � � � � �� $ � � �� (7.12)

Here � denotes the specific spacetime trajectory we carry the detector along. How-ever, the quantity as defined above is divergent. What one usually considers is thetransition probability per unit time and that concept happens to be well defined. Wethen define the response function as

� ��

� ��

� � �� $ ! � # � � � � � � � � � � � � � � � � �� $ � � �� (7.13)

where the step function� �� is inserted as a reminder that � � � � . The response

function, as defined by equation (7.13), measures the probability per unit time to getexcited to a specific energy � . If the detector has a lowest energy � � � � then it will

7.3. Discussion 61

fail to detect these low energy quantas. In order not to exclude possible low energyexcitations one should consider monopole detectors for which � �

� � .We have now a quantity

�that we can use in order to define a vacuum. If there is

a congruence of these detectors so neither detects anything, i.e does not get excited,then we say that we have a vacuum state. If there is no such congruence it may beconcluded that there are particles around.

7.3 Discussion

In Paper I we use the response function to disprove the folklore conjecture that a de-tector in a general spacetime detects no particles if and only if it moves on a geodesic.This seems at first to disprove the equivalence principle. An experimenter, equippedwith a Dewitt monopole detector, might then determine that he is, or is not, in acurved background. However, it is not fair to say that the equivalence principle isviolated. We know that not even vacuum is empty. There is always some vacuumfluctuations left. When comparing the situation in Minkowski spacetime with the onein curved spacetime, one is using two different vacua. One might then argue that oneis comparing two situations that are physically different, even locally. Therefore wehave a trivial violation of the very conditions for testing the equivalence principle,namely that we locally set up the same situation. For a more detailed discussion seePaper I.

To end this section I would like to point out that Jacobson’s interesting derivationof Einstein’s field equations from thermodynamics [31], apparently makes use of theabove erroneous application of the equivalence principle. In that paper he arguesthat locally in a general spacetime, a uniformly accelerated observer will see a ther-mal bath. However, we have no reason to believe that this is true, as seen in PaperI. Despite this apparent problem, Jacobson does derive the Einstein field equations.It would be very interesting to study this subject further and see exactly what wasassumed in Jacobson’s derivation.

On General Covariance‘Why were another seven years required for the con-struction of the general theory of relativity? Themain reason lies in the fact that it is not so easy tofree oneself of the idea that coordinates must have adirect metric significance.’ A. Einstein

Inspired by Machian ideas, Einstein convinced himself that the theory of gravita-tion had to be invariant under arbitrary coordinate transformations. While the equa-tions of special relativity were only invariant under the Poincare group, he wantedthe equations of general relativity to be invariant under the whole diffeomorphismgroup, that is the group of all differentiable coordinate transformations with a differ-entiable inverse. This invariance principle is called the principle of general covari-ance.

However, Einstein soon convinced himself that any such generally invariant the-ory must be in conflict with determinism. The field equations, with suitable bound-ary conditions, did not seem to be able to uniquely pick out one solution but ratherclasses of solutions. Einstein thought that the different solutions in such classes rep-resented physically distinct situations. Thus the field equations seemed to fail to bedeterministic, in the sense that no solution in that class is picked out as special. Thisis Einstein’s notorious ‘hole argument’ [65].

It took him two years to realize that his conclusion of indeterminism was not accu-rate. It rested on the assumption that the coordinates in a generally invariant theoryhas the same status as those in the special theory of relativity. But, as Einstein cameto realize, that is not the case. In the special theory, the way Einstein presented it inhis 1905 article (see e.g ref. [41] p. 37-65), the coordinates one uses to write downequations, are operationally defined, and therefore understood as actual readings ofphysical objects such as clocks and rulers. For example, the symbol � in special rela-tivity is not to be understood as an arbitrary label, but as the reading of a clock. The

63

64 8. On General Covariance

situation in general relativity is however different. As we shall see, the coordinates� � one uses to write down Einstein’s field equations are devoid of any such physicalmeaning. Not even when a solution is at hand do these coordinates have a physicalmeaning, i.e being in a unique correspondence with the readings of some physicalobjects.

