Fisica de La Emergencia y La Organizacion

Physics of Emergence and Organization

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N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

Physics of Emergence and Organization

editors

Ignazio LicataInstitute for Scientif ic Methodology, Palermo, Italy

Ammar SakajiAjman University, Abu Dhabi

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

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ISBN-13 978-981-277-994-6ISBN-10 981-277-994-9

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PHYSICS OF EMERGENCE AND ORGANIZATION

Lakshmi - Physics of Emergence.pmd 7/22/2008, 10:01 AM1

v

Foreword

Gregory J. Chaitin

IBM Research, P O Box 218,Yorktown Heights, NY 10598, USA

[email protected]://www.umcs.maine.edu

When I was a child the most exciting field of science was fundamental

physics: relativity theory, quantum mechanics and cosmology. As we all

know, fundamental physics is now in the doldrums. The most exciting field

of science is currently molecular biology. We are being flooded with fasci-

nating data (for example, in comparative genomics), and the opportunities

for applications seem limitless.

A mathematician or physicist looking with admiration at the exciting

developments in molecular biology might be forgiven for wondering how

theory is faring in this flood of data and medical applications. In a way,

molecular biology feels more like software engineering than a fundamental

science. Nevertheless there are fundamental questions in biology. Here are

some examples:

a) Is life pervasive in the universe, or are we unique?

b) What are consciousness and thought, and how widespread are they?

c) Can human intelligence be greatly amplified through genetic engineer-

ing?

These are extremely difficult questions, but as the articles in this

special issue attest, the human being is a theory-building as well as a

tool-using animal. The desire for fundamental understanding cannot be

suppressed.

Someday we will have a theoretical understanding of fundamental bio-

logical concepts. In order to do that we will probably have to drastically

change mathematics and physics. This is already happening, as the notions

of complexity, computation and information develop and spread.

vi Foreword

What the mathematics and physics of complex systems may ultimately

be like, nobody can say, but as the articles in this issue show, we are starting

to get some interesting glimpses. The only thing I can safely predict is that

the future is unpredictable. There will no doubt be a lot of surprises.

Gregory Chaitin

vii

Preface

It was Philip W. Anderson in his famous 1972 paper More is Different

who first threw doubts upon the reductionist approach as the fundamental

methodology in Physics. Since then, the problem to build a Theory of Emer-

gence has grown more and more as a matter of great importance both in

Physics and in a broad interdisciplinary area. If, initially, emergence could

be framed within a naıve, “objective” world scheme and thus easily seized by

mere formal and computational models, today widespread, epistemological

awareness has rooted the idea that the most genuine and radical features of

emergence cannot be separated from the observer’s choices. In this way, the

fundamental lesson of Quantum Physics extends to the whole theoretical

field transversally crossing old and new subjects. A general theory of the

observer/observed relationships thus represents the ideal framework within

which the connections among different areas can be discussed. Such kind

of theory can actually be regarded as an authentic “Theory of Everything”

in systemic sense and can go side by side with the more traditional unified

theories of particles and forces.

From the Physics viewpoint, the matter implies many fundamental

questions connected to the different ways in which classical and quantum

systems exhibit emergence. The formation and evolution of structures find

their natural, conceptual context in the theories of critical phenomena and

collective behaviors. It is getting clear that information takes on different

connotations depending on whether we consider a phase transition from

the classical or quantum viewpoint, as well as that the connection between

Physics and computation finds its deepest significance in regarding physi-

cal systems as systems performing effective computation. This is the reason

why the recent researches on Quantum Computing are just one eighth of an

iceberg which will lead not only to new technological perspectives, but —

above all — to a different comprehension of the traditional questions about

the foundations of Physics. Another relevant problem concerns the relation-

ships among the different description levels of a system in that complex and

viii Preface

not completely colonized middle land represented by the mesoscopic realm,

where Physics meets the other research fields. Among such questions, one

emerges as the most drastic: whether to extend the syntax of both Quantum

Theory and Quantum Field Theory in order to build a general theory of the

interaction between the observer and the external world, or whether such

action will rather lead to include Quantum Theory itself as a particular

case within a more general epistemological perspective.

This project has been conceived within the yearly tradition of the

Special Issues of Electronic Journal of Theoretical Physics. We intended

to give birth to an open space hosting quite different conceptions so as to

offer the researchers a significant state of the art scenario in such exciting

topics. Such a project would have never turned into a book without the

fruitful discussions with my friend Ammar J. Sakaji, who shared both the

conceptual choices and hard work.

I would also like to thank the authors who perfectly grasped the spirit

of our project and have presented precious contributions — our debate will

not stop! I extend my thanks to the EJTP Editorial Board, and Teresa

Iaria whose artwork on science and art relationships has provided the cover

image. Finally, a warm acknowledgement to all my friends and colleagues

who — during these years — have developed these topics with me . . . I am

glad I can say that all of them are — directly or indirectly — present in

this volume.

Ignazio Licata

November 2007

ix

Contents

Foreword v

Gregory J. Chaitin

Preface vii

Ignazio Licata

Emergence and Computation at the Edge of Classical and

Quantum Systems 1

Ignazio Licata

Gauge Generalized Principle for Complex Systems 27

Germano Resconi

Undoing Quantum Measurement: Novel Twists to the Physical

Account of Time 61

Avshalom C. Elitzur and Shahar Dolev

Process Physics: Quantum Theories as Models of Complexity 77

Kirsty Kitto

A Cross-disciplinary Framework for the Description of

Contextually Mediated Change 109

Liane Gabora and Diederik Aerts

Quantum-like Probabilistic Models Outside Physics 135

Andrei Khrennikov

Phase Transitions in Biological Matter 165

Eliano Pessa

Microcosm to Macrocosm via the Notion of a Sheaf

(Observers in Terms of t-topos) 229

Goro Kato

x Contents

The Dissipative Quantum Model of Brain and Laboratory

Observations 233

Walter J. Freeman and Giuseppe Vitiello

Supersymmetric Methods in the Traveling Variable: Inside

Neurons and at the Brain Scale 253

H.C. Rosu, O. Cornejo-Perez, and J.E. Perez-Terrazas

Turing Systems: A General Model for Complex Patterns

in Nature 267

R.A. Barrio

Primordial Evolution in the Finitary Process Soup 297

Olof Gornerup and James P. Crutchfield

Emergence of Universe from a Quantum Network 313

Paola A. Zizzi

Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 327

Joseph P. Zbilut

Order in the Nothing: Autopoiesis and the Organizational

Characterization of the Living 339

Leonardo Bich and Luisa Damiano

Anticipation in Biological and Cognitive Systems: The Need

for a Physical Theory of Biological Organization 371

Graziano Terenzi

How Uncertain is Uncertainty? 389

Tibor Vamos

Archetypes, Causal Description and Creativity in Natural World 405

Leonardo Chiatti

1

Emergence and Computation at the Edge of Classicaland Quantum Systems

Ignazio Licata

ISEM, Institute for Scientific Methodology, Palermo, [email protected]

The problem of emergence in physical theories makes necessary to build a gen-eral theory of the relationships between the observed system and the observingsystem. It can be shown that there exists a correspondence between classi-cal systems and computational dynamics according to the Shannon–Turingmodel. A classical system is an informational closed system with respect tothe observer; this characterizes the emergent processes in classical physics asphenomenological emergence. In quantum systems, the analysis based on thecomputation theory fails. It is here shown that a quantum system is an infor-mational open system with respect to the observer and able to exhibit processesof observational, radical emergence. Finally, we take into consideration the roleof computation in describing the physical world.

Keywords: Intrinsic Computation; Phenomenological and Radical Emergence;Informational Closeness and Openness; Shannon–Turing Computation; Bohm–Hiley Active InformationPACS(2006): 03.50.z; 03.67.a; 05.45.a; 05.65.+b; 45.70.n; 45.05.+x; 45.10.b;03.65.w; 03.67.a; 03.70.+k; 89.75.k; 89.75.Fb

1. Introduction

The study of the complex behaviors in systems is one of the central prob-lems in Theoretical Physics. Being related to the peculiarities of the systemunder examination, the notion of complexity is not univocal and largely in-terdisciplinary, and this accounts for the great deal of possible approaches.But there is a deeper epistemological reason which justifies such intricate“archipelago of complexity”: the importance of the observer’s role in detect-ing complexity, that is to say those situations where the system’s collectivebehaviors give birth to structural modifications and hierarchical arrange-ments. This consideration directly leads to the core of the emergence ques-tion in Physics. We generally speak of emergence when we observe a “gap”between the formal model of a system and its behaviors. In other words,

2 Ignazio Licata

the detecting of emergence expresses the necessity or, at least, the utilityfor the creation of a new model able to seize the new observational ranges.So the problem of the relationship among different description levels is putand two possible situations arise: 1) phenomenological emergence, wherethe observer operates a “semantic” intervention according to the system’snew behaviors, and aiming at creating a new model—choosing the statevariables and dynamical description—which makes the description of theobserved processes more convenient. In this case the two descriptive levelscan be always—in principle—connected by opportune “bridge laws”, whichcarry out such task by means of a finite quantity of syntactic information;2) radical emergence, where the new description cannot be connected to theinitial model. Here we usually observe a breaking of the causal chain (com-monly describable through opportune symmetries), and irreducible formsof unpredictability. Hence, the link between the theoretical corpus and thenew model could require a different kind of semantics of the theory, such asa new interpretation and a new arrangement of the basic propositions andtheir relationships.

Such two distinctions have to be considered as a mere exemplification,actually more varied and subtler intermediate cases can occur. The rela-tionships between Newtonian Dynamics and the concept of entropy canbe taken into consideration as an example of phenomenological emergence.The laws of Classical Dynamics are time-reversal, whereas entropy definesa “time arrow”. In order to connect the two levels, we need a new modelbased on Maxwell–Boltzmann statistics as well as on a refined probabilis-tic hypothesis centered, in turn, on space-time symmetries—because of thespace-time isotropy and homogeneity there do not exist points, directionsor privileged instants in a de-correlation process between energetic levels.So, a “conceptual bridge” can be built between the particle descriptionand entropy, and consequently between the microscopic and macroscopicanalysis of the system. But this connection does not cover all the facetsof the problem, and thus we cannot regard it as a “reduction” at all. Infact in some cases, even within the closed formulation of classical physics,entropy can decrease locally, and after all the idea to describe a perfect gasin molecular terms would never cross anybody’s mind!

Another example regards the EPR-Bell correlations and the non-localityrole in Quantum Mechanics. Within the Copenhagen Interpretation thenon-local correlations are experimentally observed but they are not con-sidered as facts of the theory. In the Bohm Interpretation the introductionof the quantum potential makes possible to bring non-locality within the

Emergence and Computation at the Edge of Classical and Quantum Systems 3

theory. It should not be forgotten that historically the EPR question comesout as a gedanken experiment between Einstein and Bohr about the “el-ements of physical reality” in Quantum Mechanics. Only later, thanks toBohm analysis and Bell’s Inequality on the limits of local hidden variabletheories, such question developed into an experimental matter. Nor Ein-stein neither Bohr would expect to observe really the “ghost-like-action-at-a-distance”. It is useful to remember that in Bohm theory the introductionof non-locality does not require any additional formal hypotheses but thestandard apparatus provided by the Schrodinger equation. Besides, if onone hand the new interpretative perspective provides a different compre-hension of the theory, on the other hand it puts some problems about theso-called “pacific coexistence” between Restricted Relativity and QuantumMechanics.

In both of the briefly above-examined cases we can see how the phe-nomenological and radical features of emergence are strongly intertwinedwith the development dynamics of physical theories and how the generalproblem of emergence points up questions of fundamental importance forthe physical world description, such as the updating mechanism of the the-ories and the crucial role of the observer in choosing models and their inter-pretations. In particular, it is worth noticing that the relationship betweenthe observer and the observed is never a merely “one-way” relationship andit is unfit to be solved in a single direction, which would lead to epistemo-logical impoverishment. This relationship has rather to be considered as anadaptive process in which the system’s internal logic meets our modalities toacquire information about it in order to build theories and interpretationsable to shape up a system’s description.

The problems related to the emergence theory, conceived as a generaltheory of the relationships between observing system and observed system,will be here taken into consideration, and will be tested on some evolutionmodels of both classical and quantum systems. Finally, we will develop someconsiderations about the logic limits of the theories and the computabilityrole in describing the physical world.

4 Ignazio Licata

2. The Observers in Classical Physics: Continuous Systems

For our aims, an informational or logical closed formal system will be in-tended as a model of physical system such that: 1) the state variables andthe evolution laws are individuated; 2) it is always possible to obtain thevalues of the variable states at each instant; 3) thanks to the informa-tion obtained by the above mentioned two points, it is always possible toconnect univocally the input and the output of the system and to forecastits asymptotic state. So, a logical closed formal system is a deterministicsystem with respect to a given choice of the state variables.

Let us consider a classical system and see how we can regard it as logicalclosed with respect to the observation procedures and its ability to showemergent processes. To be more precise, the values of the state variablesexpress the intrinsic characteristic properties of the classical object andthey are not affected by the measurement. Such fact, as it is known, canbe expressed by saying that in a classical system the measurement madeon all state variables are commutative and contextually compatible, i.e. allthe measurement apparatuses connected to different variables can alwaysbe used without interfering one with the other and without any loss ofreciprocal information. This assumption, supported by the macroscopic ob-servations, leads to the idea of a biunivocal correspondence between thesystem, its states and the outcomes of the measurements. Hence, the logicof Classical Physics is Boolean and orthocomplemented, and it formalizesthe possibility to acquire complete information about any system’s state forany time interval. The description of any variation in the values of the statevariables, at each space-time interval, defines the local evolutionary featureof a classical system, either when it is “embedded” within the structure ofa system of differential equations or within discrete transition rules.

The peculiar independence of a classical system’s properties from theobserver has deep consequences for the formal structure of classical physics.It is such independence which characterizes the system’s local, causal deter-minism as well as the principle of distinguishability of states in the phasespace according to Maxwell–Boltzmann distribution function. This all putsvery strong constraints to the informational features of classical physics andits possibility to show emergence.

The correspondence between the volume in a classical system’s phasespace and Shannon information via Shaw’s Theorem (Shaw, 1981) allows tocombine the classification of the thermodynamic schemes (isolated, closed,open) in a broader and more elaborate vision. Three cases are possible:


a) Information-conserving systems (Liouville’s Theorem);b) Information-compressing systems, ruled by the second principle of ther-

modynamics and consequently by the microscopic principle of correla-tion weakening among the constituents of the system individuated by theMaxwell-Boltzmann statistics (see Rumer & Ryvkin, 1980). These sys-tems, corresponding to the closed ones, have a finite number of possibleequilibrium states. By admitting a more general conservation principle ofinformation and suitably redefining the system’s boundaries for the (b)-type systems, we can connect the two kind of systems (a) and (b), and socoming to the conclusion that in the latter a passage from macroscopicinformation to microscopic information takes place;

c) Information-amplifying systems which show definitely more complex be-haviors. They are non-linear systems where the variation of a given orderparameter can cause macroscopic structural modifications. In these sys-tems, the time dependence between the V volume of the phase space andthe I information is given by: dI/dt = (1/V ) dV /dt. The velocity of infor-mation production is strictly linked to the kind of non-linearity into playand can thus be considered as a measure of complexity of the systems.Two principal classes can be individuated: c-1) Information-amplifyingsystems in polynomial time, to which the dissipative systems able toshow self organization processes belong (Prigogine, 1994; Haken, 2004);c-2) Information-amplifying systems in exponential time; they are struc-turally unstable systems (Smale, 1966), such as the deterministic chaoticsystems. Both the information-amplifying types belong to the open sys-tem classes, where an infinite number of possible equilibrium states arepossible.

Despite their behavioral diversity, the three classical dynamic systemsformally belong to the class of logical closed models, i.e. because of thedeep relationships between local determinism, predictability and computa-tion, they allow to describe the system by means of recursive functions. Ifwe consider the evolution equations as a local and intrinsic computation,wewill see that in all of the three examined cases it is possible to character-ize the incoming and outgoing information so as to define univocally theoutput/input relationships at each time (Cruchtfield, 1994). In dissipativesystems, for example, information about the self-organized stationary stateis already contained in the structure of the system’s equations. The actualsetting up of the new state is due to the variation of the order parameter inaddition to the boundary conditions. Contrary to what is often stated, eventhe behavior of structural unstable systems — and highly sensible to initial

6 Ignazio Licata

conditions — is asymptotically predictable as strange attractor. What islacking is the connection between global and local predictability, but wecan always follow step-by-step computationally the increase of informationwithin a predefined configuration. Our only limit is the power of resolutionof the observation tool and/or the number of computational steps. In bothcases we cannot speak of intrinsic emergence, but of emergence as detectionof patterns.

3. The Observers in Classical Physics: Discrete Systems

The discrete systems such as Cellular Automata (CA) (Wolfram, 2002)represent interesting cases. They can be considered as classical systemsas well, because the information on the evolution of the system’s states isalways available for the observer in the same way we saw for the continuoussystems. On the other hand, their features are quite different than those ofsuch systems in relation to emergent behavior.

The Wolfram–Langton classification (Langton, 1990) identifies four fun-damental classes of cellular automata. At the λ parameter’s varying — asort of generalized energy — they show up the following sequence:

Class I (evolves to a homogeneous state)→ Class II (evolves to sim-ple periodic or quasi-periodic patterns)→Class IV (yields complex pat-terns of localized structures with a very long transient, for ex. ConwayLife Game)→Class III (yields chaotic aperiodic patterns).

It is known that cellular automata can realize a Universal Turing Ma-chine (UTM). To this general consideration, the Wolfram–Langton classifi-cation adds the analysis of the evolutionary behaviors of discrete systems,so building an extreme interesting bridge between the theory of dynamicalsystems and its computational facets.

The I, II, III classes can be directly related to the information-compressing systems, the dissipative-like polynomial amplifiers and thestructural unstable amplifiers, respectively.This makes the CA a power-ful tool in simulating physical systems and the privileged one among thediscrete models. The correspondence between a continuous system and theclass IV appears to be more problematic. This class looks rather like anintermediate dynamic typology between unstable systems and dissipativesystems, able to show a strong peculiar order/chaos mixture. It suggeststhey are systems which exhibit emergence on the edge of chaos in specialway with respect to the case of the continuous ones. (Bak et al., 1988).

Although this problem is still questionable from the conceptual and for-mal viewpoint, it is possible to individuate at least a big and significant


difference between CA and continuous systems. We have seen that infor-mation is not erased in information-compressing systems, but a passagefrom macroscopic to microscopic information takes place. Therefore, thenot time reversal aspects of the system belong more to the phenomenologi-cal emergence of entropy at descriptive level than the local loss of informa-tion about its states. In other words, for the conservation principle and thedistinguishability of states, the information about any classical particles isalways available for observation. It is not true for CA, there the irreversibleerasing of local states can take place, such as in some interactions amonggliders in Life. The situation is analogous to the middle-game in chess,where some pieces have been eliminated from the game. In this case, it isimpossible to univocally reconstruct the opening initial conditions. Never-theless, it is always possible to individuate at least one computational pathable to connect the initial state to the final one, and it is possible to show,thanks to the finite number of possible paths, that one of such paths mustbe the one that the system has actually followed. Consequently, if on theone hand the erasing of information in CA suggests more interesting possi-bilities of the discrete emergence in relation to the continuous one, on theother its characteristics are not so marked to question about the essentialclassical features of the system. In fact, in more strictly physical terms, itis also possible for the observer to locally detect the state erasing withoutlosing the global describability of the dynamic process in its causal features.

Some interesting formal analogies between the unpredictability of struc-tural unstable systems and the halting problem in computation theory canbe drawn. In both cases, there is no correlation between local and globalpredictability, and yet the causal determinism linked to the observer’s pos-sibility to follow step-by-step the system’s evolution is never lost. Far fromsimply being the base for a mere simulation of classical systems, such pointilluminates the deep connection between computation and classical systems.Our analysis has provided broad motives for justifying the following def-inition of classical system: a classical system is a system whose evolutioncan be described as an intrinsic computation in Shannon–Turing sense. Itmeans that any aspect of the system’s “unpredictability” is not connectedto the causal structure failing and any loss of information can be individu-ated locally by the observer. All this is directly linked to what we called theclassical object’s principle of indifference to the measurement process andcan be expressed by saying that classical systems are informational closedwith respect to the observer.

Such analysis, see in the following, is not valid for the quantum systems.

8 Ignazio Licata

This is one of the reasons which makes the introduction of the indetermin-ism in discrete systems quite problematic (Friedkin, 1992; Licata, 2001).

4. Quantum Events and Measurement Processes

4.1. The Centrality of Indeterminism

The Heisenberg Uncertainty Principle is the essential conceptual nucleusof Quantum Physics and characterizes both its entire formal developmentand the debate on its interpretations.