Once this was understood by Einstein, he saw that the indeterminism in gener-ally invariant theories is not a physical one. Instead of identifying one solution witha physical situation, one should identify the class of diffeomorphically related so-lutions with a physical situation [67]. Today we understand this feature of generalrelativity as being an expression of a gauge symmetry.

However, in 1917, only a year after the publication of the general theory of rela-tivity, Kretschmann [68] objected that any geometric theory can be put in a generallycovariant form1 and argued therefore that the principle of general covariance mustbe devoid of physical implications (see e.g ref. [29], p. 92). Ever since, the principleof general covariance has been subject to confusion and controversy. For a thoroughtreatment see [33]. The following will serve as an introduction to Paper IV.

8.1 General invariance as a mathematical symmetry

We shall now try to clarify Kretschmann’s objection and why it misses the point. Weshall see that indeed general invariance puts restrictions on the field equations and istherefore not physically vacuous.

8.1.1 The Klein-Gordon field

To warm up, consider the Klein-Gordon equation.

��

� � � ��

� � � ��

� � �� (8.1)

As we all know, this equation is invariant under the Poincare group. By invariancewe mean that the equation written down in coordinates looks the same (except forsome primes) after a coordinate transformation. For example, after a general Poincaretransformation �

� � � $ � � � � � ��

, the new equation looks like

�� $� � $ �

� � � � $� � $ �

� � �� $� � $ � � � � � $

� � $ �� $ � � (8.2)

which is identical to equation (8.1) except for the primes. This is not a trivial propertysince not all equation are invariant under the Poincare group. Furthermore, it implies

1Note that I have written covariant instead of invariant. That distinction will be important later inorder to see why Kretschmann’s objection does not touch the real issue.

8.1. General invariance as a mathematical symmetry 65

a certain structure of the solution space, i.e the set of all scalar functions � � � � thatsatisfies equation (8.1). If we have found a � � � � satisfying equation (8.1) it is alsoguaranteed that the different scalar field � � � � � � � is also a solution to the same equation(8.1).

The above should be contrasted to merely writing down equations in a coordinateindependent way. The Klein-Gordon equation can be written as

� � �� (8.3)

where�� is the covariant derivative related to the Minkowski metric �

� . Trivially,

this equation looks the same in all coordinate systems, only because it is written ina coordinate independent way. Let us call this trivial property of tensor equationsfor covariance to contrast it to invariance. These two concepts have little in commonand should not be confused. The concept of invariance of a physical law has a phys-ical significance while that of covariance is a mathematical identity as Kretschmannaccurately pointed out. Any equation can be put in a coordinate independent form(by introducing absolute geometrical objects such as � � � ) and hence any equation canbe said to be covariant with respect to an arbitrary coordinate transformation. How-ever, the concept of invariance is not a triviality as should be clear from the above.Equations may be invariant with respect to Galilean transformations but not Lorentztransformations and vice versa.

8.1.2 Einstein’s vacuum field equations

Let us now proceed to discuss Einstein’s field equations. For simplicity we restrictour attention to the vacuum equations�

� � � (8.4)

The following discussion may easily be generalized to include matter fields (e.g scalarfields � , electromagnetic fields � �

, etc.). As with the Klein-Gordon equation (8.1) letus now write down the field equations in a specific coordinate system2

� � ��

��

��

� � �� (8.5)

This equation has an interesting mathematical invariance property. By making anarbitrary coordinate transformation � � � � $ � � � � � � � one ends up with

� $� �� $ �� $� � $ � � � $ � $� � � � $� � $� � � � � $ � �� $ �� $� � $�� $� � $� � � � $� � $� � � �

2Since coordinates cannot in general cover the whole manifold we are restricting ourselves to atopologically trivial part of the manifold.