For a quantum system, the set of values of the state variables is notaccessible to the observer at each moment. It is so necessary to substitutethe “state variable” concept with the system state notion, characterizedby the wave-function or state vector |Ψ〉. The relation between the systemstate and the variables is given by the Born rule. For example, given astate variable q

′we can calculate its probability value through the state

vector’s general form. It leads to explicitly introduce in the formalism themeasurement operations as operators which are — differently from the clas-sical ones — non-commutative and contextually incompatible. So, QuantumLogic differs from the Classical one because it is relatively orthocomple-mented and not Boolean, thus orthomodular (D’Espagnat, 1999). A directformal consequence is that for the quantum objects there exists a Principleof indistinguishability of states which characterizes the Fermi–Dirac as wellas the Bose–Einstein quantum statistics.

In this way, the observer takes on a radical new role, which can be clari-fied by means of the relational notion of quantum event (Healy, 1989; Bene,1992; Rovelli, 1996; for the notion of quantum contextuality as a frame seealso Griffiths, 1995; for the collapse problem from the logic viewpoint see:Dalla Chiara, 1977).

Let us consider a quantum system S and an observer O. The interac-tion between the two systems, here indicated as O+S, will give a certainvalue q of the corresponding state variable q. So, we can say — by para-phrasing Einstein — that the quantum event q is the element of physicalreality of the system S with respect to O. If we describe S and O as twoquantum systems, for ex. two-state systems — another system O’ will findthe value q′relative to the system S+O. Therefore, a quantum event alwaysexpresses an interaction between two systems and, according to the rela-tional approach, this is all we can know about the physical world, becausea hypothetical comparison between O and O’ will be a quantum event, too.

Despite the fact that it cannot be fully considered as an


“interpretation” yet, the relational view clearly shows the irreducibilityof the role of the observer in QM as well as the new symmetry with theobserved system. In spite of such radically not classic feature, the so-called“objective reality” described by the quantum world has not to be ques-tioned at all; it is precisely the peculiar nature of the quantum objectswhich imposes a new relationship with measurements on the theory andleads to a far different notion of event than the classical one.

4.2. The Collapse Postulate

When we consider the “collapse postulate” of the wave function, a radi-cal asymmetry will be unavoidable. In this case, we have to consider onthe one hand the temporal evolution of the wave function U, provided bythe rigorously causal, deterministic and time-reversal Schrodinger equation,and on the other the reduction processes of the state vector R, where themeasurement’s outcome is macroscopically fixed, such as the position of anelectron on a photographic plate. The processes R are not-causal and timeasymmetrical.

Different standpoints are possible about the role of the processes Rin QM, it recalls the 1800s mathematical debate about the structure ofEuclidean geometry and in particular the fifth postulate position. Withoutany claim to exhaustiveness, we can individuate here three main standpointsabout R:

A) The wave function contains the available information on the physicalworld in probabilistic form; the wave function is not referred to an“objective reality”, but — due to the intrinsically relational features ofthe theory — only to what we can say about reality. Consequently, the“collapse postulate” is simply an expression of our peculiar knowledgeof the world of quantum objects;

B) The wave function describes what actually happens in the physicalworld and its probabilistic nature derives from our perspective of ob-servers; R, as well as the entire QM, is the consequence of the fact thatthe most part of the needed knowledge is structurally unavailable;

C) The wave function partially describes what happens in the physical pro-cesses; in order to comprehend its probabilistic nature and the postulateR in particular, we need a theory connecting U and R.

The above three standpoints are subtly different and yet connected.To clarify these interrelations we will give some examples. (A) reflects thetraditional Copenhagen interpretation. We are not interested here in the

10 Ignazio Licata

old ontological debate about reality, but rather we are going to focus onthe formal-logical structure of each theory. According to the supportersof (A) — or at least one of its manifold forms — QM is coherent andcomplete, and the collapse postulate assures the connection between Uand the observations, even if this connection cannot be deduced by U.It means that in order to provide a quantum event, the “Von Neumannchain” breaks at the observer level. The price to pay for such readingsof the theory is that the non-local correlations come out as a compromisebetween causality (conservation laws and Relativity) and quantum casuality(R irreducibility).

In the class (B) quite different interpretations are taken into considera-tion, such as the Everett Theory on the relative states (DeWitt & Graham,1973; Deutsch, 1985) and the Bohm–Hiley theory of Implicate Order (Bohm& Hiley, 1995). The Everett Many Worlds Theory regards U as the descrip-tion of the splitting of the branches in the multiverse (Deutsch, 1998), andR simply as the observer’s act of classically detecting its own belonging toone of the branches of the quantum multiverse by the measurement. TheEverett Theory can be considered to lie at the bottom of the current re-lational approaches as well as, indirectly, the so-called decoherent historiesand, in general, many of the quantum cosmology conceptions (Gell–Mann& Hartle, 1993; Omnes, 1994). Unfortunately, the idea of the multiversedoes not solve all the R-related problems. The real problem is to clar-ify why the universes interfere one with the other and show entanglementwith respect to any observer at a given scale. So the idea to reduce thequantum superposition to a classical collection of observers fails. That iswhy, in the decoherent histories, the problem shifts to the emergence — acoarse-grained one, at least — of classicality from quantum processes.

For our aims it is particularly relevant the recent observation that a co-herent theory of multiverse implies the existence of objects carrying notclassical information, such as the ghost-spinors (Palesheva, 2001; 2002;Guts, 2002). Deutsch and Hayden have shown that non-locality can beregarded as locally inaccessible information and such conception can clar-ify the decoherence processes (Deutsch and Hayden, 2000; Hewitt–Horsman& Vedral, 2007). In Bohm–Hiley Theory — which has to be kept separatefrom the minimalist and mechanic reading often made (Berndl et al., 1995)the radical not-classical aspects are likewise considered as a kind of in-formation which is unknown in the classical conception and linked to thefundamentally not-local structure of the quantum world. The irreducibilityof R is thus the consequence of the introduction of quantities depending


on the quantum potential and representing not casual events, but rather“potentialities” in a formally very strict sense.

The class (C) instead includes all those theories which tend to reconcileU with R by introducing new physical process, the so-called OR (Objec-tive Reduction) (see for ex. Longtin & Mattuck, 1984; Karolyhazy 1974;1986; Pearle, 1989; Ghirardi et al., 1986; Diosi, 1989; Percival, 1995; Pen-rose, 1996a). The OR theories focus on the relationship between quantumobjects and classical, macroscopic measurement apparatuses (i.e. made upof a number of particles equal to at least 1023). The structure of U is modi-fied by introducing new terms and parameters able to obtain a spontaneouslocalization. The current state of these theories is very complex. In theseapproaches some conflicts with Relativity and suggestions for the quantumgravity stay side by side, so making difficult to say what is ad hoc and whatwill turn out to be fecund. Nevertheless, an acknowledgement is due to thetheories of class (C) because they identify the asymmetry between U and Ras a radical emergence within the structure of physics and it is a demand fornew theoretical proposals able to comprehend the border between quantumand classical systems. However, we have to say that the theories of sponta-neous localization do not introduce any new element useful to understandthe non-local correlations, but they make the question much more intricate.In fact, if we consider an EPR-Bell experiment with quantum erasing, itis difficult to reconcile it with the thermodynamic irreversibility implicit inthe localization concept (see Yoon-Ho et al., 2000; and also Callaghan &Hiley, 2006a ; 2006b).

Obviously, the standpoints on the singular role of R in the axioms of QMare much more elaborate, and there are also theories with intermediate fea-tures with respect to the above mentioned three classes. In the Cini–Servatheory of correspondence between QM and classical statistical mechanics,for instance, the measurement effect is just to reduce the statistical ensem-ble and consequently our uncertainty on the system’s conditions as well. Itis thus eliminated a physically meaningless superposition so as to find theminimum value of the wave packet compatible with quantum indeterminacy(Cini & Serva, 1990; 1992). This theory comprises some typical features ofboth the class (B) “realism” — yet rejecting its analysis about the natureof the “hidden” information — and the class (C), since it is centered onthe localization problem at the edge between micro and macro physics,but it does not introduce any new reduction mechanism. Finally – like inthe standard interpretation of the class (A) – , in the Cini–Serva theory,the not-separability aspects of quantum correlations are regarded not as a

12 Ignazio Licata

background dynamics, but as being due to the intrinsical casual characterof quantum behaviors. The naıve (i.e. pre-Bell) theory of hidden variablescan be considered as the precursor of the class (B) philosophy, even if theidea of restoring the classical mechanical features has been replaced todayby a not-mechanical conception in Bohm sense (Bohm, 1951).

4.3. Quantum Observers and Non-local Information

Let’s try now to focus on the formal facets of QM with respect to theobserver. In the quantum context it is impossible for the observer a biu-nivocal correspondence between the notion of system state, described bythe state vector, and the value assignment to the state variables at anyinstant; it imposes to give up the indifference principle of the system statewith respect to the observer, typical of the classical systems, and it leadsto the breakdown of the causal, local determinism. In QM formalism, suchaspect is expressed by the fact that a closure relation is only valid forthe eigenvalues (see for ex. Heylighen, 1990). So we can naturally charac-terize the quantum systems as informational open systems with respect tothe observer. Now the problem is what meaning we have to give to suchlogical openness. To be more precise, if we want both no modification inthe formalism and a broader comprehension of the QM non-local features,we have to focus our attention on the (B)-type theories and to define asuitable non-local and classically irreducible information able at the sameto extend the concept of intrinsic computation to quantum systems, too.Once again, computation theory shows to be very useful in characterizingthe physical systems. In fact, considering the irreducibility of an outcomeR to the evolutionary structure of U, we can state that it is impossibleto apply an algorithmic causal structure to two quantum events relativeto an observer O at different times. We can so conclude that a quantumsystem is a system where the quantum events cannot be correlated one withthe other by means of a Shannon-Turing-type computational model. Thisproves the intrinsically “classic” roots of the computation concept. We willsee in the following that the Bohm-Hiley active information is a really use-ful approach. But first we have to briefly review the emergent processes inQuantum Field Theory (QFT).

5. Emergence in Quantum Field Theory (QFT)

The idea according to which the QFT distinctive processes are those ex-hibiting intrinsic emergence and not mere detection of patterns is widely


accepted by the community of physicists by now (Anderson and Stein, 1985;Umezawa, 1993; Pessa, 2006; Vitiello, 2001; 2002). The central idea is thatin quantum systems with infinite states, as it is known, different not uni-tarily equivalent representations of the same system, and thus phase tran-sitions structurally modifying the system, are possible. This takes placeby means of spontaneous symmetry breaking (SSB), i.e. the process whichchanges all the fundamental states compatible with a given energy value.Generally, when a variation of a suitable parameter occurs, the system willswitch to one of the possible fundamental states, so breaking the sym-metry. This causes a balance manifesting itself as long-range correlationsassociated with the Goldstone–Higgs bosons which stabilize the new con-figuration. The states of bosonic condensation are, in every respect, formsof the system’s macroscopic coherence, and they are peculiar of the quan-tum statistics, formally depending on the indistinguishability of states withrespect to the observer. The new phase of the system requires a new de-scriptive level to give account for its behaviors, and we can so speak ofintrinsic emergence. Many behaviors of great physical interest such as thephonons in crystal, the Cooper pairs, the Higgs mechanism and the mul-tiple vacuum states, the inflation and the “cosmic landscape” formationin quantum cosmology can be included within the SSB processes. It is soreasonable to suppose that the fundamental processes for the formation ofstructures essentially and crucially depend on SSB.

On the other hand, the formal model of SSB appears problematic inmany respects when we try to apply it to systems with a finite numberof freedom degrees. Considering the neural networks as an approximatemodel of QFT is an interesting approach (Pessa & Vitiello, 1999; 2004).Thegeneralized use of the QFT formalism as a theory of emergent processesrequires new hypotheses about the system/environment interface. The mostambitious challenge for this formalism is surely the Quantum Brain Theory(Ricciardi & Umezawa, 1967; Vitiello, 2001).

A formally interesting aspect of QFT is that it allows, within limits, toframe the question of reductionism in a clear and not banal way, i.e. to givea clear significance to the relations among different descriptive levels. Infact, if we define the phenomenology linked to a given range of energy andmasses as the descriptive level, it will be possible to use the renormalizationgroup (RG) as a tool of resolution to pass from a level to another by meansof a variation of the group’s parameters. We thus obtain a succession ofdescriptive levels — a tower of Effective Field Theory (EFT) — having afixed cut-off each, able to grasp the peculiar aspects of the investigated

14 Ignazio Licata

level (Lesne, 1998; Cao, 1997; Cao & Schweber, 1993). In this way eachlevel is connected to the other ones by a rescaling of the kind Λ0 → Λ (σ) =σΛ0, where Λ0 is the cut-off parameter relative to the fixed scale of theenergies/masses into play. The universality of the SSB mechanism is deeplylinked to such aspect, and the possibility to use the QFT formalism asa general theory of emergence puts the problem of extending the EFT“matryoshka-like” structures allowed by the RG to different systems.

In what sense can the intrinsic emergence of SSB be compared to thephenomenological detection of patterns and which are the radically quan-tum features in the same sense we pointed out for ordinary QM? Also in thecase of the SSB processes, the phase transition is led by an order parametertowards a globally predictable state, i.e. we know that there exists a valuebeyond which the system will reach a new state and will exhibit macro-scopic correlations. Once again a prominent role is played by the boundaryconditions (all in all, a phonon is the dynamic emergence occurring within acrystal lattice, but it is meaningless out of the lattice). Moreover, in the SSBthere exists a transient phase whose description is widely classic. Where theanalogy falls down and we can really speak of an irreducibly not-classic fea-ture is in bosonic condensation which is a non-local phenomenon. While ina classical dissipative system is possible, in principle, to obtain informationabout the “fine details” of a bifurcation and to know where “the ball willfall”, in a process of SSB it is impossible for reasons connected to the natureof the quantum roulette! In this sense the radical features of emergence inQFT and in QM are of the same nature and they demand a new informationtheory able to take into account the non-local aspects.

6. Quantum Information from the Structure of QuantumPhase Spaces

The concept of active information has been developed by Bohm and Hileyand the Birbeck College group within a wide research programme aimedat comprehending the QM non-local features (see for ex. Bohm & Hiley,1995; Hiley, 1991; Hiley et al., 2002; Monk & Hiley, 1993; 1998; Brown &Hiley, 2004). This theory is formally equivalent to the standard one anddoes not introduce any additional hypothesis. The features often describedas “Bohmian mechanics” and aimed at a “classic” visualization of the tra-jectories of quantum objects are not essential. The theory rather tends tograsp the essentially not-mechanic nature of quantum processes (Bohm,1951), in the sense of a topology of not-separability quite close, in for-malism and spirit, to non-commutative geometries (Demeret et al., 1997;


Bigatti, 1998). It is known that the theory first started from splitting theSchrodinger ordinary equation in real and imaginary part in order to ob-tain the quantum potential Q (r, t) = − (

~2/2m

) (∇2R (r, t)/R (r, t)

). This

potential QP, unknown in classical physics, contains in nuce the QM non-local features and individuates an infinite set of phase paths; in particular,the QP gets a contextual nature, that is to say it brings global informationabout the quantum system and its environment. We underline that suchnotion is absolutely general and can be naturally connected to the Feynmanpath integrals. Our aim is to briefly delineate the geometrical aspects of thequantum phase spaces from which the QP draws its physical significance.

Let us consider an operator O and the wave function ψ (O). By usingthe polar form of the wave function we can write the usual equations ofprobability and energy conservation in operatorial form:

idρ

dt+ [ρ,H]− = 0; ρ

dS

dt+

12

[ρ,H]+ = 0, (1)

where ρ = Ψ∗ (O)〉〈Ψ(O) is the density operator which makes availablethe entropy S of the system as S = trρ ln ρ, i.e. as the maximum obtainableinformation from the system by means of a complete set of observations(Aharonov & Anandan, 1998). Now let us choose a x-representation for the(1) and we will get:

∂ρ

∂t+∇rj = 0;

∂S

∂t+

(∇rS)2

2m+ Q (r, t) + V (r, t) = 0. (2)

Instead, in a p-representation the (1) take the form:

∂ρ

∂t+∇pj = 0;

∂S

∂t+

p2

2m+ Q (p, t) + V (∇pS, t) = 0. (3)

From the probability conservation in (2) and (3), two sets of trajectories inthe phase space can be finally derived, with both R and Re for real:

∇rS = Re [Ψ∗ (r, t)PΨ(r, t)] = pR;∇pS = Re [Φ∗ (p, t)XΦ(p, t)] = xR.

(4)The conceptual fundamental point is that we obtain the quantum po-

tential only when we have chosen a representation for the (1). Such a con-struction is made necessary by the non-commutative structure of the phasespaces of the conjugate variables, and it implies that the observer has tofollow a precise process of information extracting via the procedure statepreparation→state selection→measurement. So the quantum potential canbe regarded as the measure of the active information extracted from what

16 Ignazio Licata

Bohm and Hiley have called implicate order to the information preparedfor the observation in the explicate order. The process algebra (SymplecticClifford algebra) between implicate and explicate order is thus a dynamicsof quantum information.

It is worth noticing that in the (4), given a representation, the conjugatevariable is a “beable” variable (Bell, 1987), i.e. a construction depending onthe choice made by the observer. Such choice is in no way “subjective”, butit is deeply connected with the phase space structure in QM. The Bohmand Hiley’s reading restores the natural symplectic symmetry between x

and p, but, unlike the classical case, it does so by geometrically justifyingthe complementarity notion.

To better understand the dynamics of quantum information and its rela-tion with Shannon–Turing classical information, let us consider the notionsof both Implicate and Explicate Order and enfolded and unfolded informa-tion. (Licata & Morikawa, 2007).

For our aims, it will suffice here to define the Implicate Order as thenon-commutative structure of the conjugate variables of the quantum phasespace. Therefore, it is impossible to express such structure in terms of space-time without making a choice within the phase space beforehand, so fixingan explicate order. Let us designate the implicate order with E and theexplicate order with E’; we will say that E is a source of enfolded informa-tion, whereas E’ contains the unfolded information extracted from E. Therelation between the two orders is given by:

E′ (t) =∫

implicate

G (t, τ)E (τ) dτ, (5)

where G (t, τ) is the Green’s Function and τ is the unfolding parameter. Theinverse operation, from the explicit space-time structure to the implicateorder, is so given by:

E (τ) =∫

exp licate

G (τ, t)E′(t)dt. (6)

The passage from (5) to (6) — and vice versa — has a simple physicalsignificance. The unfolding corresponds to the state selection, and the en-folding to the state preparation. Let us consider a typical two-state system,for ex. a spin 1/2 particle:

|Ψ〉 = a |↑〉+ b |↓〉 . (7)

Here a and b are a measure of the active information extracted from theimplicate order by choosing the state variable. It has to be noticed that the


state preparation itself , as well as the measurement, modifies the systemcontextual information, so defining a new relationship between the back-ground order E (τ) and the foreground one E′ (t). In other words, beingchosen a variable, the information of the previous explicate order vanishesinto the implicate order.

During the measurement, information becomes enactive, that is to saythat the information contained in the superposition state (7) is destroyed,thus selecting between a and b via an artificial unfolding. Here the term“artificial” means that the measurement — or the “collapse” — has not aprivileged position within the theory, but it is only one of the ways whichthe configuration of the system active information can change into. Forexample, a creation and annihilation process of particles can be consideredas a “spontaneous” process of unfolding/enfolding. What is really importantis the relationship between the contextuality of active information and thenon-commutativity of the phase space. We can sum up all this by sayingthat in Bohm–Hiley theory a quantum event is the expression of a deeperquantum process connecting the description in terms of space and time withthe intrinsic non-local one of QM. We can say that the implicate order is arealization of the Wheeler “pre-space” (Wheeler, 1980).

The quantum potential, as well as the appearing of “trajectories” in the(4), in no way restores the classic view. In fact, the algebraic structure ofQM clearly tells us that both the sets of trajectories are necessary to com-prehend the quantum processes, and describing them in the explicate orderimplies a complementarity, and consequently a structural loss of informa-tion. Any pretence about the centrality of x-representation is arbitrary andit is only based on a classical prejudice, i.e. the “position” regarded as the“existence”. In no way a quantum system has to be considered as less “real”and “objective” than a classical one. It is the observer role which changesjust in relation to the peculiar nature of quantum processes. So the trajec-tories have not to be regarded as “mysterious” or “surreal” lines of forceviolating Relativity, but as informational configurations showing the intrin-sic not-separability of the quantum world in the space-time foreground.

From what we have said above, it clearly appears that the problem of theemergence of the classical world from the quantum one cannot be dealt withby using a classical limit for the quantum potential, despite the statisticalinterest of such pragmatic approach (see for ex. Allori & Zanghı, 2001).The limit of classicality has rather to be faced as a problem of relationshipsbetween algebra and metric in the sense of the principle of reconstructionof Gel’fand (Demaret et al., 1997). Another significant point, which we can

18 Ignazio Licata

only mention here, is the importance of the unfolding parameter in quantumcosmology, for example in passing from a timeless quantum De Sitter-likeuniverse to a post inflationary time evolution (Callander & Weingard, 1996;Nelson Pinto-Neto, 2000; Lemos & Monerat, 2002; Licata, 2006; Chiatti &Licata, 2007).