� � �� $ � � � � $� � $ � � � $ � $� � � � $� � $� � � � �� $ �� $� � $�� $� � $� � � � $� � $� � � �

� � �� $ � � � � $� � $� � � � $� � $�� $� � $� � � � � �� $ � � � � $� � $ � � � $ � $� � � � $� � $ � � � � �

which looks the same, except for the primes. Thus, the situation is very similar to theone with the Klein-Gordon equation, the only difference being that now the equationis invariant not only under Poincare transformations but also under the whole dif-feomorphism group, i.e under arbitrary changes of coordinates. And this propertyis certainly not shared by all equations, in particular not the Klein-Gordon equation(8.1).

If one uses the short hand notation�� it is easy to get confused. One might

then claim that this equation trivially looks the same in all coordinate systems andthat it is merely a mathematical identity. But this is merely the trivial covariance prop-erty. The failure to distinguish between invariance and covariance has generatedneedless confusion so I better state it clearly: When we speak of general invariance it isnot this trivial property of covariance we are addressing.

The above type of general invariance is mathematically not very different fromLorentz invariance. The only difference is how big the invariance group is. If wetake the principle of general covariance to mean this and nothing else it is far fromvacuous. Certainly, not every equation has this property. All equations in pre-generalrelativity physics violate this principle.3

8.2 Einstein’s hole problem

As with the Klein-Gordon equation, the invariance property of Einstein field equa-tions implies a certain structure of the corresponding solution space: if � � � � � is asolution, then since the equations are generally invariant

� � � � � � � $� � � � � � � ��

��

� � � � � � � � � � � (8.6)

is a new solution to the Einstein equations (8.5). Technically this is called an auto-morphism. In mathematical terms this is a map from the manifold � to itself, � �� , that induces a well-defined transformation on any tensor � � � $ � � � � .It should be clear from the definition that an automorphism is mathematically distinctfrom a coordinate transformation.

3One can make a somewhat artificial objection here. If we write the Klein-Gordon equation, notas in equation (8.1), but as ��

�� , and 0 �� for the metric. Since the latter ‘fieldequation’ only has one solution with signature �� (which is the Minkowski metric), this differentmathematical formulation is physically the same as the standard formulation. It is then made gen-erally invariant, just as Einstein’s field equations. This objection seems to me as technically correct.However, one can do the same artificial reformulation of the non-relativistic heat equation, making itLorentz invariant [84].

8.2. Einstein’s hole problem 67

Let us now consider a particular automorphism. Let � � � � � be a specific solutionof Einstein’s field equations. In Fig. 8.1 the automorphism is taken to be the iden-tity in the spacetime region � and nontrivial in the region � . Since the solutions arethe same in the (extended) spacetime region � they must have the same initial valuedata. However, in the spacetime region � they differ by the construction of the auto-morphism. Since both are patently solutions to the Einstein equations we are facingEinstein’s problem of indeterminism: the same initial data may evolve into two dis-tinct solutions.

2

1

PSfrag replacements

� $� � � � � � � � � �

� $� � � � � � � � � �

�

� � � �

� � � �

� � � �

Initial data

��

�

��

Spacelikehypersurfaces

Figure 8.1: The figure illustrate the mathematical consequence of the general invari-ance of Einstein field equations. Suppose we make an automorphism which is theidentity in the spacetime region � but nontrivial in region � . Both solutions are thenequal in spacetime region � and therefore they must have the same initial data. Butsince they are different in spacetime region � it is then clear that no initial data can beenough to uniquely specify a solution. This is a particular version of Einstein’s holeproblem.

Further, it is not merely the metric that is non-unique: any quantity derivablefrom the metric is equally non-unique, in particular the Riemannian curvature scalar � �

� � � � � � � . Let us suppose that we have two points �

� and �� in the spacetime

region � . At point � � , is different from zero but at the other point � � , is equal tozero. Now we design the automorphism such that: � � � �

� �� and � � � � � �

� . Thenwe may see how such an automorphism acts in the Riemann curvature scalar:

� � �� $� � � �

��

� ��

� �� (8.7)

� � �� $� � � �

��

� ��

� � � � (8.8)

This means that Einstein’s field equations in this case does not even make any com-mitment as to which point the Riemann curvature scalar should be zero or nonzero.