The Shannon–Turing information comes into play when a syntactic,local analysis of the system is possible on a well-defined channel. In QM,it is possible only when a system has been prepared in a set of orthogonalwave functions, thus making a selection of the enfolded information whichcan be analyzed in terms of usual quantum bits. Elsewhere we have pointedout how such crucial difference between contextual active information andq-bit information can be used to explain forms of quantum computationmuch more powerful than the one based on quantum gates (Licata, 2007).

A different and more immediate way to understand the limits of theclassical computation theory in QM is to take into consideration again thetrajectories in the (4). As the two sets are complementary, the thought ex-periment of “rewinding” the trajectories of an unmeasured quantum systemis structurally unable to provide information about the details of evolution.In fact, let us suppose we have chosen, for example, a family of trajectoriesin x-representation and we — yet ideally — have made a computationalanalysis on it, all the same we can say nothing on the computation of thetrajectories in the p-representation: just like the variable on which it iscentred on, the computation on p-representation appears to be a “beablecomputation”! It brings out a deep connection between non-commutativity,non-locality and the new kind of quantum time-asymmetry related to thefact that the logic of the process of unfolding/enfolding described by (5)and (6) — that is to say τ → t and vice versa — implies an uncomputablemodification of the system information.

We can conclude that the features of informational openness of QMare a direct consequence of the non-commutative algebra which imposesupon the observer to make a choice on the available information and itprevents from describing the intrinsic computation of a quantum systemsvia a Shannon–Turing model because of the contextual nature of the activeinformation.

7. On the Relationships Between Physics and Computation

In the last years an important debate on the relationships between phys-ical systems, formal models and computation has been developed. Inour analysis, we have used the classical computation theory not only as


simulation means, but rather as a conceptual tool able to let us understandthe formal relationships between the system’s behaviours and the informa-tion that the observer can get about it. We have seen that it is possibleto give a computational description of the classical systems centred on thepossibility to be always able to identify the information in local way. So,emergence in classical systems is fundamentally of computational kind andthere exists a strict correspondence between classical systems and compu-tational dynamics (Baas & Emmeche, 1997). This is not in contradictionwith the appearing of non-Turing features in some classical systems (seefor ex. Siegelmann, 1998; Calude, 2004) because it has been proved thatsets of interactive Turing Machines (TMs) with oracles can show computa-tional abilities superior than those of a TM in the strict sense (Kieu & Ord,2005; Collins, 2001). Actually, if we look at the “oracles” as metastableconfigurations fixed during the evolution of a system, this is the minimumrequired baggage to understand the biological evolution and mind. QM doesnot seem to be necessary to comprehend the essential features of life andcognition, in contrast to what Penrose claims (Penrose, 1996).

In QM, we deal instead with radical processes of observational emer-gence which cannot be overcome by the construction of a new model. TheBohm–Hiley theory helps us to understand such radical features of QM bythe Implicate/Explicate Order process algebra and the non-commutativestructure of the quantum phase spaces. The failure of a TM-based observerin describing quantum systems and the consequent recourse to a proba-bilistic structure are related to the fact that any observer belongs to theexplicate order and it has to use a space-time structure to build up the con-cept of physical event. From this view, the role of computation in physicsappears to be even more profound than what the Turing Principle sug-gests (Deutsch, 1998). In fact, the Shannon–Turing theory here appears asa computability general constraint on the causal structures which can be de-fined in space-time and as the extreme limit of any description in terms ofspace-time (Markopoulou–Kalamera, 2000a; 2000b; Tegmark, 2007).

Processing information is what all physical systems do. The way to pro-cess information depends on the system’s nature, and it is not surprisingthat the Shannon–Turing information naturally fits to not-quantum sys-tems or, at the utmost, to q-bit systems because it has been conceivedwithin a classical context. In quantum systems instead, the breakdown ofthe classical computation model calls into play a new concept of infor-mation, in the same way as more than once in the history of physics, aconceptual crisis has required new strategies to comprehend the unity of

20 Ignazio Licata

the physical world. The Bohm–Hiley concept of active information allowsto extend the concept of intrinsic computation to quantum systems, too.In order to make such notion efficacious, a great deal of new ideas will benecessary. Already in 1935, Von Neumann wrote in a letter to Birkhoff: “Iwould like to make a confession which may seem immoral: I do not believein Hilbert space anymore.” (Von Neumann, 1935). In fact the separableHilbert spaces are not fit for representing infinite systems far from equilib-rium. It is thus necessary a QFT able, at least, to represent the semigroupsof irreversible transformations — i.e. operators for the absorption and dis-sipation of quanta — by not-separable Hilbert spaces. Here, a new scenarioconnecting non-commutativity, irreversibility and information emerges.

The “virtuous circle” of systems-models-computation finds its ultimatesignificance not only in the banal fact that we build models of the physicalworld so as to be “manageable”, but also because we are constrained to builtsuch models based on family of observers in the explicate order. It suggeststhat an authentic “pacific co-existence” between Relativity and QM willrequire a general mechanism to obtain space-time transformation groupsvia an unfolding parameter as a boost from a more complex underlyingbackground structure. In this context, the hyperspherical group approachappears to be interesting (see Licata, 2006; Chiatti & Licata, 2007).

The physical world has no need for observers for its structure andevolution. But the nature of the quantum processes makes the relation-ship between the observer and the observed irreducibly participatory, suchthat the description of any physical system is necessarily influenced byunbounded and contextual information. Such kind of information is notcooped up in the “Hilbert cage” and has less to do with what is usuallymeant by quantum computing. Far from being a “shadow” information, itfixes the fundamental non-local character of the quantum world and thelimit of Shannon–Turing information in describing the intrinsic computa-tion of quantum systems. In contrast, it is just this radical uncomputablefeature which individuates the Boolean characteristics of the observers inthe space-time and allows, thanks to the spontaneous symmetry breakingprocesses, the increasing complexity of the physical universe.

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27

Gauge Generalized Principle for Complex Systems

Germano Resconi

Catholic University, Via Trieste,17, Brescia, Italy

[email protected]

In this paper we want to extend the traditional idea of the gauge to differentparts of the physics. And also to complex system of agents and also to thebiological systems. After the introduction of the fundamental definitions of theGauge Generalized Principle , we move to the different domain as physics,agents , symmetry and so on to show the applicability of the gauge generalprinciple.

Keywords: Complex Systems; Biological Systems; Gauge Generalized PrinciplePACS(2006): 89.75.k; 89.75.Fb; 87.10.+e; 11.10.z; 11.15.q

1. Compensation as a Prototypical Cognitive Operation

The first step toward an understanding of the general properties of the cog-

nitive process “compensation” requires taking into consideration the mental

path which leads from an explicit description of events taking place in the

surrounding environment to an implicit description of events themselves,

realised through the introduction of some invariance principle. To this re-

gard , the conceptual structure, and the history itself, of classical mechanics

is very instructive. As it is well known, the explicit description of motion

of, let us say, a point particle, is given by Newton’s dynamic equations:

F

m=d2x

dt2(1)

That is the invariant form for the transformation

[

1 −v0 1

] [

x

ct

]

=

[

x′

ct′

]

.

For the Lorenz transformation 1√1−β2

[

1 −β−β 1

] [

x

ct

]

=

[

x′

ct′

]

, β = vc .

The (1) is false and

d2x

dt2is different from

F

m.

28 Germano Resconi

Now we want to restore the original invariant form (1) also when the velocity

is near to the speed of light ( Lorenz transformation ) . To obtain this result

we “compensate” the mechanical system by the change of the definition of

the derivative operator in this way

D =d

dτ=

d

dt+ [

1 −√

1 − β2

√

1 − β2]

d

dt=

1√

1 − β2

d

dt.

So we have D2 = ddτ ( ddτ ) = 1√

1−β2

ddt (

1√1−β2

ddt ) and the equation

F

m= D2x =

1√

1 − β2

d

dt(

1√

1 − β2

d

dt)x (2)

is the relativistic form of the Newton equation. We remark that formally

the (2) is equal to the (1). So the (2) is a “compensate” expression of

(1). The (2) formally is the copy of (1). With the compensate principle

we can use the Newton equation as a prototype theory for the relativistic

phenomena. In conclusion from ordinary mechanics we can generate the

relativistic mechanics without loss the previous knowledge but with use of

the suitable compensation for which rules in classical mechanics is again

true in the relativistic mechanics.

Now we know that Newton’s equation (1) can be easily cast in the form

of a conservative principle, such as

∑

k

m

2(dxk

dt)2

+ U = const (3)

provided that we suppose the existence of a function U such that

Fi = − ∂U

∂ xi. (4)

Even if , from a formal point of view, (4) is perfectly equivalent to (1), its

meaning and role in mechanics is deeper. Namely, by assuming as starting

point a conservative principle like (4), we don’t obtain, as consequence, a

single particular system of dynamic equation (1), but a whole class of such

systems. Besides, it is easy to think of a generalisation of (4), by introduc-

ing a Hamiltonian function, which lets us deal with a class of mechanical

phenomena much wider the one which could be studied by reasoning only

to original Newton’s equations (1).

In conclusion the (3) is true for a set of trajectories obtained by a set

of homomorphism transformations C. The coherence condition in (3) does

not take into consideration possible incoherence in the Newton low given


by (1), but more general incoherence related to the connections among all

possible trajectories.

We remember that for the relativistic case the Newton law is not true,

but as we know we can again write the (3), the conservative law exists

also for the relativity dynamics. In turn, this generalisation let us, via the

methods of Calculus of Variations, to discover more powerful Extremum

Principle lying behind the whole apparatus of analytical mechanics, such

as the celebrated least action Hamilton Principle. In the Conservative Me-

chanical Systems the canonical transformations change one trajectory to

another trajectory that is again solution of the differential equation that

represents the conservative condition, Hamiltonian differential equation. So

if we consider natural all the trajectories that are solution of the Hamil-

tonian equations, conservative system, the canonical transformation moves

from one natural transformation to another natural transformation. The

mechanical system is always close because we cannot consider any type of

trajectories but only the natural trajectories. In the Extremum Principle

we open the mechanical system to no natural mechanical trajectories. In

this way all possible geometric transformation in the space-time can be con-

sidered. The Extremum Principle introduces another new transformation

that changes the general transformation in the canonical transformation .

In the previous cases It has to be underlined that here we deal with two

different types of invariance: one is connected to the numerical value of a

particular function such as, in the case of mechanics, the Hamiltonian one,

and another is connected to the so called “form invariance” of the func-

tion appearing in the extremum principle ( in the case of mechanics, it is

the Lagrangian action function). To each type of invariance is associated a

particular type of transformations: these latter are the canonical transfor-

mations when we deal with the numerical value of the Hamiltonian, and

generic global space time or dependent variable transformations, when we

deal with form invariance of the action. Besides, to each type of invariance

is associated a particular equivalence relation between different mechani-

cal systems. In conclusion the fusion of the concept of Invariance and the

Calculus of Variations generate a lot of new types of incoherence and also

many different types of compensation.

From the Newton law, to the Invariance and the Calculus of Variation

there is a continuous movement of increase of generality and at the same

time the logic structure of coherence and compensation become more and

more complex. To describe completely all the logic structure from New-

ton to the Calculus of Variation it is necessary a logic structure of five

30 Germano Resconi

dimensions. The Calculus of Variation is at so high level that we can create

the general relativity and also the quantum mechanics by this mathematical

instrument.

In this way we enlarged the framework originally adopted by Newton.

Namely, in this latter case we had only one possible type of transformations,

those given by the application of the time evolution operator to the initial

data. In analytical mechanics, instead, we have two more transformations

available on a same particular mechanical system: the one corresponding

to a representational change as regards canonical variables, and the one

corresponding to a reference frame change, or to a change of dependent

variables.

We must ask ourselves why the mental path which goes from dynamic

equation (1) to Hamilton’s Principle was so advantageous for the progress

of science knowledge that it is considered the prototype itself of general-

ization processes introduced in every domain of theoretical physics. The

answer is that the existence of an Extremum Principle, besides the realiza-

tion of an economy of thinking ( the implicit description it gives is much

more compact than the explicit one offered by dynamical equations), gives

us the possibility of reasoning about phenomena at higher level, where

the details of the particular functional expressions appearing in (1) become

irrelevant. In other words, we can discuss about general features of mechan-

ical systems, prove universal theorems regarding them, being at the same

time able to derive, through Euler–Lagrange equations, the consequences

of these theorems within particular mechanical systems.

2. Sources as Compensate Term for Change of Fields

At first sight, the logical structure of mechanics we sketched in the previous

paragraph appears as the best one to capture both general and particular

feature of dynamical behaviours. For this reason it could be seen as the

prototype of every theoretical construction designed to link together both

abstract and concrete aspects of phenomena taking place in a given domain.

The only problems to be solved, in order to grant for an efficient operation

of such a structure, seem to be of technical mathematical nature. However,

behind this apparent perfection a serious drawback is hidden: the need for

the introduction of external sources, as an indispensable tool for forecasting

the details of dynamical evolution of mechanical systems.

As regards traditional classical mechanics, where the only macroscopic

field is substantially the gravitational field, without the introduction of

sources we would obtain trivial models, unable to account for experimental


data, such as the one e.g. relative to planetary motion. The introduction by

hand of suitable external sources is nothing but a process of adaptation of

models to the state of fact observed in the external world. In other words,

it is a measure of our ignorance in forecasting all possible phenomena from

a global point of view when we lack for an explanation of an observed

dynamical trajectory, we place an external source in a suitable space-time

region to give us the possibility of reconciling the structure of our model

with what we observed. For this reason the introduction of external sources

into a theory, such a classical mechanics, cannot be considered as a true

process of “compensation” for our ignorance, or lack of information, about

the system we are studying, because it is not connected to some well-defined

transformation between our old model and a new model. On the contrary,

the introduction of external sources is nothing but an “adaptation”. Which

leads to a model logically different from the one without external sources.

Besides, this introduction is done in an essentially local way, without taking

into account the true nature of the model and its transformations.

To explain the “compensate” meaning of the external sources, we refer

to M. Jessel before (1963) and after G. Resconi (1986) who have discovered

the possibility to use the sources as adaptation at the real worlds. Given

the equation of the mechanical waves

∂2ϕ

∂x2−

1

c2∂2ϕ

∂t2= S (5)

where c is the velocity of the mechanical waves, ϕ(x,t) is the field of the

waves and S are the sources that generate the field. The wave equation (5)

can be decomposed in two differential equations of the first order in this

way

c−2 ∂ϕ∂ t + ρ ∂η

∂ x = Q

ρ ∂η∂ t + ∂ϕ∂ x = F

(6)

where η is an auxiliary function, Q is a monopole source and F a dipole

source (Jessel 1973) It is easy to prove that

S = ∂xF − ∂tQ. (7)

Now we introduce the transformation T in the invariant form (4) or (5). So

we obtain the new form

c−2 ∂ (Tϕ)

∂ t + ρ∂ (Tη)∂ x = TQ

ρ∂ (Tη)∂ t +

∂ (Tϕ)∂ x = TF.

(8)

32 Germano Resconi

Now because the (8) can be written in this way(

c−2 ϕ∂T∂ t + ρ η ∂T∂ x + T (c−2 ∂ϕ

∂ t + ρ ∂η∂ x) = TQ

ρ η ∂T∂ t + ϕ∂ T

∂ x + T (ρ ∂η∂ t + ∂ ϕ

∂ x) = TF

the (8) is true for all the transformations T where (9) is true(

c−2ϕ ∂ T∂ t + ρ η ∂ T

∂ x = 0

ρ η ∂ T∂ t + ϕ ∂ T

∂ x = 0(9)

when we multiply the first equation by ∂ T∂t and the second by ∂ T

∂ x we

obtain from the (9) the condition

(∂ T

∂ x)2 −

1

c2(∂ T

∂ t)2 = 0. (10)

The differential equation (10), as we know gives the characteristic func-

tion of the wave equation, The field (ϕ ,η) does not change its structure

after the transformation T.

The transformation T for which (9) is true changes one solution of (5)

in another solution of (5). So for T the fields are inside the same set of

solutions where (5) is invariant.

When (9) is not true to restore the invariance form (5) we change the

sources in a way that

∇2Tϕ =∂2Tϕ

∂x2− 1

c2∂2Tϕ

∂t2= T∇2ϕ+ SC = S + SC . (11)

We will introduce the word “compensation” to denote every cognitive

operation able to use an old model in a new domain, by maintaining the

previous formal structure, except for some compensating terms. So we use

the compensation mechanism to set up the fracture between two domains

of research.

3. Quantum Mechanics and Compensation by Gauge

Transformation

Let us consider the simplest case in which each structure is associated to

only one operator, denoted by T in the first structure and by T ′ in the

second structure. Besides, let us denote by x a generic element belonging

to the first structure and by ϕ the law of biunivocal correspondence between

the objects of the two structures. Then, the two structures are isomorphous

if Tx is the object corresponding, through ϕ , just to T ′(ϕ x ). Therefore, an

isomorphism between the two structures exists if and only if the following

relation is satisfied (see Figure 1):


ϕ (Tx) = T ′(ϕ x ) ,

in turn implying:

T ′ = ϕ T ϕ −1 .

Fig. 1. Graphical representation of an isomorphism.

As regards gauge transformations, the simplest (abelian) case assumes

the form: ψ’ (x, t) = ψ (x, t) exp ( i ϕ ) where the symbols ψ (x, t ) and

ψ’ (x, t ) denote, respectively, a (scalar) field and its gauge-transformed

version, whereas ϕ is a space-time dependent phase and i denotes the

imaginary unit. The importance of the gauge transformations case stems

from the fact that, if the phase is space-time dependent, the compensa-

tion mechanism imposing invariance (isomorphism) with respect to gauge

transformations gives rise to the introduction of the concept of force and,

most generally, of interaction. In other words, the presence of an interaction

compensates for the changes induced by a gauge transformation.

A simple example is related to the behavior of a system in Quantum

Mechanics, which is described through a wave function ψ, which is a solution

for a dynamical equation prescribing its space-time evolution: the celebrated

Schrodinger equation for a single particle (h = 1),

i∂ψ

∂t=

(

1

2m

)

(−i )2∑

j∂2j ψ + Uψ,

The wave function carries no physical meaning per se: only its square modu-

lus is associated to a measurable quantity, interpreted as a probability. This

implies that the wave function can be represented by a complex scalar, func-

tion of space-time coordinates, which has the form ψ = ρ eiθ, where ρ and θ

are, respectively, its modulus and its phase. As only ρ has a physical mean-

ing, we expect that the quantum-mechanical description of phenomena is

invariant with respect to phase transformations, leaving ρ unaffected, and

34 Germano Resconi

defined by ψ’ = ψ eiϕ. What happens if we let ϕ depend on space-time

coordinates? It is easy to see that we will obtain a new equation no longer

invariant with respect to phase transformations, that is:

i∂ψ∂t = e(

∂ϕ∂t

)

ψ =

=(

12m)

(−i )2

∑

j ∂2j ψ − 2ie

∑

j (∂j ϕ)∂j ψ +

−[

ie∑

j ∂2j ϕ+ e2

∑

j (∂jϕ)2]

ψ + Uψ.

However, the difference between the two equations will disappear when we

deal with a particle interacting with an electromagnetic field. In this case

the Schrodinger equation takes the form

i∂ψ′

∂t=

(

1

2m

)

∑

j

DjDj ψ′ + (eφ′ + U)ψ′,

where

Dj ψ′ = −i∂j ψ′ − eA′

j ψ′

in which A′

j is the usual 3-dimensional vector potential and φ is the scalar

electric potential. If we define the transformed values of A′

j and φ by:

Aj = A′

j + (∂j ϕ ), φ = φ’ - (∂ϕ/∂ t), (sometimes called gauge trans-

formations of second kind ) we will find that the Schrodinger equation for

the transformed wave function ψ will assume the form:

i∂ψ′

∂t=

(

1

2m

)

∑

i

DjDj ψ + (eφ+ U)ψ,

where

Dj ψ = −i ∂j ψ − eAj ψ.

The generalized derivative Dj coincides with the old derivative of the classi-

cal physics plus a compensating term that is proportional to the electromag-

netic potential. Associated to the local change of the phase, we have a new

interpretation of the four-potential Aj . It appears as a compensating term

whose role is that of keeping unchanged the previous formal Schrodinger

structure. Moreover, the generalized derivative Dj is the analogous of the

covariant derivative ∇λ introduced in General Relativity, and is known as

”gauge-covariant derivative”.

Summarizing the previous arguments, we can conclude that the dynam-

ical evolution equation (that is the Schrodinger equation) of a suitable basic

field (substratum), described by a wave function ψ, is invariant in form with

respect to local gauge transformations of the first kind if and only if it is


subjected to an electromagnetic interaction whose field potential transforms

according to a gauge transformation of the second kind. In other words, this

gauge transformation ”compensates” for the effects of the gauge transfor-

mation of the first kind on the substratum itself. As this ”compensating”

transformation is possible only in presence of an electromagnetic field, we

can say that this latter acts as a ”compensating” field.

We thus have two equivalent representations of the dynamical evolu-

tions of ψ. Alternatively, we can say that there is no electromagnetic field,

but that the Schrodinger’s equation is not invariant with respect to gauge

transformations of the first kind, or we may say that Schrodinger’s equation

is invariant in form with respect to these latter, but that such an invariance

is due to the presence of an electromagnetic field. Starting from the latter

representation, it is possible to build a general theoretical apparatus for

describing all physical interactions and for formulating unified theories of

these interactions. As it is well known, such an approach was very success-

ful in the last forty years and opened the way to a number of new exciting

discoveries within the domains of particle physics and condensed matter

physics (see, e.g. Frampton, 1987).

4. Compensation of Fields in the Physical Medium

(Coupling Constant)

Despite the successes obtained by such an approach, it needs a suitable

generalization. Namely it can be applied only to fixed-structure systems,

that is to systems in which the number of possible interactions is fixed in

advance and the coupling parameters are to be considered as constants,

determined once and for all, like the electron charge or the gravitational

constant. On the other hand, in many physical and nonphysical complex

systems we need a generalized description able to take into account the

occurrence of a variable number of interactions and of space-time dependent

coupling parameters.

To bear witness to the interest of physicists in this kind of complex

systems, we remind that in recent times, such a description was introduced

in Quantum Field Theory under the name “Third Quantization” (Maslov

and Shvedov, 1998). Whereas the expression “Second Quantization” refers

to a situation in which we allow a nonconservation of particle number,

associated to processes of particle creation or destruction, the expression

36 Germano Resconi

“Third Quantization” refers to a more general situation in which even

the number and the kind of physical fields (that is of possible interactions)

are allowed to vary.

The practical implementation, however, of a Third Quantization proce-

dure is very difficult, owing to the fact that we still lack a description of

its classical counterpart, that is of a classical situation in which the kinds

of physical fields can vary as a consequence of suitable interactions. From

a general point of view, this would require a mathematical approach based

on functionals instead of ordinary functions; however, even if someone tried

to explore such a possibility (cfr. Kataoka and Kaneko, 2000, 2001), it still

appears as implying considerable technical difficulties. To this regard, the

simplest way to describe the “kind” of a physical field is to focus on its

coupling constant with the other fields (e.g. with the substratum field as

in the previous section). If this constant were no longer a constant, but a

space-time dependent quantity, we had a possibility of introducing a simple

description of a kind-variable field.

Inspired by these developments and these considerations we then in-

troduced a generalized gauge transformation: ψ’ = ψ e iqϕ where q is the

(space-time dependent) “charge” of the gauge field (that is its “coupling

constant”) and ϕ is the gauge field itself.

In the particular case in which q is a function only of the “compensating

field” Aµ we can develop the general compensating field in term of a Taylor

series in Aµ, thus obtaining, at the first order, a gauge derivative of the

form:

Dµ = ∂µ∑

ρχµ,ρAρ

From the expression of such a gauge derivative we can obtain a general-

ized field equation describing the evolution of the compensating field Aµ.

Such an equation is based on a number of logical steps (for the details

see Mignani, Pessa & Resconi, 1999) which characterize all gauge theo-

ries which, in a sense, generalize Maxwell’s theory of electromagnetic field.

These steps can shortly be summarized as follows:

(1) definition of a generalized gauge derivative and whence of the field

potentials;

(2) computation of the commutator of the generalized gauge derivatives

defined in the previous step; the obtained result lets us find the rela-

tionship between the field strength and the field potentials;

(3) the Jacobi identity applied to the commutator gives the dynamical

equations satisfied by field strength;


(4) the commutator between the generalized gauge derivative and the pre-

vious commutator (triple commutator) lets us define the field currents

and find the connection between field strength and field currents.

When applying this procedure to our previous definition of generalized

gauge derivative, we obtained, after a number of calculations, that the step

(4) gave rise to the following complicated equation connecting the field

potentials with the field currents:

Aν = −Jν

where

−Jν = [(∂µµχδν)Aδ − (∂µν χ

γµ)Aγ ] + [χδν(∂

µµAδ) − χγµ(∂

µνAγ)]

+[2(∂µχδν)(∂µAδ) − (∂νχγµ)(∂

µAγ) − (∂µχγµ)(∂νAγ)].

From such an equation it is possible to derive a number of consequences,

the most important of which is that, in presence of space-time dependence of

the “charge” (or “coupling constant”) of the gauge field, the compensating

field undergoes a spontaneous spatial decay in crossing the physical medium

in which the field itself is embedded. Such a circumstance follows from the

fact that the previous expression of the field current contains terms which

are directly proportional to the field potentials themselves. This means that

the dynamical equation ruling the behavior of field potentials has solutions

decaying to zero.

Such an effect is analogous to the Meissner effect in superconductors

and shows that in the case of variable-structure systems the variability of

the interactions prevents from a complete transmission of the information

(carried by the gauge field) within the system itself. In other terms, inner

coherence opposes to the propagation of disturbances mediated by the in-

teractions themselves and can preserve the stability of the system itself;

related, in turn, just to the variability of the interactions.

5. Gravitational Maxwell Like Physical System and

Compensation (Dark Matter and Energy)

5.1. Introduction

Cosmic microwave background (CMB) temperature anisotropies have and

will continue to revolutionize our understanding of cosmology. With the

CMB Wayne Hu and Scott Dodelson [1] With the (CMB) had established

a cosmological model for which a critical density universe consisting of

38 Germano Resconi

mainly dark matter and dark energy. In this paper we present two com-

plementary non-conservative gravitational theories. For these theories the

energy and matter dynamically disappear into the dark matter and the dark

energy and vice-versa the dark mater and energy appear as ordinary energy

and matter. So the ordinary energy and matter are not conserved but are

in communication with the dark matter and energy. The new approach to

gravity includes the dark energy and matter as intrinsic part of the theory

itself. The dark matter and energy are quantum gravitation physical phe-

nomena described as cosmological constant. In the Universe the quantum

wave function or the quantum field are not separate from the gravitational

field. The interaction between Gravitation field and Quantum field in the

vacuum is a source of the gravitational field. In the Einstein equations,

the dark matter and energy are introduced by the cosmological constant

as external elements without any theoretical justification. The geometric

interpretation in G. Resconi [2] of the dark matter and energy came from

a pure geometric extension of the Maxwell equations and of the Einstein

equations to the quantum theories. The field approach in Ning Wu [3,4 ]

represents the dark matter and dark energy in the ordinary gravitational

field in a flat space with interaction with quantum field. The G. Resconi

work and the Ning Wu work are gauge theories where the geodetic line is

interpreted in two different ways. In G. Resconi the geodetic line is the same

as in the Einstein theory. In Ning Wu the geodetic line is the trajectory of

a massive body in a flat space as in the Newton theory. Ordinary matter

and energy are elements that include ordinary gravitational phenomena and

dark matter and energy include quantum phenomena. Quantum field and

gravitational field are connected one with the other. Only total matter and

energy that include dark and ordinary are conserved. We can experiment

that ordinary matter disappear into the dark matter and that dark matter

disappear into ordinary matter.

5.2. Maxwell Equation

We know that the Maxwell equations in the tensor form are

Fαβ/γ + Fβγ/α + Fγα/β = 0

Fαβ/β = −4πJα(12)

where

Fαβ = (∂Aβ

∂xα− ∂Aα

∂xβ) and Fαβ/γ =

∂Fαβ

∂xγin the flat space-time


and Aβ is the electromagnetic potential. Using the covariant derivative

notation defined by

Dµ = ∂µ − ieAµ (13)

we have the classic relation

[Dµ, Dν ] ≡ DµDν −DνDµ = −ieFµν . (14)

For the (14) we have

Fµν = −Fνµbecause

Fαβ/βα = F01/10 + F02/20 + F03/30 +

+F10/01 + F12/21 + F13/31 +

+F20/02 + F21/12 + F23/32 +

+F30/03 + F31/13 + F32/23 = 4π(j0/0 + j1/1 + j2/2 + j3/3) = 0.

The divergence of the current is equal to zero. We know that for the ten-

sor form of the Maxwell equations, the Maxwell equations are invariant for

every affine transformation. In particular the Maxwell equations are invari-

ant for the Lorenz transformations. We remember that from pure electro-

static field with a suitable Lorenz transformation, we can obtain a general

electromagnetic field. For more general transformations, the Maxwell equa-

tions are not invariant. The aim of this paper is to rewrite the Maxwell

equations so as to be invariant for any transformations of the space time.

5.3. Maxwell Scheme

To write the extension of the Maxwell equations, we must define the deriva-

tive operator in a general way. If Ω is a general continuous transformation

we have

Dγ = Ω∂γΩ−1 or DγΩ = Ω∂γ (15)

where Dγ is the covariant derivative for the gauge transformation Ω

The expression

DγΩ = Ω∂γ can be written also in this way

Dγ = ∂γ + [Ω, ∂γ ]Ω−1. (16)

When Ω = eie∫

x2

x1Aµdxµ we have for the (16) the ordinary covariant

derivative (13).

40 Germano Resconi

For any type of transformations Ω we assume that the expression (14)

is true. In this case we have

[Dµ, Dν ] ≡ DµDν −DνDµ = −ieFµν (17)

where Fµν is the general form for any field generated by the transformation

Ω.

For the commutative property

[Dµ, [Dη, Dν ]]ψ + [Dη , [Dν , Dµ]]ψ + [Dν , [Dµ, Dη]]ψ = 0 (18)

the (7) can be written in this way

[Dµ, Fην ]ψ + [Dη, Fνµ]ψ + [Dν , Fµη ]ψ = 0 (19)

because

[Dγ , Fαβ ]ψ = [∂γ + [Ω, ∂γ ]Ω−1, Fαβ ]ψ = [∂γ , Fαβ ]ψ + [[Ω, ∂γ ]Ω

−1, Fαβ ]ψ

= (∂Fαβ

∂xγ+ [[Ω, ∂γ ]Ω

−1, Fαβ ])ψ (20)

for ∂Ω∂xγ

= 0 we have [ Ω , ∂∂xγ

] ϕ = 0 and

[Dγ , Fαβ ]ψ =∂Fαβ

∂xγψ (21)

when the covariant derivative is the (13) then

[Dγ , Fαβ ]ψ =∂Fαβ

∂xγψ (22)

and the (8) is the same first equation as in (12). For the (22) the equation

[Dγ , [Dα, Dβ]]ψ = [Dγ , Fαβ ]ψ = χJαβγψ (23)

when (21) and (13) are true , χ = -4π and γ = β. The (23) becomes the

second equation of the (12).

In conclusion the Maxwell Scheme for gauge transformation Ω is

[Dγ , Fαβ ] + [Dα, Fβγ ] + [Dβ, Fγα] = 0

[Dγ , Fαβ ]ψ = χJαβγψ(24)

where

Fαβ = [Dα, Dβ] and Dα = ∂α + [Ω, ∂α]Ω−1 or Dα = Ω∂αΩ−1 (25)

and Jαβγ are the currents of the particles that generate the Gauge Field F.

For the (24) the currents have this conservation rule

Jαβγ + Jβγα + Jγαβ = 0. (26)


5.4. Non-Conservative Equation of Gravity by Maxwell

Scheme

Given the affine transformation Ω

xη → xµ = xµ(xη).

Given the field φ ( Scalar Field ) and xη → ∂φ∂xη

we have

∂φ

∂xη→ ∂φ

∂xµ=∑

η

∂xη

∂xµ∂φ

∂xη= Ω(xη)

∂φ

∂xη.

For the previous gauge transformation Ω we have

DkΩ∂φ

∂xη= Ω∂k

∂φ

∂xη

and

Dk∂φ

∂xµ= ∂k

∂φ

∂xµ+ [Ω, ∂k]Ω

−1 ∂φ

∂xµ= ∂k

∂φ

∂xµ+ Γµkj

∂φ

∂xj

where Γµkj are the Cristoffel symbols and

Fk,h = [Dk, Dh]∂φ

∂xi= R

jihk

∂φ

∂xj

where Rjihk is the Riemann tensor. In this case the Maxwell scheme gives

us the set of equations

[Dγ , Rαβ ] + [Dα, Rβγ ] + [Dβ , Rγα] = 0

[Dγ , Rαβ ]ψ = χJαβγψ

that can be written in this way

Rabjk +Rabhk +Rabkj = 0

DkRabjk +DjR

abhk +DhR

abkj = 0

DhRaijkDµφ+RahjkDaDµφ = χJhjkDµφ.

Where the first equation is the well known Riemann tensor symmetry, the

second is the Bianchi identity and the last is the equation for the Gravity

Field. The current can be given by the form

Jijk = DiTjk −DjTik −1

2(gjkDiT − gikDjT ) for which DkJijk = 0

and Tjkis the energy-momentum tensor. When Dγφ = 0 the equation for

the Gravity Field is DhRih = χJαij for the expression of the current we have

Rij = χ(Tij −1

2gijT )

42 Germano Resconi

that is the Einstein field equations. When Dγφ is different from zero the

field equations are

DhRaijkDaφ = χJhjkDaφ−RahjkDaDaφ.

The new source of gravitational field

Λhij = −RahjkD2

aφDaφ

is a negative source comparable with the cosmological constant.

In the general coordinate the cosmological constant is proportional to

the second derivative of the quantum scalar field φ and is proportional to

the curvature of the space-time.

For the Kline–Gordon equation we have

ηαβ∂αφ∂βφ = m2φ and in the general coordiante ηαβDαφDβφ = m2φ

so the cosmological constant can be written in this way

Λhij = −Rahjkm2φ

( ∂φ∂xa+ Γαβγ

∂φ∂xα

).

For weak and nearly static gravitational field we have

R44 = −∆G

c2and Γα,44 ' 1

2(∂g44

∂xα) with g44 ' 1 − G

c2

and

Λ 'm2φ∆G

c2 + ∂G∂xµ

∂φ∂xη

'm2φ∆G

c2=SQSG

c2

where G is the gravitational field, SQ = m2φ is the source of the quantum

waves and SG are the sources of the scalar gravitational field.

Because

SG ≈ GN ρ = 6.673 10−11 m3 kg−1s−2 3 10−22 kg m−3= 1.8 10 −32 ,

1/c2 ≈ 10−17 s2 m−2.

Where ρ is the density of local halo, we have Λ ≈ 10−49 [m2φ ] that is

comparable with the experimental value of Λ ≈ 2.853 10−51 h−20 m2 where

h0 ≈ 0.71, The cosmological constant is a physical effect of interaction

between the gravitational field and the scalar quantum field. The quantum

gravity phenomena of the cosmological constant are represented by the

black matter and energy.

In [3] we show that the derivative coupling with the quantum field in-

troduces an extra-mass term or black matter in the standard Schwarzschild


metric. The application of such a result to perihelion shift and light deflec-

tion yields results comparable with those obtained in General Relativity. It

is also shown that the non conservative theory of gravity implies a cosmo-

logical model with a locally varying, non zero black matter and energy or

cosmological “constant”.

6. Network of Intelligent Agents and Compensation

6.1. Introduction

Any problem solving can be modeled by actions or methods by which, from

resources or data, one agent makes an action to obtain a result or arrive at

a task. Network of actions can be used as a model of the behavior of the

agents. Any sink in the network is a final goal or task. The other tasks are

only intermediate tasks. Any source in the network is a primitive resource

from which we can begin to obtain results or tasks. Cycles in the network

are self-generated resources from the tasks. Now we denote agent at the first

order any agent that can make one action or can run a method. Now we

argue that there also exist agents at the second and at the more high order.

Agents that copy the agents of the first order are agents of the second order.

To copy one agent of the first order means to copy all the properties of one

agent or part of the properties. This is similar to the offspring for animals.

The network of resources, action, tasks in the new agent has the same prop-

erties or part of the properties of the original network. In the copy process

is possible that the new agent has new properties which are not present

in the prototype agent. This is similar to the genetic process of the cross

over. Agent of the second order uses the prototype agent as reference to

create new agent in which all the properties or part of the properties of the

original agent are present. When all properties of the prototype network

of agents are copied in the new network of agents, we have a symmetry

between the prototype network of agents in the new network. When agents

are permuted in the same network, agents can change their type of activi-

ties without losing the global properties of the network. The properties are

invariant for the copy operation as permutation. We remember that also if

two networks of agents have the same properties they are not equal. When

in the copy process only part of the properties does not change, and new

properties appear, in this case we say that we have a break of symmetry.

For example in the animals in the clone process one cell is generated from

another. The new cell has the same properties of the old cell. In this case

we have symmetry among cells. In fact because any cell is considered as

44 Germano Resconi

a network of internal agents (enzymes) two cells are in a symmetric posi-

tion when the internal network of both the cells has the same properties.

With the sexual copy process is possible that we lose properties or we gen-

erate new properties. In this case the cellular population assume or lose

properties. We break the symmetry in the cellular population. Adaptation

process can be considered as a copy process triggered by the environment.

For example to play chess is a network of possible actions with resources

and tasks. Any player is an agent of the second order which can change the

network of the possible actions. The player can copy the schemes or network

of actions located in the external environment in his mind. A physicist who

makes model of the nature is a second order agent which makes copy of the

agents network of actions in a physical nature into the symbolic domain of

the mathematical expressions. Agents as ants can share resources from one

field generated by other agents or ants. This field is a global memory that

is used by the agents. For example ant pheromone field, generated by any

ant, is used by all ants. In this way ants are guided to obtain their task

(minimum path). In this case the pheromone field is an example of global

memory resource. Network of connection among ants and its field is shown

in the paper.

Agents which take care of copying one network of agents in another

network of agents are second order agents. Because we can copy also the

network of agents of the second order by agents, these agents are at the

third order. In this way agents of any order can control and adapt network

of agents at a less order.

6.2. Agents and Symmetry (Coherence)

Any agent [10, 11,12 ] at the first order can be considered as an entity

which, within the domain of the resources, uses action to obtain a wanted

range of tasks. Resources can be any type of information or entities as

physical resources, functions, table of data or any type of information that

is necessary to make the action and obtain the wanted task. Action is a

method or a set of methods or in general any activity by which we solve

our problem and obtain our task. In a symbolic way we can represent any

simple agent at the first order by the figure 2.

Now agent of the first order uses the resources, generates actions and

obtains the tasks. Agents of the first order cannot generate other resources,

cannot change the actions or have new tasks. Only agents of the second

order can change the resources and the tasks and implement the suit-

able actions. To introduce agents of the second order we must extend the


Fig. 2. Resource, Action, Task of the agents or first order.

traditional paradigm of input/output structure in the figure 1. To built

the second order of agents we use the theory described in the papers [1-8].

In this theory denoted General System Logical Theory, we begin with the

graph

Fig. 3. Graph support of the agent coordinate ( coherent ) actions.

On the figure 3 we locate the resources and the task represented sym-

bolically in this way: Resources Sk, tasks Th. The network of agents of the

first order, one for each arrow is

Fig. 4. Resources, Actions and tasks for a network of five collaborative agents of thefirst order, one for any arrow.

In figure 4 agent whose action is Action1,2 and agent whose action

Action1,4 share the same resources S1. In this case agents are dependent on

the same resource.

Again in figure 4 agents whose actions are Action2,3 and Action4,3

46 Germano Resconi

collaborate to obtain the same task T3. One agent is also subordinate to

another agent when he must wait the action of the other agent. For exam-

ple the agent whose action is Action2,3 is subordinated to the agent whose

action is Action1,2. the graph in figure 3 with the agents of the first order is

an holonic system [12] and shows all possible connections among agent to

obtain global task obtained by the fixed interconnection of resources and

tasks. We remark that tasks for one agent can become resources for other

agents. Cycles inside the network of agents are homeostatic systems which

regenerate the initial resources after an interval of time. When the graph

has no cycle the network of agents at the first order stops its process when

obtain all the tasks without regeneration of the resources.

Symmetry principle:

In a point of the intelligent network the entity is a resource and at the

same time can be also a task. So we have a fundamental symmetry principle

“The task in one node is the same entity of the resource in the same

node”.

Agent of the second order is responsible for the allocation of the source,

task and actions. Only the agent of the second order can change the tasks

and resources inside the network of agents.

Now we describe the possible changes obtained by the agent at the

second order..

The first change is of this type

The agent of the second order changes the tasks and the resources with

the same transformation. In this way the new intelligent network preserves

the symmetry principle. For example we have this change

Fig. 5. Transformation of the resources, tasks by agent of the second order with thepreservation of the symmetry principle.

In figure 5 we show a possible change of tasks and sources where we have

the same structure. The new graph obtained by the second order of agents

is similar to the previous one but is not equal. Similar means that have the

same properties. In fact the first network and the second network in figure 5


have the same graph in common that is an invariant of the transformation.

Because the graph is the same, all the properties of the first network are

copied in the second network. For the relativistic principle, the new network

in figure 5 can also be obtained by changing the graph but without changing

the tasks and the resources. In figure 6 we show the same transformation

as in figure 5 without changing the tasks and the resources.

Fig. 6. Dual representation of the transformation in figure 4 where the tasks and re-sources are in the same position but we change the graph.

Formally the transformation in figure 5 can be given by the operator P

in this way

P =

(

1 2 3 4

2 3 4 1

)

andP [Resources] = P [Tasks].

The transformation P by the agent of the second order in figure 5 can

be represented in figure 7.

Fig. 7. Action P of the agent of the second order that copies one network in another.

In a symbolic way the transformation in figure 6 can be represented in

this way.

48 Germano Resconi

Fig. 8. Symbolic representation of the transformation in figure 6.