If matter fields are included the situation does not change. The above sort of in-determinism is a fundamental feature of general relativity. But why is this a potentialproblem? To understand why one must keep in mind that Einstein thought of theparameters � � as readings of physical objects, as is the case in special relativity.

Suppose that the coordinates � � have operational significance, i.e they are notmere parameters but represent ‘readings’ of physical pbjects. However, as Einsteinrealized, a generally invariant theory will then fail to predict a unique correlation be-tween the coordinate � � and the value of the Riemann curvature scalar, for example.That is not acceptable for a physical theory. Although a generally invariant theorywill fail to predict a unique correlation between the readings � � of some physical ob-jects, an experimenter will measure a unique value of the Riemann curvature scalar atthe point � � . Unless one is prepared to accept indeterminism, it is clearly somethingwrong here. Therefore Einstein first thought that generally invariant theories shouldbe discarded as unphysical.

8.3 Solution to the hole problem

The solution of the above dilemma lies in Einstein’s insight that the coordinates � �do not have the same status as the coordinates in special relativity.

In a measurement of some field, be it either the gravitational field or some other,one always aims to establish a correlation between one field and another. Roughlyspeaking, in order to measure a scalar field, suppose we lay out a physical grid. Ateach vertex corresponding to a position in space, a clock is situated. Then, a mea-surement amounts to establishing a specific correlation: at the point in space wherethe vertex denoted is located and at the time when the clock at that vertex showedthe time � (e.g the clock reads � � : � � $ : � � $ $ ), the scalar field had a certain value � . Onemay then continue to measure the scalar field in all the other vertices and at othertimes. This yields a function � �� . It is important to realize that �� are not mereparameters. They represent ‘readings’ of physical objects.

However, there is nothing that says that one must think of the � � as representingreadings of some physical objects. Let us instead accept that they are merely parame-ters devoid of any operational meaning, and see how one can deal with this situation.

Consider then a specific solution to the Einstein field equations (supplementedwith extra equations for the other fields), i.e � � �� where � � � � collectivelystands for the the matter content (e.g scalar fields, electromagnetic fields, etc.). Thenconstruct somehow from � , four coordinate fields �� . The precise physical nature ofthese four fields is irrelevant. They might be the curvature scalar invariants, or thephysical construction with the grid and clocks used above. Note that � is not a tensorindex. It is merely a label � � �� of the four coordinate fields.

The coordinate fields are defined, by construction, to be invertible so that, givena particular solution �� , one can express � � in terms of the coordinate fields: � � �� .

8.4. General invariance vs Lorentz invariance 69

Using the usual tensor transformation rules one can now replace the � � in � � � � �yielding a new quantity

��

� �

� � � � � � � �� (8.9)

which is a collection of 10 quantities. Again, the indices � and � are not tensor indicesbut mere labels. All of these 10 quanties are, in fact, physically measurable.

An important property of�� is that it is independent of which of all diffeo-

morphically related solutions we start from. Let � � �� $� � � � �� $ � � � � be another solu-tion related to the former by a diffeomorphism. Then if the corresponding coordinatefield is inverted � � �� $ � and then used to replace the � � in � $� � � � , then one ends up withthe same function

�� . Therefore�� is a gauge independent quantity since it is

left invariant under diffeomorphisms, and thereby measurable. One may, of course,apply the same trick to any other field, in the end producing only measurable gaugeindependent quantities. Similar have been made in [85].

Note that the parameters � � , that did not represent anything physical, are all elim-inated by the above ‘inversion’ procedure. It is in this way that general relativity es-capes the indeterminism that was apparently established by the hole argument. Thecoordinates � � in general relativity are indeed very different from those in specialrelativity.