The symbolic graph in figure 8 is denoted elementary logical system of

the order two. See papers [1- 8].

Now the agent of the second order can generate more complex transfor-

mations of the resources and tasks. In fact the agent of the second order

can collect tasks in clusters and consider all the tasks in the same cluster as

equal to the same for the resources. In this case we have the transformation.

Fig. 9. Agent of the second order makes clusters of tasks and resources so as to havethe same structure but with repetition.

In figure 9 the cluster approach generates a new graph which appears

equal to the initial graph but with the property that the parts of the graph

inside the clusters are equal, we cannot make any distinction (granulation).

Fig. 10. The agent on the same structure creates clusters of equal tasks and sources.

Formally the transformation in figure 10 can be given by the operator

P in this way

P =

(

1 2 3 4

1 1 3 3

)

and P [Resources] = P [Tasks].

The agent of the second order has another possibility to change the net-

work of agents of the first order. The possibility is to breach the symmetry

between the resources and the tasks.


In this case formally we have one transformation for the tasks and an-

other different transformation for the resources. For example we have

P [Resources] =

(

1 2 3 4

2 3 4 1

)

and Q[Tasks] =

(

1 2 3 4

4 1 2 3

)

.

So we have the transformation represented in figure 11.

Fig. 11. Agent of the second order allocates the sources and the tasks, in the samegraph, but with different operations P and Q.

We remark that in the same node of the graph the resources and the

tasks have not the same index. In this case we break the fundamental

principle of symmetry. The entity in the node is not one entity but there

are two entities: one for the resources and the other for the tasks. In this

way we lose the correspondence between node and entities. The break of

symmetry symbolically is represented by the elementary logical system in

figure 12.

Fig. 12. Symbolic representation of the transformations P and Q in figure 11.

We remark that in the transformation of the figure 10 when we locate

only one entity in one node we have always a degree of conflict among

entities (Resources, tasks). In fact for example when in the nodes we put

the entity whose index is equal to the index of the resources we have the

network.

50 Germano Resconi

Fig. 13. At any node we assign one entity whose index is equal to the index of theresources.

From figure 12 we see that we have for any node a degree of conflict

that cab be shown in this way[

node node 1 node 2 node 3 node 4

degree of conflict 12

23

12

13

]

.

To solve the conflict we have different possibilities. The first possibility is

to define in the same node two entities and separate one from the other as

we show in figure 14.

When we separate the two entities (one for the tasks and the other for

the resources) inside the same node in two different nodes as we show for

example in figure 13.

Fig. 14. Separation of the entities in the transformation of the tasks and resources.

When we separate the entities in the same node, the network that we

obtain is shown in figure 14 where the conflict is completely eliminated.

Another way to solve the conflict without changing the original graph

is to define a compensatory network whose arrows are inside the node in

this way.

The internal transformation moving from one entity to another inside

any node is a compensation transformation which eliminates the original

conflict. Symbolically represented in figures 16-17.


Fig. 15. Entities encapsulated in the same node are separate so as to restore the prim-itive symmetry between tasks and resources and eliminate any conflict.

Fig. 16. Separation of the entities and internal connection to solve the conflict.

6.3. Natural Example of the Coherence in Ant Minimum

Path

In this part we describe examples of Bio-inspired Algorithm by agents [15].

Given a fully connected graph G where n is the set of points and e is the set

of connections between the points (a fully connected graph in the Euclidean

space) see figure 18.

The problem is finding a minimal length closed tour that visits each

point at a time. The length of the path between points i and j

di,j =√

(xi − xj)2 + (yi − yj)2.

52 Germano Resconi

Fig. 17. Compensation transformation to solve the conflicts.

Fig. 18. Fully connected graph.

Initially all m ants are uniformly distributed in all points. Each ant is a

simple agent with the following characteristics:

(1) It chooses the point to go to with a probability that is function of

the point distance and of the amount of trail present on the connecting

edge;

(2) When it completes a tour, it lays a substance called trail on each edge

(i,j) visited.

Let in document be the intensity of trail on edge (i,j) at time t. Each ant

at time t chooses the next point, where it will be at time t+1. When time

t + n each ant has completed a tour. Then the trail intensity is updated

according to the following formula.

τi,j(t+ n) = ρτi,j(t) + ∆τi,j (27)

ρ is a coefficient such that ( 1 - ρ ) represents the evaporation of trail

between time t and t+ n,

∆τi,j =

m∑

k=1

∆τki,j (28)

where ∆τki,j is the quantity per unit of length of trail substance (pheromone


in real ants) laid on edge (i,j) by the k-th ant between time t and t+n. The

coefficient ρ must be set to a value <1 to avoid unlimited accumulation of

trail and set initial intensity to be small positive constant.

The field of trial substance can be represented in a space of the relations.

Any point in this space is a connection edge. At any connection edge we

assign the intensity of the trial substance generated by all the ants in the

movement inside the graph.

Fig. 19. Field of trial substance inside the fully connected graph

The source, task and actions of the agent ants to generate the field F is

given by the simple graph in figure 17.

Fig. 20. Graph sources and tasks to generate the field S.

In figure 19 we have the feedback by which any ant generates the field

of the trial substance. The graph is repeated in the same way for all the

edge of the full connected graph. In the graph 19 at the first node we have

the source S1 that is the value of the trial substance that any ant in a

particular edge can measure. With the value of S1 the ant p has the task

54 Germano Resconi

Tp to decide to move inside the edge.

The task of the decision of the ants is given in this way. Given the

visibility ηi,j = 1di,j

. This quantity is not modified during the run of the

Ant System, as opposed to the trail that instead changes according the

previous formula (24).

The normalized decision index from point i to point j for the k=th ant

is

Di,j(t) = ταi,j(t)ηβi.j (29)

where α and β are parameters which control the relative importance of

trail versus visibility. When the ant decides to generate the Sp trial with

the task T to change the field S from S the task T1 is to be maintain the

field S in time in a way to be used from the other ants. The coherence

of the ant system consists in the invariance of the graph in figure 19 with

different assignment of the sources and task as we had discussed. Therefore

the decision index is a trade-off between visibility (which says that close

points should be chosen with high decision index, thus implementing greedy

constructive heuristic) and trail intensity at time t (that says that if on edge

(i,j) there has been a lot of traffic then it is highly desirable).

An ant algorithm presents the following characteristics:

(1) It is a natural algorithm since it is based on the behavior of ants in

establishing paths from their colony to feeding sources and back;

(2) It is parallel and distributed since it concerns a population of agents

moving simultaneously, independently and without a supervisor;

(3) It is cooperative since each agent chooses a path on the basis of the

information, pheromone trails, laid by the other agents which have

previously selected the same path; this cooperative behavior is also

autocatalytic, i.e. it provides a positive feedback, since the probability

of an agent choosing a path increases with the number of agents that

previously chose that path;

(4) It is versatile that it can be applied to similar versions the same prob-

lem; for example, there is a straightforward extension from the traveling

salesman problem (TSP) to the asymmetric traveling salesman problem

(ATSP);

(5) It is robust that it can be applied with minimal changes to other combi-

natorial optimization problems such as quadratic assignment problem

(QAP) and the job-shop scheduling problem (JSP).

In the discrete case we have a graph of actions that connect different entities.


Now for a continuous set of entities we have a field of entities. The action

in this case is an operator that connect the resource and the task fields.

Immune System equation [13, 14].

df(ak)dt =

∑

j,k

mj,kajak + pkak

f(x) = 12 − ln( 1−x

x )(30)

See the behaviour of function f in figure 21.

Fig. 21. Behavior of the function f.

where gj,k can be positive or negative the same for pk. The f(a)k is the

stimulus of the antibody and the ak are the concentration of the antibodies.

Because we have∑

j,k

mj,kajak + pkak = ATMA+ PAT . (31)

For the transformation

A = B + C

and

ak = bk + ck

(32)

we have

ATMA+ PAT = (B + C)TM(B + C) + (B + C)TP

= BTMB + CTMB + BTMC + CTMC +BTP + CTP.(33)

Now for

CTMB +BTMC +BTP = BTMTC +BTMC +BTP

= BT [(MT +M)C + P ] = 0(34)

we have

C = −(MT +M)−1P (35)

56 Germano Resconi

and the (3) becomes

ATMA+ PAT = BTMB + CTMC + CTP

= BTMB + CTMC + CTP = BTMB + CT (MC + P ).(36)

In this case we eliminate in the quadratic form the term of the first degree.

So we have∑

j,k

mj,kajak + pkak = BTMB +CT (MC + P ) =∑

j,k

mj,kbjbk + hk. (37)

So the (1) becomes

df(bk + ck)

dt=∑

j,k

mj,kbjbk + hk. (38)

Now the matrix M can be represented by a network among the cellular

concentration bk.

Fig. 22. Relation among four different types of antibody cells where 1, 2 , 3, 4 are theconcentrations b1, b2 , b3, b4.

All the permutations of the types of the antibody give a similar network.

The network 20 can be interpretd in this way. One antibody is considered

the source of the interaction with another antibody or task antibody with an

assigned affinity m. The same antibody can have two different functions.

One antibody is the cause (source) of the change of the other antibody

(task) but at the same time the same antibody can be the task for other

antibody cells (sources). Every Immunity System can change its structure

so as to have similar behavior given by the (9).The structure in figure 21

is the copy of the structure in figure 17. In the copy process we obtain

two different structures for which the interaction among antibodies has the

same properties even if the affinity m (relationship) among the antibody

cells is different.


Fig. 23. We break the symmetry between the antibody cells. When one type of cells(task, in the first box) is under the influence of another antibody type (source), the sametype (task) is not the source but we move to another antibody type (second box) toobtain the source of the positive or negative influence.

In figure 23 we break the ordinary symmetry for which the same anti-

body cell is under the action of other antibody and at the same time acts

as source of influence on other antibody. In fact in figure 22 for the same

structure one type of antibody cells are the task and another type of the

antibody is the source of the positive or negative influence. So we can have

the same structure but with change of the antibody type in the two function

as task or source.

We know that any parent has cell with half genome. Now in figure 24

we show four parts of the genome or genes connected one with the other by

a network. We assume that in the genome the genes are not independent

one from the other but there is a correlation between one and the other.

The operator P is the ordinary mutation process for which we change the

genes.

Fig. 24. Change of the genes (1, 2, 3, 4) of the genome in the source parent.

We fuse the source and the task parents and we have the children in

figures 24-25

58 Germano Resconi

Fig. 25. Change of the genes (1, 2, 3, 4) in the genome of the task parent.

Fig. 26. Genome of the children obtained by the fusion or crossover of the source parentand task parent. We remark that even if the network is the same both for children andparents, due to symmetry breaking the children are different from the parents.

7. Conclusion

We show that agents are connected in an intelligent network where at any

arrow we associate one agent at the first order whose work is to trans-

form, by action, one resource in a goal or task. When the task is to change

the network of agents of the first order, we introduce the agents of the

second order. Agents at the second order see the agents of the first order

in an holonic way [12, 16] as a totality. The agents of the first order can

collaborate, can share common resources or can be subordinated one to the

others. Agents of the second order have many different types of transforma-

tion so as to adapt the intelligent network to the global different tasks. A

deeper connection with natural agents as Ant or Immune System or Genetic

System is represented by the adaptation of the intelligent network by agents

of the second orders. We can also define networks of agents of the second

order whose tasks are networks and resources are networks. The adaptation

of network of agents at the second order is realised by agents of the third

order. We can continue in this rank of orders of agents to elaborate very

high abstraction in the realisation of tasks by resources.


References

1. Penna M.P.,Eliano Pessa,G.Resconi,(1996,October) General System LogicalTheory and its role in cognitive psychology, Paper presented at the ThirdEuropean Congress on Systems Science Rome.

2. Resconi G.,G.V.Tzvetkova (1991, June) Simulation of Dynamic Behaviourof robot manipulators by General System Logical Theory, Presented at the14-th International Symposium “Manufacturing and Robots”.

3. Resconi Germano, Gillian Hill (1996, October) The Language of GeneralSystems Logical Theory: a Categorical View, Presented at the EuropeanCongress on Systems Science Rome.

4. C.Rattray,G.Resconi,G.Hill (1997, March) , GSLT and software Develop-ment Process, Presented at the Eleventh International Conference on Math-ematical and Computer Modelling and Scientific Computing, GeorgetownUniversity, Washington D.C.

5. G.A.Kazakov and G.Resconi (1994) Influenced Markovian Checking Pro-cesses ,General System Logical Theory, International Journal General Sys-tem 22, pp 277–296.

6. Saunders Mac Lane (1971) Categories for Working Mathematician,Springer-New York Heidelberg Berlino.

7. M.D..Mesarovich and Y.Takahara (1989) Abstract System Theory, In Lec-ture Notes in Control and Information Science 116 (Springer Verlag).

8. Resconi G. M.Jessel (1986) A General System Logical Theory, InternationalJournal of General Systems 12, pp 159–182.

9. G.J. Marshall , A. Behrooz (1995)Adaptation Channels , Cybernetics andSystems 26, (3), pp 349–365.

10. Jacques Ferber (1999) , Multi – Agent Systems, Addison – Wesley.11. Germano Resconi, Lakhmi C. Jain (2004) Intelligent Agents, Sprinter.12. S.M.Deen (2003) Agent Based Manufacturing (Advance in Holonic Ap-

proach), Springer.13. P.M. Frank, B. Koppen-Seliger (1997) New developments using AI in fault

diagnosis, Engineer Application Artificial Intelligence. 10 (1) pp 3–14.14. G-C Luh, W.C. Cheng (2005) Immune model-based fault diagnosis, Math-

ematics and Computing in simulation pp 515–539.15. Stephan Olariu , Albert Zomaya (2006) Handbook of Bioinspired Algorithm

and Application, Chapman & Hall/CRC.16. V. Mahadevan , R.Braun and Z. Chaczko (2004) An exploratory study of

network performance evaluation of telecollaboration environments, ScienceSCI Florida.

17. Wayne Hu and Scott Dodelson (2002) Cosmic Microwave BackgroundAnisotropies, Annu.Rev.Astron.and Astrophys, pp 1–50.

18. R.Mignani , E.Pessa , and G.Resconi (1997) Non-Conservative GravitationalEquation, General Relativity and gravitation, 29, (8), pp 1049–1073.

19. Ning Wu, (2001) Commun.Theor.Phys. 36, pp 169–172.20. Ning Wu,(2001) Gauge Theory of Gravity, September 21.21. Frampton, P.H. (1987) Gauge Field Theories. Benjamin-Cummings, Menlo

Park, CO.

60 Germano Resconi

22. Kataoka, N., and Kaneko, K. (2000) Functional Dynamics. I: ArticulationProcess, Physica D 138, pp 225–250.

23. Kataoka, N., and Kaneko, K. (2001) Functional Dynamics. II: SyntacticStructure, Physica D 149, pp 174–196.

24. Maslov, V. P., and Shvedov, O. Yu. (1998)Large-N Expansion as Semi-classical Approximation to the Third-Quantized Theory. Preprint hep-th/9805089.

25. Mignani, R., Pessa, E., and Resconi, G. (1999) Electromagnetic-Like Gen-eration of Unified-Gauge Theories, Physics Essays 12, pp 61–79.

61

Undoing Quantum Measurement: Novel Twists to thePhysical Account of Time

Avshalom C. Elitzura and Shahar Dolevb

a ISEM, Institute for Scientific Methodology, Palermo, Italyb Philosophy Department, Haifa University, Haifa, Israel

a [email protected]

Since quantum measurement remains so ill-understood, its time-reversed ver-sion is bound to offer new insights. If undoing a quantum measurement isimpossible in principle, it bears on the old problem concerning the origin ofirreversibility. If it is possible, it gives new twists to basic notions like inde-terminism and the uncertainty principle. We demonstrate this approach onthe time-reversed version of the EPR experiment, where two entangled par-ticles undergo measurements followed by ”unmeasurements.” Whereas Bell’stheorem rules out local hidden variables, our experiment yields a complemen-tary proof that even nonlocal hidden variables lead to observed violations ofrelativity. We are therefore left with either genuine randomness or a subtletime-asymmetry at the basis of quantum phenomena. Both options render therelativistic account of time incomplete. Genuine emergence and novelty turnout to be the most fundamental characteristics of physical reality.

Keywords: Unmeasurement, Quantum Reversibility; Time-Reversal; QuantumErasure; Foundations Of Quantum Mechanics; Quantum Measurement Theory;Entanglement and Quantum NonlocalityPACS(2006): 01.70.+w; 03.65.Ta; 42.50.Xa; 03.65.Ud; 89.75.k

1. Introduction

Quantum measurement, after more than hundred years of research, remainsthe main obstacle to a better understanding of quantum theory. What aboutthe time-reversed process? Is it possible in the first place? And whether theanswer is positive or negative, what bearing has it on all the paradoxesbesetting quantum mechanics?

Merely raising the question makes us aware that quantum measurementis deeply related to a broader physical riddle, namely, irreversibility andtime-symmetry in general. This gives further incentive to study the issue.

This article communicates a work in progress. Some of its conclusions are

62 Avshalom C. Elitzur and Shahar Dolev

tentative, hopefully to be made more rigorous in the future and stimulatefurther research.

2. The Physics of Time-Reversal

Consider an elastic collision between two particles. It is possible to reversethe process by reversing the momenta of the particles after the collision,such that the entire process would evolve backwards, retrieving the initialstate from the final one. What about larger, macroscopic processes, in-volving myriad of particles? Mainstream physics holds that the question ismerely technical. Given a suitable technology, so goes this majority opinion,every process can be reversed. No rigorous theoretical limit to reversibilityhas been pointed out so far.

3. Introducing ‘Unmeasurement’

The quantum version of the reversibility question is even more intriguing.Can a quantum measurement be time-reversed such that a measured parti-cle will resume its initial superposed state? While opinions are more dividedon this issue, it is obviously akin to the issue of time-reversal in general.A few interpretations of quantum mechanics invoke the notion of ”wave-function collapse,” an abrupt change of the measured particle’s state thatis irreversible (see below). Other interpretations, however, avoid collapse,regarding measurement as reversible as any other process.

The issue is not entirely divorced from present-day technology. A proce-dure known as ”quantum erasure” (Scully, Englert & Walther, 1991) pur-ports to time-reverse quantum measurement. In the commonest versionof this method, a particle undergoes a double-slit experiment, but whileit passes through the slits it is entangled with another particle. Measure-ment of the latter particle can reveal which path was taken by the former.Consequently, the original particle’s interference is destroyed. However, bysubjecting the other particle to an appropriate measurement analogous tointerference, each particle gives a ”key” that enables observing the other’sinterference.

A closer inspection, however, reveals that no real undoing of measure-ment takes place in this method because no real measurement was made inthe first place. Rather, entanglement was employed, later to be countered(but not undone) by a more subtle measurement.

Elitzur and Dolev (2001) proposed a different method of unmeasure-ment based on interaction-free measurement. They showed, for the first

Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 63

time, that just like measurement, unmeasurement too exerts a non-local ef-fect in the EPR experiment. Another advance of their technique is that it isnot subject to the No-Trace Restriction (see Section 5 below). However, themethod works only for partial measurements, and its success becomes in-creasingly improbable as the measurement’s outcome approaches certainty.

Other methods, still far from technical feasibility, have been studied(Peres, 1980; Greenberger & YaSin, 1989). They indicate that, just likeearlier famous gedankenexperiments, deriving novel predictions gives anincentive to experimentalists to turn them into real experiments.

Unmeasurement, then, is of enormous importance for theoreticalphysics. If quantum measurement turns out to be irreversible for some ba-sic physical reason, then the age-old problem of the origin of irreversibilityand time-asymmetry (Savitt, 1995) will get a fresh answer. If, on the otherhand, quantum measurement turns out to be reversible, several surprisingresults would follow, bearing on other fundamental questions.

4. Collapse entails Fundamental Irreversibility

What is measurement in the first please? Rather than implicating humanobservers, consciousness, etc., let us adopt Bohr’s simple definition ‘an ir-reversible act of amplification.’ The meaning of ‘irreversible’ naturally, willbe the bone of contention.

Next, a closely related term needs to be considered. ‘Collapse of thewave-function’ is a process that several interpretations of quantum me-chanics invoke in order to explain the strange nature of measurement. It isimportant to realize that this collapse, if it indeed occurs, is irreversible inprinciple, regardless of technology.

Consider a wave-function of a single photon being split by a beam-splitter, each half going to another detector. One detector clicks will theother remains silent. In other words, the particle is detected in one of itstwo possible locations and turns out not to be in the other. The collapseinterpretation holds that, while the wave-function initially went to bothlocations, measurement has made it materialize in one and totally vanishedfrom the other location. Can the entire process be time-reversed? As longas we are talking at the gedanken level, it is possible to make the detectorthat clicked eject back the particle it has detected. But what about theother detector? Obviously, it cannot eject back the ‘nothing’ that it did notabsorb. Consequently, the wave-function will not return to its original state,failing, for example, to exhibit interference. Wave-function collapse, then,


entails a fundamental form of irreversibility, perhaps the most fundamentalone.

So, does collapse occur? Our own belief is that it does, thereby provid-ing the microscopic basis to all macroscopic manifestations of irreversibil-ity. But of course, ‘belief’ is no substitute for a rigorous proof. Hence weshall henceforth study quantum measurement as if collapse does not occur.Rather, we shall follow hidden-variables theories and assume that some‘guide wave’ always remains where the particle turns out not to be. It issuch hidden variables that make the retrieval of the entire wave-functionpossible.

5. The Sacrifice: No Record of the Measurement’sOutcome must be Left

Annoyingly, undoing a quantum measurement carries a peculiar price: Un-measurement necessitates complete elimination of any record of the mea-surement’s outcome, including the observer’s memory and knowledge. Wemust remain oblivious of the outcome that has been erased, as even thesmallest trace of a measurement left unreversed would ruin the entire pro-cedure.

Is there any point, then, to considering quantum unmeasurement if theoutcome to be undone must never be known? Fortunately, the answer is aresounding yes in the EPR case. We do not need to know the outcomes ofthe measurements that we seek to undo. Suffice it that we can see whether,following the unmeasurement, the two particles are entangled again.

6. Unmeasurement in the Context of the EPR Experiment

The EPR experiment (Einstein, Podolsky & Rosen, 1935) gave rise to oneof physics’ most famous paradoxes, in that it helped prove that quantummechanics involves interactions that violate, albeit indirectly, the relativis-tic prohibition on velocities greater than light. Consider an atom with spin0 ejecting two spin-1/2 particles that fly far apart towards two measuringdevices that measure whether each particle’s spin is 1/2 (“spin up,” ↑) or –1/2(“spin down,” ↓). Conservation laws oblige the two spins to be opposite: ↑↓or ↓↑.

Is the correlation between the spins pre-established, having been fixedwhen the particles left the atom? A celebrated proof by Bell (1964) ruledout just that: The correlation can be established only upon the particlesundergoing measurement. Hence, each particle must somehow “know”


which measurement the other particle is undergoing and what the out-come is, in order to maintain the appropriate correlations even when thetwo measurements are orthogonal.

Now let unmeasurement be considered in this case. Straightforwardly:

Let two EPR particles, A and B, undergo measurements. Then let themeasurements be time-reversed. Would the two particles resume their en-tangled state, being capable of manifesting again Bell-inequality violationsupon consecutive measurements?

This, of course, is only a gedankenexperiment, as was initially the EPRitself. Present-day technology is far from capable of time-reversing anylarge-scale process, let alone the ill-understood quantum measurement.

None of this should concern us here. We envisage an idealized, verysmall and perfectly isolated laboratory, within which the particle pair un-dergoes a pair of measurements and then the entire process is “undone”without inspecting the measurements’ outcome (see 4 below for the reasonfor this avoidance). Then let the two particles come out of the laboratoryand be subjected to yet another pair of new measurements, this time to befully inspected. Would these later measurements manifest Bell-inequalityviolations?

The stakes, no doubt, are high. A negative answer would be nothingshort of a theoretical proof that quantum measurement is inherently irre-versible, likely the source of all larger-scale irreversibilities.

What about a positive answer? Here, two assumptions need to be madein order to allow unmeasurement, and the result is far-reaching.

7. First Restriction: Reversibility entails Determinism

It is almost a tautology that, for a process to be reversible, it must bestrictly deterministic. To retrieve the process’s final conditions from theinitial ones, each and every one of the intermediate stages must strictlydetermine the next one. And when the process involves numerous molecules,the slightest deviation from determinism will result in the time-reversal’sfailure (Elitzur & Dolev, 1999). For the EPR experiment to be reversible,therefore, we must assume, in line with the “hidden variable” theories, thatthe spin values yielded by the two measurements are not genuinely randombut rather determined by causal factors, merely unknown yet objectivelyexistent.


Apparently, our quest should end at this point with the desired conclu-sion. Has Bell’s theorem not ruled out hidden variables? Experimental testsof the theorem show that the two particles could not have their spins fixedwhen they were emitted from the atom. Q.E.D.?

No, because Bell’s theorem ruled out only local hidden variables, leavingit possible that nonlocal variables play a role (this was Bell’s own belief). Inother words, the process may be deterministic as long as an event can exertits causal effect on another spacelike-separated event, i.e. instantaneously.

The stakes, then, are even higher than a quantum origin of irreversibil-ity: If we rule out the exotic possibility, never challenged so far, that theparticles’ spins are determined by nonlocal deterministic causes, we shallbanish determinism even from its last resort in quantum mechanics.

8. Second Restriction: Quantum Determinism EntailsAbsolute Time

We assume, then, that for quantum mechanics to be reversible it must obeysome hidden determinism, albeit of a nonlocal type. This leads to the sec-ond and last restriction: Nonlocal hidden variables entail an absolute timeparameter.

Consider the presumed hidden factor, F↑ or F↓, that determines thespin direction. Where can such a factor reside? Bell’s theorem has banishedit from its natural place, namely, the common source of the particles, fromwhich it could locally determine the particles’ spins. It can therefore bepresent only in the measuring apparatus. Presumably, then, the presenceof the factor F↑ or F↓ in the measuring advice causes the spin to be ↑ or ↓,respectively.

Now, in the EPR case, the same factor may often happen to be presentin both measurements of the two particles A and B. By angular momentumconservation, it is impossible for both spins to be ↑. We must thereforeassume that the causal priority goes to the “earlier” measurement, the“later” being able only to amplify the “already” determined spin.a Thequotes around “earlier,” “already” etc., remind us that the two EPR mea-surements are space-like separated, hence any temporal relations between

aA bolder assumption, namely that one EPR measurement can causally affect the ear-lier one, would immediately give rise to causal loops. Indeed, the retrocausal models,(Sheehan, 2006 and references therein) assume that measurement affects the particle’sentire history backwards down to its emission, but these models are understandably verycareful not to invoke any deterministic mechanism that would lead to such causal loops.


them can exist only within an absolute reference frame. Indeed, an abso-lute time parameter (say, that of the entire Universe’s rest frame) is theinevitable consequence of any nonrelativistic theory that allows effects topropagate faster than light.

With this minimum of two restrictions, let us proceed to the unmea-surement experiment.

9. Time-Reversing Nonlocal Hidden Variables enablesDistinguishing Absolute Motion

Let, then, two EPR particles A and B undergo spacelike separated mea-surements M1 and M2, respectively, with a small time interval betweenthem (in the absolute time frame), such that M1 occurs slightly “before”M2. Suppose now that the presumed spin-determining factor F↑ happensto be present in both M1 and M2. As M1 is slightly “earlier” than M2, onlyM1 affects A’s spin to be ↑, thereby forcing B to be ↓, in compliance withconservation laws. Consequently, M2 can only amplify the ↓ result imposedby M1, despite the presence of F↑ in M2 too.

Next, let the entire experiment be time-reversed. Let two unmeasure-ments M1 and M2 be performed on the two particles. By our above secondRestriction of Determinism, we assume that the spin-determining factor F↑that resides in both measuring devices also takes part in the two unmea-surements. As M1 occurred before M2, M2 must now occur before M1.This, by definition, is true for every deterministic time-reversal. The prooffor this requirement is simple, given in Appendix (1), Fig. 1 depicts theexperiment.

So far so good. Things do not go well, however, when the frame is amoving one, within which the experimental setup must obey Lorentz trans-formations. These transformations, in turn, clash with the absolute time-parameter which, by the above Absolute Time Restriction, is supposed togovern the nonlocal correlations.

The clash arises as follows. Let the entire experiment – the spin mea-surements and their time-reversal – be carried out in some inertial framemoving at near-c velocity with respect to the observer. Fig. 2 reveals thecrucial twist brought by this motion: The simultaneity plane t′ of the mov-ing frame becomes slanted proportionately to the frame’s velocity, deviatingfrom the absolute simultaneity plane t. Consequently, the time-interval be-tween the spacelike-separated M1 and M2 merely dilates, but the M2-M1

interval is reversed. This will ruin the overall time-reversal of the EPR ex-periment.


Fig. 1. An EPR experiment undergoing time-reversal. On each particle’s world-line, the dotted segment represents the time interval during which the particle is super-posed whereas the solid segment depicts its having a definite spin.

Conversely, if the reference frame moves to the opposite direction, the“unmeasurements” will maintain the same absolute time order as in theabsolute time, but the measurements themselves will suffer the same order-exchange, leading again to the failure of time-reversal.

One escape hypothesis, albeit far-fetched, may be proposed here insteadof the Absolute Time Restriction: Perhaps, in a moving frame, the nonlocalhidden variables do not operate in absolute time but obey the Lorentztransformations too? This possibility has been considered long ago (Elitzur,1996) and shown to clash with conservation laws (see Appendix 2). Theclash arises when the moving frame reverses its motion during the timeinterval between the two EPR measurements: The presumed cause-effectrelations between the measurements must reverse, enabling anomalous casesof ‘↑↑’ or ‘↓↓’ spins to occur, in defiance of angular momentum conservation.

10. The Alternative Left: Genuine Randomness

What, then, have we achieved in the above analysis? Bell’s theorem ruledout local hidden variables. We have now proved that nonlocal hidden vari-ables are equally impossible.

One way out of the dilemma is offered by the collapse theories: If thereis something inherently irreversible to measurement, no distinction between


Fig. 2. The same experiment in a moving frame.

rest and motion will be possible in the first place. In other words, perhapsunmeasurement, the very subject-matter of this paper, is simply impossible.But, once again, we have no decisive proof for this option, although wefavor it. Let us, then, consider the alternative and show that it has equallyintriguing consequences.

We assume, then, that measurement is reversible. In this case, a mea-surement’s outcome is genuinely random, but a subtle time-asymmetryemerges. Quantum measurement is indeterministic in one direction anddeterministic in the other: The transition from superposition to a definitevalue can lead to one out of several outcomes, but time-reversing each ofthese outcomes must lead to the same state of superposition.As far as we know, no observable reversibility can be derived from thisasymmetry. Consider a process that contains some quantum spin measure-ments. Time-reversing the process ensures that the initial superpositionstates will be retrieved. Then, re-running the process must (by our aboveproof against nonlocal hidden variables) give rise to new spins, differentfrom the earlier ones. Yet, by the No Record Restriction (Section 4), it isimpossible to know what the initial spins were.

A slight change, however, enables us to derive a surprising predictionthat is certainly observable.


11. The Consequence: Unmeasurements are Immune toNon-Commutation Laws

We have pointed out that genuine quantum randomness gives rise to asubtle asymmetry, in that measurement turns superposition into one out ofmany possible states randomly, while unmeasurement turns every possiblestate into superposition with certainty. Can this asymmetry be observed?

Yes. All we have to do in order to observe it is to make the two EPRmeasurements noncommuting, say, subject one particle to measurement ofits spin along the x direction while the other’s spin is measured along they direction.

Here, time plays an observable role: While the measurements of the spindirections are non-commuting, the unmeasurements do commute.

The reason is simple. Consider a pair of EPR spin measurements, one ofthe x direction and one of the y direction. Bell’s theorem proves essentiallythat the result of each particle’s measurement depends on the choice ofmeasurement to be performed on the other particle and its result. Hence, ifx is performed before y, the following counterfactual holds: The outcomeswould be different have the measurements been made in a reverse order.This is a direct consequence of Bell’s theorem.

It is equally clear, however, that the same cannot be said about theunmeasurements of x and y, otherwise it will be possible to distinguishbetween motion and rest as in the case of nonlocal hidden variables in theprevious section.

A subtle asymmetry between measurements and unmeasurements arises,which warrants an experimental test: If unmeasurements turn out to bepossible, they will be immune to non-commutation relations.

12. Hawking’s Information-Loss Conjecture Supported

Next we wish to point out a surprising consistency of our Genuine Ran-domness Conjecture with one of the most intriguing issues in present-dayphysics, namely, Hawking’s (1976, see Preskill, 1993) information-loss para-dox. True, Hawking himself has eventually retracted from his claim, butseveral physicists, we among them, remain unconvinced.

In essence, Hawking has shown that when a black hole swallows matter,the information associated with that matter is lost, and when the blackhole evaporates, new information is created at its event horizon. This in-formation seems to be genuinely new in that it appears to be unrelated tothe matter swallowed earlier.


But this is precisely what the Genuine Randomness Conjecture holdsto take place every time a measurement is made! If a particle is first mea-sured, then unmeasured and then measured again, the outcome of the lastmeasurement must not be determined by the first one. This is uncannilysimilar to the process in which empty space gives rise to virtual particle-pairs that form and then mutually annihilate. When such a virtual pairbecomes real,as in black hole evaporation, new information is created, notdetermined by any previous process. In mainstream physics today this re-sult is regarded as disturbing and often dismissed as erroneous. Our anal-ysis, in contrast, renders this genuine novelty, natural: New information isgenerated with every quantum measurement.

13. A New Twist to the Transactional Interpretation

Finally, let us assess our findings in terms of one of the most consistentand appealing interpretations of quantum mechanics, namely, the transac-tional interpretation (Cramer, 1986). It has gained a widening acceptanceduring the last decade, not the least because of its simplicity, elegance andexplanatory power (see, e.g. Sheehan, 2006).

In the transactional interpretation, any quantum interaction is the re-sult of two wave-functions, one retarded and the other advanced, going backand forth in time between the source and the absorber, eventually givingrise to a complete quantum exchange of a particle. EPR-Like effects get avery fresh explanation this way: The back-and-forth wave-functions form aspacetime zigzag, connecting the two particles in the present through theircommon origin in the past.

How, then, should an EPR experiment look like in this framework whenthe two particles undergo a pair of measurements and then a pair of unmea-surements? Again, the transactional interpretation faces the same choice asother interpretations: If it holds that unmeasurement is impossible, thenit is pointing out the microscopic source of irreversibility. If, on the otherhand, it allows unmeasurements, the resulting picture is very peculiar: Anentire spacetime segment undergoes ‘revision’, such that, the entire historyof the EPR pair is ‘re-written’. We have earlier (Elitzur & Dolev, 2006)derived a similar conclusion from the transactional interpretation with theaid of novel gedankenexperiments indicating that spacetime itself is sub-ject to a subtler form of evolution, not yet accounted for in the present-dayrelativistic framework.


14. Summary

What, then, has unmeasurement added to our understanding of quantummechanics? With due modesty, quite a lot. Our analysis indicates that mea-surement may be fundamentally irreversible, hence unmeasurement prob-ably cannot take place. Quantum mechanics, presently held to be a time-symmetric theory, may turn out to be the source of all asymmetries.

Conversely, if unmeasurement is possible, measurement must be gen-uinely random. In order for unmeasurement not to lead to results thatviolate special relativity, the outcome of each and every measurement mustnot be determined by local hidden variables neither by non-local ones. Thelatter restriction is a novel result, making quantum randomness absolute.Every event of measurement must introduce a genuinely new bit of infor-mation into the universe.

Which of the two alternatives eventually turns out to be correct? Wedo not know yet, but a more decisive argument will hopefully be found.

As for the interpretation of quantum mechanics, we have shown that,within the transactional interpretation, unmeasurement indicates that twoor more histories must be ‘written’ and then ‘re-written’ on top of one an-other on the same time segment. Spacetime itself must therefore be subjectto evolution in some higher time parameter. This is very likely the time ofBecoming (Elitzur & Dolev, 2005a, 2005b), so missed in present-day physics(Davies, 1995).

It seems safe to conclude, therefore, that the famous ‘uneasy truce’ be-tween relativity and quantum mechanics has never been uneasier. If thereare hidden variables beneath the quantum level, then, by an earlier proofof ours (Elitzur & Dolev, 2005a), they must be ‘forever-hidden variables’in order to never give rise to violations of relativity. But then, by the samereasoning that has lead Einstein to abolish the aether, they probably donot exist. Indeed, it has been proved long ago (Elitzur, 1992) that Godmust play dice in order to preserve relativistic locality. Hence, randomness,novelty and emergence, which for luminaries like Parmenides, Spinoza andEinstein were mere epiphenomena to be explained away, are likely the Uni-verse’s very mode of existence.

Appendix 1: The Order of the “Unmeasurements” must bePrecisely Opposite to that of the Measurements

The world-lines of particles A and B are as in Fig. 1: The dotted seg-ments represent superposition while the solid segments represent definite


spins. Each double-line represents the world-line of the measuring instru-ment that carries first the measurement, and then its undoing, on the sameparticle.

On the left is a correct time reversal of the two measurements. Al-though the deterministic F↑ factor is present in both measurements, M2

fails to force particle B to have the ↑spin because of the priority of M1.During the time-reversal, whatever causal influence M2 had, it is undoneby the time M1 occurs. Now read the whole experiment backwards in time:Every unmeasurement becomes a measurement and vice versa. Everythingseems equally consistent.

On the right is the same procedure without being strict with reversingthe order of the unmeasurements. Here, the backward-reading of the exper-iment is clearly odd: Why should M2, now turned into a measurement, failto turn the superposed particle into a ↑, getting ↓ instead? The rest of theevolution is equally inconsistent.

Fig. 3.

Appendix 2: Relativistic Nonlocal Effects Lead toConservation Laws Violations

Consider the following hypothesis: The nonlocal effects observed in theEPR experiment are exerted in accordance with special relativity, that is,the simultaneity plane changes in accordance with the system’s motion.

Next, consider an EPR experiment carried out in a frame that moves


at a near-c velocity in one direction and then in the opposite direction. Letthe two measurements be carried out with a slight time-interval, one beforeand one after the velocity reversal. In this case, a measurement that residedin the other measurement’s ‘future’ prior to the reversal will suddenly shiftto the ‘past’. The presumed nonlocal effects would therefore fail to occur,giving rise to spin conservation violations.

Fig. 4.

References

1. Bell, J. S. (1964) On the Einstein-Podolsky-Rosen Paradox. Physics, 1, pp195–200.

2. Cramer, J. G. (1986) The transactional interpretation of quantum mechan-ics. Rev. Mod. Phys. 58, pp 647–687.

3. Davies, P. C. W. (1995) About Time: Einstein’s Unfinished Revolution. NewYork: Simon & Schuster.

4. Einstein, A., Podolsky, B., & Rosen, N. (1935) Can quantum-mechanicaldescription of physical reality be considered complete? Phys. Rev. 47, 777–780.

5. Elitzur, A. C. (1992) Locality and indeterminism preserve the second law,Physics Letters A, 167, 335-340.

6. – (1996) Time anisotropy and quantum measurement: Clues for transcend-ing the geometric picture of time, Astrophysics and Space Sciences (specialissue), 244, 313-319.

7. Elitzur, A.C., & Dolev, S. (1999) Black hole evaporation entails an objectivepassage of time, Foundations of Physics Letters,12, pp 309–323.


8. Elitzur, A.C., & Dolev, S. (2001) Nonlocal effects of partial measurementsand quantum erasure, Phys. Rev. A 63, 062109-1—062109-14.

9. Elitzur, A.C., & Dolev, S. (2005a)Quantum phenomena within a new the-ory of time, In Elitzur, A.C., Dolev, S., & Kolenda, N. [Eds.] Quo VadisQuantum Mechanics? , pp 325–350. New York: Springer.

10. Elitzur, A.C., & Dolev, S. (2005b) Becoming as a bridge between quan-tum mechanics and relativity, In Saniga, M., Buccheri, R., and Elitzur,A.C. (Editors) Endophysics, Time, Quantum and the Subjective, pp 589–606. Singapore: World Scientific.

11. Elitzur, A.C., & Dolev, S. (2006) Multiple interaction-free measurementas a challenge to the transactional interpretation of quantum mechanics,In Sheehan, D. [Ed.] Frontiers of Time: Retrocausation – Experiment andTheory. AIP Conference Proceedings, 863, pp 27–44.

12. Greenberger, D. M. & YaSin A. (1989) “Haunted” measurements in quan-tum theory, Foundation of Physics, 19, pp 679–704.

13. Hawking, S. W. (1976) Breakdown of Predictability in Gravitational Col-lapse, Physical Review D, 14, pp 2460–2473.

14. Preskill, J. (1993) Do black hole destroy information?, International sym-posium on black holes, membranes, wormholes and superstrings. World Sci-entific, River Edge, NJ.

15. Peres, A. (1980) Can we undo quantum measurements?, Phys. Rev. D 22,pp 879–883.

16. Savitt, S. (1995) Time’s Arrow Today, Cambridge: Cambridge UniversityPress.

17. Scully, M. O., Englert, B.-G., and Walther, H. (1991) Quantum optical testsof complementarity, Nature 351, 111–116.

18. Sheehan, D. [Ed.] (2006) Frontiers of Time: Retrocausation – Experimentand Theory, AIP Conference Proceedings , 863.

77

Process Physics: Quantum Theories as Models ofComplexity

Kirsty Kitto

The School of Chemistry, Physics and Earth SciencesFlinders University, Adelaide, Australia

[email protected]

Originally based upon a pregeometric model of the Universe, Process Physicshas now been formulated as far more general modeling paradigm that is capableof generating complex emergent behavior. This article discusses the original re-lational model of Process Physics and the emergent hierarchical structure thatit generates, linking the reason for this emergence to the historical basis of themodel in quantum field theory. This historical connection is used to motivate anew interpretation of the general class of quantum theories as providing mod-els of certain aspects of complex behavior. A summary of this new realisticinterpretation of quantum theory is presented and some applications of thisviewpoint to the description of complex emergent behavior are sketched out.