8.4 General invariance vs Lorentz invariance

As we have seen, general covariance is, mathematically speaking, very similar toLorentz invariance, the only difference being the relevant invariance group. How-ever, physically they are very different. While Lorentz invariance implies the relativ-ity of uniform motion, general invariance does not imply a corresponding relativityprinciple. In particular, it does not imply the relativity of acceleration, which in gen-eral relativity is defined as the amount of deviation from a straight line in spacetime.We will here clarify the reason for this difference between general invariance andLorentz invariance.

The relativity principle in special relativity is an example of what is commonlycalled a dynamical symmetry [86] among philosophers of physics. The relativityprinciple states roughly that an experimenter, enclosed inside a laboratory and suit-ably shielded from the rest of the universe cannot, by any experiment carried outinside that laboratory, determine his motion relative to the rest of the universe. Forexample, he will be unable to determine if he is at rest or not with respect to the cos-mological rest frame, defined by the microwave background. As should be clear, therelativity principle involves a comparison between two physically distinct situations.

Lorentz invariance implies such a dynamical symmetry. However, general in-variance implies no such a thing. This is so because a diffeomorphism acts per


definition on everything and as we saw above�� is left invariant by a diffeo-

morphism. Therefore, a diffeomorphism does not change the physical situation, andconsequently we do not have two physically different situations to compare.

On the other hand, in special relativity the coordinates are not mere parame-ters. They represent readings of physical objects. Therefore, the situations � � � � and� � � � � � � (where �

� � � � � represent the readings of some physical objects) representtwo physically different situations. One in which the scalar field has the value � � � � �at the vertex denoted and at time when the clock at this vertex showed � , and theother when the scalar field has the value � � � � � � � at the same point.4

Lorentz invariance, as a dynamical symmetry, comes about in general relativityonly in very special circumstances. Firstly, we are restricted to a spacetime region inwhich where � � � �

� (which implies that the fields involved are only test fields that

do not alter the spacetime structure), and secondly that the coordinate fields � � (thatwe normally call � � � � in special relativity) do not enter the dynamics of the field wewish to measure.

8.5 The role of the manifold in general relativity

In general relativity, the parameters � � label points of a manifold � . The coordinatesare used to write down field equations, but once a solution has been constructedthey can completely be eliminated by the above inversion trick. Therefore the mani-fold, whose points were labeled by the parameters � � , plays a somewhat remarkablerole in general relativity. The manifold in general relativity is merely a mathemati-cal expedient, a scaffolding used in the construction of solutions, only to be thrownaway once such solutions have been obtained. When gauge invariant quantities ( e.g�� , �� , etc.) has been constructed from the solutions, the points of the manifoldare rendered useless.

At first sight the manifold may naturally be thought of as a ‘container’. In partic-ular, it seems to make sense to say that the scalar field � has such and such a value insuch and a point. One may easily adopt the view that fields (be it metrical or matterfields) ‘live’ in a container whose existence is somehow independent of the fields.However, to begin with, the hole argument effectively shows that it does not makephysical sense to speak of a scalar field having a certain value at some point on themanifold. Furthermore, once the gauge independent quantities has been constructedthere is no longer any need for the container. In this sense, general relativity declaresthe spacetime container nonexistent. Only the field values, as summarized in thegauge independent quantities, are real.

However, these gauge invariant quantities has inherited some structure from the

4One does not end up with a hole problem in special relativity because if one specifies initial data atsome time � � � � , this is enough to determine the mathematical solution for all future times. Thereforeone does not run into the hole problem raised above.

8.6. The manifold structure and quantum non-locality 71

manifold, i.e the dimensionality, topology and differentiability. By requiring that allthe fields (i.e metrical plus matter fields) are differentiable tensorial functions on � ,also the gauge independent quantities constructed from these solutions are differ-entiable. For example, when slowly varying the values of �� , the ten quantities

�� varies in a differentiable way.

8.6 The manifold structure and quantum non-locality

Let us now bring quantum mechanics into the discussion. We saw in section 6.2 thatquantum mechanics is irreducibly non-local in the sense of implying superluminalinfluences in the EPRB scenario. However, the non-locality in the EPRB scenariois more dramatic than it first seems. Neither quantum theory nor pilot wave theoryimplies that any superluminal signal goes through space, in order to produce the non-local correlations. If it were possible to perfectly shield off one particle of the EPR pairfrom any imaginable signal (going through space) from the other, quantum theoryand pilot wave theory still predict the non-local correlations and thus require non-local causal influences. Thus, it seems that space is somehow ‘short-circuited’, andthat the particles communicate without any need for a signal through space.5 How-ever, general relativity is a theory where causal influences must propagate throughspace, and also with a speed less than that of light.

I end this section by noting that a simple generalization of the manifold conceptmight be given by the adjacency matrices in ref. [80]. Presumably a manifold struc-ture together with a metric will emerge for some special matrices on a coarse-grainedlevel. One might call such matrices manifold compatible. A matrix almost manifoldcompatible could perhaps allow for causal influences that, on a coarse grained level,does not seem to go through space.

5Changing the topology of space would not solve this problem since even in a nontrivial topol-ogy there will be topologically trivial regions in which quantum theory still would predict nonlocalcorrelations.

Conclusions and Outlook‘For me then this is the real problem with quantumtheory: the apparent essential conflict between anysharp formulation [of quantum theory] and funda-mental relativity. This is to say, we have an appar-ent incompatibility, at the deepest level, between thetwo fundamental pillars of contemporary theory... Itmay be that a real synthesis of quantum and relativ-ity theories requires not just technical developmentsbut radical conceptual renewal,’ J. Bell

The above quote from Bell shows clearly how differently he viewed his own non-locality theorem as compared with many mainstream presentations. In those it isoften ‘degraded’ into a mere no-go theorem for local hidden variable theories [60].As such it can easily be brushed off as a problem only for those who were so naivein the first place to believe in hidden variable theories. Somehow the full complex-ity of the problem has gotten out of sight and somehow the crucial role of the EPRargument (beautifully simplified and clarified by Bell himself in [1], p. 52) has beenmissed and misunderstood even to the extent that it is commonly believed that Bell’stheorem constitutes a refutation of the argument. These two arguments together,Bell’s EPR argument and Bell’s theorem, constitutes a significant threat to the idea oflocal causality. As made clear by Bell ([1], p. 110) and also elaborated on by Maudlin[78], there are strong indications of superluminal causal influences in nature.

We have also seen that quantum probabilities may been seen as arising from adeeper underlying deterministic theory, such as the one of de Broglie and Bohm. Wealso saw that any such deterministic theory effectively masked the non-locality inquantum equilibrium. The locality we see in nature seems to be a contingent featureof quantum equilibrium.

On the other hand general relativity implies that that any causal influence must

73

74 9. Conclusions and Outlook

travel through space. However, as argued in section 8.6, quantum theory does notseem to respect such a structure. The instantaneous causal influences does not appearto have to go through space.

One may therefore think of the non-locality in quantum theory as an expressionof manifold incompatibility, i.e the a priori postulated neighboring structure (i.e thedifferentiability of the manifold) is violated in this case. It is interesting that this sortof incompatibility might occur in macroscopic situations. Aspect type experimentshas been carried out with the polarizers 3 kilometers apart. In [79] Sonego arguesthat a spacetime geometry in quantum theory is not operationally definable, and thatone might question the relevance of geometry at atomic scales. I share this view butin addition to being sceptical to the notion of geometry at the atomic scale I also thinkthat the manifold makes little sense there.

In order to probe along these lines it might be helpful to develop structures thatimplies a manifold structure only for special cases and then only at a coarse-grainedlevel. The ideas of adjacency matrices discussed in [80] might provide a preliminarypoint of departure.

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