Keywords: Quantum Models; Quantum Information; Process Physics; GlobalColour Model; Complex SystemsPACS(2006): 03.65.w; 03.67.a; 03.70.1k; 89.75.k; 89.75.Fb

1. Process Physics

The dominant modeling methodology of physics is static. We model theuniverse as a 3 + 1 dimensional pre-existing structure, where time shares avery similar status to that of space, despite our everyday experiences of thetwo phenomena as being remarkably different. Process Physics sprang out ofan initial desire to explain from where such a structure could have emerged(i.e. to form a pregeometric model of the Universe). A particular aim wasto explain the relationships between space, time and matter, especiallywith respect to how such phenomena could dynamically emerge from anunderlying model.

The original pregeometric modeling has proven to be, at least in itsinitial stages, rather successful,1 and the reader is encouraged to consultthat reference for more details. This paper will take a different approach. It

78 Kirsty Kitto

is becoming apparent that the dominant modeling methodology of physicsis not isolated and this has led to a search for new approaches to modeling ingeneral.2 This paper will discuss some of the work that has been performedduring this search for a more process oriented approach to modeling.

The standard modeling paradigm of science is dominated by reduction-ism. This technique, which involves breaking an apparently complex prob-lem into smaller, more manageable pieces, reduction and then combining thesolutions obtained from these smaller problems into a larger solution whichrepresents the original system, synthesis, has been remarkably successful inscience. However it appears likely that this technique is approaching theend of its general scientific validity.2–4 Many of the systems in which weare now interested cannot be so simply approached, a problem which hasled to a general resurgence of holistic and systems oriented approaches.∗

There is a general pattern emerging from this work; most of these problemsof analysis tend to appear when an attempt is made to apply reductivetechniques inappropriately to systems that are often classified as complexin some manner that is yet to be adequately defined.5 Despite this prob-lem of definition, a number of characteristics shared by such systems canbe identified which make the appropriateness of their analysis by standardreductive approaches somewhat dubious. Such characteristics include:

(1) Dynamic contingent behaviour indicative of a constant processing ofinformation, energy, matter etc. This is in contrast to the generally verystatic models currently used where temporal processes are representedby mapping them to a real number line. Such a geometric approach totime tends to lose some of the most commonly experienced temporalphenomena, and has led to a number of well known problems in physicssuch as the lack of an arrow of time in our models, and the denial of apresent moment.6

(2) Contextual responses to environmental perturbations, which mean thatthe system cannot be straightforwardly separated into a subsystem ofinterest and an environment that can be ignored.2

(3) Hierarchical organisation where generally stable subcomponents func-tioning on relatively fast timescales are incorporated (often nonaddi-tively) into larger structures operating more slowly.7,8 A process offeedback is often manifested too, where the higher level of the hierar-chical structure so formed can exert causal control over the lower levelcomponents.9

∗See for example the list of references at http://pespmc1.vub.ac.be/EVOCOPUB.html


(4) Far from equilibrium and dissipative behaviour,10 which means thattraditional models based upon static equilibria and other standard ther-modynamic concepts are not necessarily applicable.

(5) The exhibition of long range order, which is stable in the face of pertur-bations.† This is an unusual feature, perturbations have traditionallybeen thought to disrupt long range order, and this is generally the casefor classical systems, but not all systems are so unstable in such situa-tions. For example, living organisms are not significantly disrupted byincoming energy, instead they convert such energy into chemical formand transport it over far greater distances than are predicted by linearmodels, leading to the so-called “bioenergetic crisis”.11

(6) Relational structure and evolution. The traditional modeling of suchsystems as objects exhibiting a number of interactions between them-selves tends to limit the emergent behaviour that can be exhibited bythem.12 For example, in attempting to model the Universe using a num-ber of fundamental objects we lose our ability to describe those veryobjects.

Interestingly, an analysis of the low level relational model of ProcessPhysics (which will be discussed in the next section) reveals that it sharesa number of these characteristics,6 a result that led to an in-depth analysisof the reasons why this might be the case.2 This paper will discuss someaspects of this journey of analysis. It will start with a quick review ofthe original low level relational model of Process Physics, illustrating theway in which it appears to be capable of generating interesting emergentbehaviour. An analysis of the structures emergent from within this modelwill reveal that it is capable of dynamically generating apparently complexhierarchical behaviour. This will lead us to an examination of the historicalroots of the model, which lie in quantum field theory (QFT), and to thehypothesis that quantum theories are capable of describing a far widerclass of system than is generally considered to be the case. In particular,the claim will be made that QFT is capable of describing complex emergentbehaviour. The remainder of this work will consist of a brief survey of somefuture promising avenues of research that follow from such a conclusion.

†At least within the limits of the systems tolerances. Outside this range catastrophicfailure generally results.

80 Kirsty Kitto

1.1. The Low Level Process Physics Model

The initial low level model constructed within the Process Physics paradigmattempts to work around the static object driven methodology of reduction-ism by adopting a more relational approach. This is achieved by startingwith a set of nodes for the sake of analysis, but instead of focussing upon theobjects themselves, the connections between individual nodes are analysed.Thus, considering a set of N nodes, we assume that they are connected insome way, with a connection strength between node i and node j given bythe real value Bij (see figure 1). We represent a set of these relational val-

2

1

3

4

5

B25

B12B13

0 1 1 0 0

2 3 4 51

1

2

3

4

5

−1 0 0 0 1

−1 0 0 0 1

0 0 −1 0 1

0 −1 1 −1 0

(a)

= [B ] = Bij

(b)

node

Fig. 1. A simple relational structure of nodes and their connections. (a)Nodes i and jare considered connected if they have a non-zero Bij value. Arrows indicate the sign ofthe Bij value. (b) This structure of nodes and their connections can be represented inan antisymmetric matrix, zero values represent no connection between nodes i and j,while a nonzero value indicating a connection is represented here as ±1 depending uponthe direction of the arrow drawn in (a).

ues as a square antisymmetric‡ matrix B. Notice that while the pitfalls ofobject driven methodologies have been explicitly recognised above, we arestill driven to talk in terms of nodes; it is very difficult to leave an object-based methodology behind. This is due to the very nature of our analytictechniques which have always been based upon the reductionist methodol-ogy. However, any system simply positing such a priori objects cannot beconsidered fundamental when modeling the Universe, as the explanation ofthese objects must lie outside the system being modeled. A partial solutionto this dilemma arises if we recognise that the nodes can in turn be defined

‡Antisymmetry ensures that Bii = 0 thus avoiding explicit node self connection. Theinternal structure of nodes will be incorporated shortly.


for the purposes of the model as a system of nodes.§ It becomes apparentthat all nodes can be thought of as composed of collections of nodes inturn, and in particular that the start up nodes can be viewed as names forsubnetworks of relations. This result is ensured if the constructed systemexhibits self-organised criticality (SOC), which enforces a fractal structureon the system.13,14

The SOC requirement is intrinsically linked with a new processing no-tion of time in this system. This is because the relational fractal structureis generated by a noisy non linear iterative map displaying SOC behavior.Thus in attempting to construct a model that does not postulate a priorifundamental objects, we find the need to introduce a time-like process. Incontrast, standard physics with its use of a priori objects is linked with thestandard geometrical model of time and the associated problems mentionedabove.

The particular map used was suggested by the Global Colour Modelof quark physics.15 Details of the history of this map will be discussed insection 4, but for now we simply assume that the B matrix representingthis system of nodes updates discreetly according to the following iterativeprocess:

Bij → Bij − α(B + B−1)ij + ωij , (1)

where i, j = 1, 2 . . . , N and N →∞. The constant α is an arbitrary tuningparameter,16 while the term ωij represents an additive noise term, whichprovides a sense of openness to the system.

At each iteration, the action of the noise term leads to the creation ofnew Bij links, incorporating a sense of innovation and contingency into thesystem. The noise term, when used iteratively in equation (1) is respon-sible for the notion of time that arises in the model. The dynamics areirreversible, with one particular past, which can be recorded as a history,but not relived. Future states of the system cannot be known. Howevercertain sets of ensemble predictions can be made. In this sense a dynam-ical notion of time is captured by the system, with a markedly differentontology from the static four dimensional spacetime of standard physics.This leads to the identification of this system and its associated modelingtechniques under the broad categorisation of Process Physics. We note fornow that this modeling of time is far more appropriate in the modelingof living systems, providing a sense of contingency and dynamism that is

§B is assumed to be a very large (→∞) matrix.

82 Kirsty Kitto

generally lacking from more standard approaches.The nonlinear matrix inversion term also performs a critical role in the

system. It causes separate structures brought into existence by the noiseterm to link up, modeling a process of self-assembly. It is interesting toexamine the dynamics of this process in detail.

The system can be started with B ≈ 0 which represents the absence ofany significant relational information. Under successive iterations of equa-tion (1) the B matrix assumes a sparse structure that can be organised intoa block diagonal form. This suggests that there is a tendency for modes toorganise into some sort of modular structure, a point that we shall returnto in Section 2.1.

TTTTTTTTTT

··

··

··

··

··

TTTTTTT

··

··

i D0 ≡ 1

D1 = 2

D2 = 4

D3 = 1

t

t t

t t t t

tFig. 2. A N = 8 spanning tree for a random graph (not shown) with L = 3. Thedistance distribution Dk is indicated for node i.

Assuming that the large ωij arise with fixed but very small probabilityp, the geometry of the structures formed can be revealed by studying theprobability distribution of minimal spanning graphs with Dk nodes and k

links from an arbitrary node i where D0 ≡ 1 (see figure 2). This probabilitydistribution is given by17

P[D,L, N ] ∝ pD1

D1!D2! . . . DL!

L−1∏

i=1

(q∑i=1

j=0 Dj )Di+1(1− qDi)Di+1 (2)

where q = 1−p, N is the number of nodes and L is the maximum depth fromnode i. The most likely pattern can be found by numerically maximisingP[D, L,N ] for fixed N with respect to L and Dk. This procedure has beenperformed,18 and the following results are from that analysis.


Figure 3 shows the set of Dk (distance distribution) values obtainedfrom one of these numerical experiments, where the log of the probabilityof a large noise value is set at log10 p = −6, and the number of nodes isfixed at N = 5000. Also shown in the figure is a curve

Dk ∝ sind−1 (πk/L) , (3)

with best fit to the data when L = 40 and when the dimensionality of thefit, d = 3.16. This same curve is obtained from the surface area of a n-dimensional sphere. Figure 4 shows the range in d for fixed N = 5000 andvarying p values. We see that for p below some critical value log10 p < −5,d ≈ 3.

0 10 20 30 40k

0

50

100

150

200

250

Dk

Fig. 3. The distance distribution Dk between nodes for a typical emergent structureunder equation (1). This is obtained by numerically maximising P[D, L, N ] where N =5000 and log10 p = −6. A curve is also drawn showing that Dk ∝ sind−1 (πk/L), withbest fit d = 3.16 and L = 40.

This indicates that the connected nodes have a natural embedding in aS3 hypersphere, which is very suggestive of the 3-dimensionality of space.Thus this model goes some way towards predicting the observed three

84 Kirsty Kitto

-8 -7 -6 -5 -4Log10p

2

3

4

5

6

d

Fig. 4. Range of d values for N = 5000 as a function of the probability p of large noisevalues.

dimensional structure of space as an emergent phenomenon, a feature thatis usually assumed in physics. Notice that the nodes are not exactly em-beddable (which would require d = 3), there is a proportion of extra links.This is a key observation which forms the basis of a theory of matter as aquantum foam that is currently under development.1 We shall discuss theunderlying ideas of this theory shortly.

While they are not the only structures generated by the system, theirmaximum likelihood makes the hypersphere structures the most common.Splitting the large B matrix into its constituent independent submatricesBsub we realise that each is almost singular (det(Btree) ≈ 0) but that thenoise term ensures extra Bij terms which lead to a small valued determi-nant. This small valued determinant implies that the next iterative stepof the system will lead to new large valued Bij entries upon inversion ofthe B matrix. Hence tree structures are sticky, at each iteration cross-linksform between the structures which act to join them, and to produce largerstructures. This behaviour has been examined in detail in a recent PhDthesis.16

Thus, under the influence of the iterative equation (1) the system canbe seen to ‘grow’ with a steady increase in relational structure. As the


tree structures stick together, they become less easily embeddable in a S3

structure. This phenomenon can be seen in figure 4, where beyond thecritical value of d ≈ 3 the dimension of the structures quickly becomes veryhigh; they are no longer embeddable in S3 and should instead be thought asdefects in the system. These defect graphs gradually lose the ability to formnew links, which tend to arise only at the level of ‘leaves’, or the endpoints ofthe graphs. As a result the defects become unable to sustain themselves andeventually they fade away from the system, or ‘die’. However, the systemitself generally grows faster than it loses such structures (see Ref. 16 formore details of these dynamics).

The nonlinear term in (1) is self-referencing; all elements of B thatarise in a previous iterative step are required in the computation of thenext value of each Bij element. Thus, this term can be seen to incorporatea weak notion of internal self-observation into the system. In particular,any node has the capacity to profoundly affect the rest of the system if itrandomly receives a large ωij value at the next iteration of the map. Thefact that one node is not ‘close’ to another in a particular S3 embeddingdoes not mean that it cannot be affected strongly by it. This leads to avery strong form of contextual behavior in the system, no one element canbe considered as isolated, in fact, at the next iterative step it may becomestrongly linked to a node which was previously not considered importantto its dynamics.

2. Emergent Structures within the Process PhysicsParadigm

There is a sense of emergence in this system. Both the embeddable and theassociated non embeddable defect structures which appear to be formingare not explicitly present in either the update equation (1) or the originalvery simple relational structure of nodes and connections. An examinationof this claim within the framework of hierarchy theory7,8,19–23 helps toclarify the form of emergent behavior realized by this system.

2.1. The Hierarchical Nature of the Model

An immediately apparent characteristic of this system is the block diagonalform of the matrix B which arises under the influence of the update equa-tion (1). This block diagonal form also arises in Simon’s near-decomposablesystems,22 making it possible to form an understanding of this system as aset of nearly-decomposable modules. This result makes sense. If nodes are

86 Kirsty Kitto

seen as themselves made up of structures of connected nodes, then clearlysome sort of modular hierarchical system has been created.

It is also possible to understand the relational model as forming at leasta 2nd order hyperstructure according to a theory of emergent hierarchicalstructure proposed by Baas.20 This framework depends fundamentally uponthe notion of observation used in the identification of new, higher orderstructures. Thus, in order to register the emergence of new structures orentities within a system it is necessary to identify a mechanism capableof observing those entities. Given that this system is aimed at modelingthe Universe, we might ask what such a mechanism could be. How couldan observational mechanism external to the Universe be identified? Such amechanism can be found through the adoption of what might be regardedas an external perspective, but this does not imply that an entity existsexternally to the Universe, merely that observation generally has to beimplemented at a different modeling level. This point has been recognisedby a number of different researchers,24–26 many of whom claim that it isnecessary to develop internal perspectives in order to understand complexphenomena such as the mind and life. The term endophysics24,27,28 hasbeen coined to refer to the study of systems which have observers enclosedwithin them. Such a view is participatory; how we look determines whatwe see, and with the development of new internal perspectives we mightbegin to understand the way in which the context of such an observer canaffect the observations they make of the system itself.2 Through a propertreatment of context we lose much of the confusion that often surrounds anexamination of the Universe; an observer of the Universe can exist withinthat Universe. However, it will be necessary to adopt more than one modelof the Universe in order to explain its complex emergent behaviour. Weshall return to this point in Section 3.

To construct a Baas hierarchy, we consider the original nodes used inthe construction of the relational structure to be a very simple family of N

first-order structures Ξ1 = Ξ1r : r = 1, 2, . . . N, where a first-order obser-

vational mechanism O1 is defined as membership in the set Ξ1 . Now, underthe influence of equation (1) which we represent as an update functional R,we find that these nodes join up, forming a second order structure:

Ξ2 = R(Ξ1i , O

1, Bij) i, j = 1, 2, . . . , N (4)

where Bij is the connection strength between nodes and Ξ2j designates the

structure that appears in the B matrix in the previous sections. The familyof second order structures consists of the maximum likelihood structures


discussed above that are embeddable in the three-dimensional hypersphere(ES3), along with the non-embeddable defect structures. Considering thespace-type structures alone, we can define the property of being embeddablein S3, (ES3) as a second order observationally emergent property,

PES3 ∈ O2(Ξ2). (5)

That is, this property is not present in the simple set of nodes that westarted with,

PES3 /∈ O2(Ξ1), (6)

rather, it becomes apparent under the influence of the update functional(1).

The analysis of the remaining structure of this system is very complexand only in the preliminary stages, but it is expected that stable patternsidentifiable as third order structures will emerge in the system. The fulltheory of this process is currently under development, but the general ar-gument can be sketched out.

We can somewhat artificially classify the structures in the systemas being either exactly embeddable in S3 (the ES3 structures), or non-embeddable structures which we will term defects (D). It is expected thatthe system contains stable defects, which we shall term Topological Defects(TD). The stability of these structures will be structural, and constitutesa new third order observation mechanism, O3, within the system. Thus itis expected that the stable TD’s will form a set of K new emergent struc-tures Ξ3 = Ξ3

v : v = 1, 2, . . . K within the system. Equation (1) ensuresa constant updating of their components — rather like the units of an or-ganism which are being constantly regenerated. These TD’s (if rigorouslyidentified) would be third order structures in the hierarchy. That is

PTD ∈ O3(Ξ3v), and PTD /∈ O3(Ξ2

r) (7)

because their emergent stability within the system is defined by their struc-tural stability, i.e. it constitutes a new observation mechanism.

Thus, there is a sense of non-computational, or rather observationalemergence (as defined by Baas20) in this system. More than one mode ofanalysis is required in order to understand its behaviour. This suggeststhat this system is truly exhibiting some sort of very complex behaviour,12

a concept that will be discussed more in the following section.

88 Kirsty Kitto

So far, a third order hyperstructure has been identified (but not rigor-ously at the third order).

nodes > S3 + D︸︷︷︸O2(Ξ2)

> stable TD︸︷︷︸O3(Ξ3)

> . . . . (8)

It is expected that more structure will emerge from the system. This isbecause the identified stable structures of this model have a deep connectionwith the more standard object driven methodologies of particle physics.In Ref. 29 a link to the theory of preons¶ was discovered, and thereforeit appears that these systems can recover much of the behavior of theStandard Model.

Despite this promise, the analysis of this system from this bottom uplevel is extremely challenging, both computationally and analytically. Thisproblem has made it necessary to attempt a more top down understandingof this system2 ; why does it generate such interesting emergent behavior?

One set of clues as to why this simple model is behaving in such amanner can be found by referring to the discussion from section 1. Summa-rizing that discussion: this system was designed to be more relational thanis normally the case, thus it avoids some of the most obvious traps of anobject driven modeling approach; it exhibits hierarchical behavior; it hasbeen designed to be far from equilibrium and dissipative; SOC behaviouris intrinsic to the model; there is a sense of long range order, even thoughmost strongly connected nodes will tend to associate closely (i.e. embedwell in a 3D hypersphere) the defects associated with this model allow forlong range connections between nodes, and some of these are expected toexhibit stability due to their topological nature. Perhaps most importantly,this system exhibits contextual responses; the dynamical, process-type un-derstanding of time in this model means that different runs of the system,while exhibiting similar structure formation, will always exhibit contingent“once-off” events that will depend strongly upon both the random noiseterms that arise at each iteration, as well as upon the pre-existing valuesof the Bij .

Thus, there is every reason to suppose that this system is exhibitingsome sort of complex emergent behaviour.

¶Preons are one of the proposed replacement ‘fundamental particles’ which in boundstates form quarks.30,31


3. High End Complexity

The complex emergent behavior realized by this simple relational systemis not generally displayed by models of complex systems. For example, it isgenerally accepted that models capable of generating open ended evolution,or more than two levels of hierarchical behavior have yet to be realized32 .Yet there is every reason to suppose that such behavior is being exhibitedby this model (although this is very difficult to verify). Is the behavior ofthis model complex? We must be careful with our conclusions. There isan ongoing debate in the complex systems science community about whatexactly complex behavior entails, and the search for a single unifying under-standing of complexity may not in fact lead to one obvious definition.5,33

It has recently been proposed that complexity should itself be regardedas a complex concept, with complex systems exhibiting a range of differ-ent behaviors and being appropriately modeled by a number of differenttechniques and concepts12 , more details can be found there. In essencethat discussion introduces a notion of high end complexity to describe thosesystems that cannot be modeled using our more standard reductive tech-niques; systems exhibiting high end complexity tend to require more thanone mode of analysis for a complete understanding2,12 . Such systems alsodisplay highly contextual responses to perturbations, which often meansthat how we analyze them can determine what we see. This suggests thatthe scientific ideal of an objective reality immune from our observations doesnot exist. However, it does not imply that reality is subjective. Generally,the contextual responses of such systems make existing models of their gen-eration from presumably simpler subcomponents unsatisfactory, and theirdynamics and evolution once created is not usually well understood either.We require new theoretical tools capable of properly incorporating suchcontextual dependency into our models. Is it possible that such tools mightresult from the Process Physics approach?

As an example of a system exhibiting high end complexity we mightconsider the process whereby a fertilised egg can divide, differentiate andeventually develop into a complete organism, specified by the informationencoded in its DNA. It is important to realise that this specification is notsufficient for a complete description of the finished organism. Our under-standing of nuclear transfer cloning illustrates the fact that DNA alone willnot undergo the complex molecular processes associated with development,it must be placed in the context of a cell. Also, the DNA cannot be placedin any cell, it must be placed in an appropriate, as well as viable cell in or-der to begin replication. Thus, it is not possible to separate genetic content

90 Kirsty Kitto

from its surrounds and retain any meaningful sense of a functional system.Computational models in, for example, the field of Artificial Life tend toemphasise DNA alone, and the systematic interdepencies of this process ofdevelopment are not ususally considered. This tends to result in a modelthat is capable of exhibiting only very simplistic behavior. A full model ofsuch a process would have to be capable of exhibiting contextual responseslike phenotypic plasticity where organisms with the same genotype may, ifplaced in a different environment, reveal significantly different phenotypes,to the extent that they may even be identified as different species34 .

Additional examples of high end complexity include:

(1) The formation and modeling of ecosystems,8 where, for example, dif-fering understandings of the same system can be generated from ananalysis of it at varying time and spatial scales.

(2) Language evolution and structure is difficult to correctly model, despitewhat is often a very detailed knowledge of its syntax. For example,the process whereby meaning is generated from initially meaninglessutterances is very difficult to understand,35 and postmodern and de-constructive theories suggest that we should be careful in developingtheories of meaning that are too simplistic.36

(3) The impact that culture has upon mental well-being appears tobe significant. For example, culture mediates nearly every aspect ofschizophrenia, from its identification and definition to symptom for-mation, and even to the eventual course and outcome of disorders.37

This suggests that such conditions cannot be analysed reductively withan emphasis on biological factors or upon social factors alone. Rather,the complex interplay of these factors must be incorporated. This is aremarkably common problem, consider for example the nature versusnurture debate.38

(4) A more modern problem is the incorporation of a user into computingmodels, where different services might be required by different usersfrom what is in essence the same system. Often, solutions are too hard-wired, and do not lend themselves to extension or adaption to newsystems as they emerge.

It is interesting to notice that many of the systems exhibiting high endcomplexity have an evolutionary form. While we have a general under-standing of the effect of the process of evolution in the biological realm,our formal and computational models of this process are sadly lacking. Thefact that the more traditional modeling approaches have been unable to


generate open ended evolution2 suggests that our current methodologiesare somehow flawed when it comes to the generation of interesting complexstructure and behavior. What is wrong with our models? A general modelcapable of generating novel hierarchical structure with different interactionsand components at the different levels of understanding, as well as with apossibility for the open ended generation of emergent behavior would go along way to answering this question. Apparently the simple low level rela-tional model of Process Physics offers a first step in that direction, but howdoes it differ from the more standard modeling methodologies?

A number of problems with our current modeling methodologies havebeen identified2,12 which, while not problematic when we consider some ofthe more traditional systems analysed by science, become almost impossibleto circumvent when it comes to the modeling and the generation of systemsexhibiting high end complexity.

These include a tendency to associate emergence with epistemol-ogy, or claim that new behavior is only evident with respect to anobserver. This generates a feeling among a number of complex systemsscience workers that emergence is not a phenomenon per se, rather thatit exists only in the mind of an observer. Although there is undeniablyan aspect of observer dependence in our modeling of high end complex-ity, this does not imply that the phenomenon of emergence is necessarilydependent upon the observer; our models might depend upon the ob-server, but the system itself may exhibit genuine emergence. For exam-ple, according to the most generally accepted theories of, for example,biological evolution, the Earth did not originally house life of any form,but gradually living forms arose, mutated and changed, forming neworganisms, species, phylogenies etc. This is a process of the dynamicalemergence of complex behavior. While there are some borderline casesof species identification, it is generally agreed that species now existwhich did not do so in the past, and this has a noticeable effect notjust in our understanding of organisms, but upon the behavior of thebiotic components of the Earth’s environment. Similar processes canbe found in the evolution of linguistic ability, religion, and the gener-ation of new technology. We require models capable of describing thisdynamical process whereby new structures and interactions can comeinto existence.

The barrier of objects consists of the assumption that our models mustnecessarily describe things, which effectively rules out our ability to

92 Kirsty Kitto

generate novelty. For example, in the standard agent based approachesthere is a tendency to define an agent as an object capable of under-going some number of interactions with its environment. However, thismeans that rather than generating novelty as agents move around somealready defined environment, their modes of interaction are effectivelyused up, resulting in a point where no new behavior is forthcoming.

The barrier of objectivity arises when we assume that objectivity is acore criterion of science despite a remarkable number of indicationsto the contrary which have been arising, often for centuries, in thegenerally ‘softer’ sciences, such as sociology, anthropology and history.Indeed, that last bastion of reductionism and objective measurements,physics, seems to be falling under a similar curse as quantum systemsappear to be rather systematically revealing a tendency to display dif-ferent outcomes, each depending upon different choices of experimentalarrangement.

All of these barriers are discussed more fully in a recent paper,12 whichexamines the way in which they impact upon our ability to understandand to generate high end complexity in our models, before proposing somepossible resolutions to such barriers of modeling.

When we reconsider the model discussed in Section 1.1, we see thatit appears to be circumventing some of these problems: the Universe istaken to be real; a shift is attempted from more traditional object-basedapproaches to a more relational framework; and all analyses of higher levelstructures appear to be dependent in some way upon an observer (whosets thresholds in the low level relational model and then makes secondarymodels in order to characterise emergent behaviour within the system).However, none of these characteristics was actively sought when the initialmodel was investigated39 , rather they have been identified later when anattempt was made to understand the reasons for the success of the model2 .Thus, we might ask at this stage how the historical basis of Process Physicsdiffers from that of the more traditional reductive approaches. Can thisbasis form a foundation for a new modeling methodology capable of bothmodeling and generating complex emergent behavior?

4. The Historical Roots of Process Physics

The fact that the very minimal relational system discussed in Section 1.1both exhibits observational emergence and generates a contextually de-pendent hierarchical structure suggests that it may have some sort of


fundamental characteristic that could be used in a more general model-ing of complex emergent behavior. In order to fully explore this possibilitywe will look at the historical roots of the model.

The iterative map (1) was suggested by the Global Colour Model (GCM)of quark physics.15,40 This model uses the functional integral method, andapproximates low energy hadronic behaviour from the underlying QFT ofquark-gluon behaviour, quantum chromodynamics (QCD). This section willdiscuss the way in which the iterative map (1) was extracted from from theGCM.

4.1. The Global Colour Model (GCM) and the Functional

Integral Calculus (FIC)

QCD is the quantum field theoretic model of quarks and gluons. Startingfrom six quark fields, q, six antiquark fields, q, and eight gluon fields, A, aclassical action can be defined (in the Euclidean metric) as

SQCD[q, q, Aaµ] =

∫d4x

(14F a

αβFαβa +

12ξ

(∂µAaµ)2

+q(γµ(∂µ − ig

λa

2Aa

µ) +M)q

)(9)

where F aµν = ∂µAa

ν−∂νAaµ +gfabcAb

µAcν describes the self-interaction of the

gluons (fabc are the QCD structure constants, and g is the colour charge),λa

2 are the eight SU(3) colour generators in the Gell–Mann representation,γµ are the Dirac matrices, and M = mu,md, . . . are the quark currentmasses. This action can be quantized via a QCD generating functional

Z =∫DqDqDA exp (−SQCD[A, q, q]) (10)

from which correlators of the general form

G(. . . , x, . . . ) =∫ DqDqDA . . . x . . . e−SQCD[A,q,q]

∫ DqDqDAe−SQCD[A,q,q](11)

can be extracted. These are then used to derive the predictions of QCD,however this procedure is very complicated and for this reason a numberof approximations have been developed.

The GCM15,40 is a low energy approximation which is used to extractthe behaviour of hadrons from QCD. The model is motivated by the ob-servation that at low energy (or long wavelength) we observe only thosedegrees of freedom associated with hadron behaviour; individual quarks

94 Kirsty Kitto

and gluons are not directly observable. This suggests that a reasonable ap-proximation to low energy behaviour could be achieved by choosing a set ofvariables which reflect only the behaviour that we can observe in an exper-imental setting. The choice of new variables is not arbitrary, it is achievedusing what are termed Functional Integral Calculus techniques41–43 whichchange variables in a dynamically determined way using analogues of thevarious techniques used in ordinary integral calculus. This process, whichhas been termed action sequencing39 changes the variables of the QCDgenerating functional from the very simple expression in (10) to one thatdescribes the hadron via a sequence of steps:

Z =∫DqDqDAexp (−SQCD[A, q, q]) (QCD) (12)

≈∫DqDqDAexp (−SGCM [A, q, q]) (GCM) (13)

=∫DBDDDD∗exp (−Sbl[B, D, D∗]) (bilocal fields) (14)

=∫DNDN . . .DπDρDω..exp

(−Shad[N , N, .., π, ρ, ω, ..]). (15)

We might consider FIC as the first documented implementation of actionsequencing. The details of this procedure are very complicated, the inter-ested reader is referred to15,40 for further information. As an illustration ofthe power of this approach, we can examine the hadronic action obtainedat the end of the complete action sequencing process (to low order) as,

Shad [ N , N, . . . , π, ρ, ω, . . . ] =∫d4xTr

N

(γ.∂ + m0 + ∆m0 −m0

√2iγ5π

aT a + . . .)

N

+∫

d4x

[f2

π

2[(∂µπ)2 + m2

ππ2] +f2

ρ

2[−ρµ(−∂2)ρµ + (∂µρµ)2 + m2

ρρ2µ]

+f2

ω

2[ρ → ω]− fρf

2πgρππρµ.π × ∂µπ − ifωf3

πεµνστωµ∂νπ.∂σπ × ∂τπ

−ifωfρfπGωρπεµνστωµ∂νρσ.∂τπ

+λi

80π2εµνστTr (π.F∂µπ.F∂νπ.F∂σπ.F∂τπ.F ) + . . .

]. (16)

We shall not list the new notation used in this action; details can be foundin the review article15 or PhD thesis.40 However, even without the detailsof the notation, when we compare this expression to that of the QCD ac-tion (9) we see a vast difference in the complexity of the two equations.Equation (16), which is is only a low order expansion suggests that there is


a very rich set of behavior evident in the behavior of the nucleon; it is anextremely complex system, but its description is well approximated by thistheory. Thus, some very complicated emergent behavior has been extractedusing this technique. Is it possible that this technique could be generalisedin some way? A first step towards answering this question was taken whenthe iterative equation (1) was extracted from this model.

4.2. The Extraction of the Iterator Equation

The richness of the behavior exhibited by this model of hadrons led tothe hypothesis that it may be possible to regain much of the dynamics inphysics without incorporating it into our models axiomatically. We takeas our starting point an action that arises in the action sequencing of thegenerating functional (in equation (14)). This action describes the dynamicsof bilocal fields B (which can be considered very similar to the connectionsbetween nodes in (1)) and appeared in an early‖ GCM paper41

S[B] = −TrLn

[/∂δ(x− y) +

Mθ

2Bθ(x− y)

]+

∫d4xd4y

B(x, y)B(y, x)2g2D(x− y)

,

(17)where /∂δ(x − y) represents an essentially localized momentum term, theMθ term is due to a bilocal meson field, Bθ(x − y) is a hermitian bilocalfield and D(x−y) is a complex bilocal field. See the early paper for details,or the more modern and very comprehensive PhD thesis.40

The stochastic quantisation (SQ) procedure of Parisi and Wu44,45 is usedto quantize this action. SQ consists of introducing a 5th time τ in additionto the usual 4 space-time points xµ and postulating that the dynamics ofsome field represented by φ in this extra time τ is given by the Langevinequation:

∂φ(x, τ)∂τ

= −δS[φ]δφ

+ η(x, τ), (18)

where η is a Gaussian random variable. The stochastic average of all fieldsφn satisfying equation (18) is calculated where τ1 = τ2 = · · · = τl, and

‖This paper was an early attempt at using FIC to bosonize QCD but made use of a lessappropriate change of variables than the later work utilising this technique. The changeof variables used here was to 1c and 8c bilocal qq variables which worked well in theextraction of meson observables but was less physical, due to the 8c fields which arerepulsive for for qq states. The later work was based upon 1c meson variables coupledwith 3c and 3c diquark variables.42 The less physical equation was used as a basis forthe derivation because it is slightly simpler than its equivalent in the later work but stillresults in rich behaviour.

96 Kirsty Kitto

finally the limit τ1 → ∞ is taken. Parisi and Wu proved perturbativelythat in this limit the average is equal to the correlation function of the fieldof interest, i.e., an l-point correlation can be defined where

limτ1→∞

〈φη(x1, τ1)φη(x2, τ1)..φη(xl, τ1)〉η =∫ Dφ φ(x1)φ(x2)..φ(xl)e−S[φ]

∫ Dφe−S[φ].

(19)Thus, at least in the perturbative sense, SQ provides an alternative routeto quantization, and in particular to the correlation functions of QFT.

We make use of this procedure to quantize the action (17), substitutingit into the Langevin equation (18) to obtain

∂B(xµ, τ)∂τ

= −δS[B]δB

+ η(xµ, τ). (20)

Thus, the B field will dynamically update in this time parameter as B →B + δB, that is:

B −→ B − δS[B]δB

+ η(xµ, τ). (21)

Returning to the action (17), we make a number of simplifications whichstrip away structure considered unimportant to the model. Namely we setany variables that are deemed irrelevant equal to 1, drop the gluon interac-tion term D(x−y) and set the variables of the B terms in the integral termas equivalent (i.e., assuming no significant interaction between the fields atthis level of the description). The term /∂δ(x− y) represents an essentiallylocalised momentum term and is ignored. Similarly the trace operation isdiscarded as it simplifies too much of the behaviour of a system that hasalready been dramatically simplified. This results in a simpler action:

S[B] = Ln [B] +∫

d4xd4yB(x, y)2. (22)

Now, making a transition to a matrix based lattice representation in placeof the continuous representation above, we obtain

S[Bij ] = B2ij + Ln [Bij ] . (23)

Substituting (23) into (21), and changing the symbol η to ω, an equivalentnoise term which operates over the matrix representation of the B fieldresults in an update equation:

Bij → Bij −(

δ

δBijB2

ij +δ

δBijLn [Bij ]

)+ ωij . (24)

Finally, we make use of the calculus identities δB2 → B and δlnB → B−1

to extract the structure of the desired equation(1).


It is worth emphasising that the use of the stochastic quantization pro-cedure in this process suggests that a system emergent from this equationshould exhibit quantum behaviour in the limit of sufficient numerical ex-periments that run for a long enough time. Note also that this is more ofan extraction than a derivation; there is almost a sense of observation inthe derivation itself, where we choose which variables to consider relevantand which to ignore.

It is remarkable how much richness of behavior is retained in the behav-ior of (1), this is despite the dramatic loss of standard physical structurethat is characteristic in this derivation. In this case it appears that a setof key characteristics are driving this behavior. Some of these include thenonlinear aspect of the equation and its grossly ‘nonlocal’ and holistic form,as well as the noise term that pushes its behavior away from equilibrium.Most important perhaps is the structure of the general equation, it is closeenough to the bilocal action that it incorporates its key physical propertiesin some way.

Indeed, it has been argued2 that the key structure remaining after thisextraction is related to the Nambu–Goldstone (NG) modes30 that are usedin the action sequencing of the QCD action. Indeed, the NG-modes formthe basis of the choice of physical variables in this process.15 The GCMis particularly effective in revealing the NG phenomena that follows fromthe dynamical breaking of chiral symmetry.30 Indeed, the GCM results inthe complete derivation of the Chiral Perturbation Theory phenomenology,but with the added feature that the induced NG effective action is non-local, so that the usual non-renormalisability problems do not arise. It ishighly likely that NG-modes are particularly important in the modeling ofcomplex emergent behaviour,2 a point that we shall return to shortly.

It is expected that there will be a class of equations all of which exhibitbehaviour similar to that discussed in section 1.1, but at this point in time(1) is the only one known. There is some reason to suppose that such aclass of models is already being discovered in the field of complex systemsscience, consider for example a very simple network model on the rise andfall of societies46 that behaves very similarly to (1). Is it possible that a classof equations such as (1) can be used in the modeling of complex emergentbehaviour? What can be determined about such a general class of models?

5. Quantum Theories as Models of Complexity

It is very likely that an essential characteristic of such a class of modelswill lie in in quantum theory. This section will briefly discuss this link

98 Kirsty Kitto

between quantum theories and models of complex behaviour, a more in-depth discussion is currently in preparation.

All quantum theories have the same fundamental structure. Indeed theimplementation of any quantum theory involves following the same generalrecipe:

(1) First, a map from a classical state space, S to a complex number C isfound. This map is often written in the form of the symbol ψ.

(2) Depending upon the system under examination some time evolutionequation is chosen from a set of possibilities including the Schrodingerequation, the Klein-Gordon equation, the Dirac equation etc., each ofwhich map to another set of complex numbers.

(3) Steps 1 and 2 are mathematically well defined and understood. How-ever, complex numbers are never revealed when a measurement is per-formed on a quantum system. Instead, the system is found to be insome classical state, related to the configuration of the experiment per-formed47,48 . The dynamics of this process are not understood, andthere are a number of competing theories of quantum measurement,each of which lead to a different interpretation of quantum mechan-ics47,49 . Mathematically, the process of measurement is implementedby mapping the inner product of ψ at the point of time in which weare interested to some sort of probability space from which a set ofpredictions about the system are obtained, in a basis determined bythe act of measurement itself.

Even the implementation of QFT’s involves the same procedure, albeit ina far more complex form2 , a point that becomes obvious with a shift tothe modern path integral formulation50 .

Although traditionally the quantum formalism has only been applied toa very particular set of systems, a wide variety of more novel applicationsare starting to appear, where quantum theories of macroscopic systems arebeing created, often quite successfully51 . For example, different varietiesof the quantum formalism have been applied to situations such as: stockmarket analysis52 ; quantum models of the brain53,54 ; models of cognitivefunction and concepts55–57 ; modeling of the process of decision making insituations of ambiguity58 etc. This general use (some might argue abuse) ofthe quantum formalism suggests that it is indeed far more generally appli-cable than is traditionally considered to be the case, and indeed the aboveconsideration of the form of the quantum formalism suggests a reason forthis; there is no mention of macroscopic detectors or microscopic particles


in this formalism, and there is no reason to suppose a priori that they arenecessary. This is merely an historical bias resulting from the discovery ofthe quantum formalism as a description of a specific class of systems (i.e.microscopic ones).

A clue to this apparent generality of the quantum formalism lies in thetheorems, generally attributed to Bell, of nonlocality and contextuality.48

Each of these theorems rely upon showing that when a quantum systemis entangled∗∗ there exists a set of observables for which it is impossibleto consistently assign an eigenvalue i.e. the outcomes of measurements ofapparently independent experiments are incompatible. The resolution tothis incompatibility lies in a proper consideration of the experimental ar-rangement; performing one experiment always results in a change of thequantum system and rules out the possibility of performing an alternativeone. Thus, it is impossible to completely describe a quantum system with-out reference to its context. Entangled quantum systems exhibit a form ofnonseparable behaviour and should not be considered independently of theset of measurements performed upon them.

This situation shares much similarity with systems exhibiting high endcomplexity. Such systems should not be considered independently of theircontext, and may show incompatible results depending upon the measure-ments to which they are subjected. For example, as was discussed in Sec-tion 3 the social context in which schizophrenia occurs can have a dramaticeffect upon the course of a patients illness. Indeed, different patients may beclassified as schizophrenic or not depending upon the culture in which theyare being diagnosed.37 This situation is analogous to the incompatible mea-surements occurring in the quantum formalism, and hence it is expectedthat the very well developed quantum formalism could be used to providemodels of such contextual dependency during measurement.

This suggests that the quantum theoretic formalism can be understoodas modeling generic situations of contextuality where a system cannot beconsidered reductively as a set of separable subcomponents uninfluencedby their environment, even in cases where two subsystems are spread overa distance.2

∗∗An entangled state consists of at least two noninteracting systems A and B, representedby the states ψA and φB on the Hilbert spaces HA and HB , the composite state of whichis written ψA⊗φB which cannot itself be separated in a meaningful manner as a productψAφB .

100 Kirsty Kitto

This contextual dependence often manifests itself as randomness arisingfrom a lack of knowledge about the outcome of experiments, which canbe used to explain the appearance of randomness in systems exhibitingcontextual behavior, including quantum ones59 .

Two specific examples of the way in which the quantum formalism couldbe used to model such systems can be found quite quickly. Firstly, it is pos-sible to formulate Bell-type inequalities which can be used to test whether asystem is exhibiting contextual responses.12 Another possibility would con-sist of incorporating the effect that an observer might have upon a systemthrough finding a set of basis states to represent the system that are com-patible with the basis states available to an observer and then performing‘measurements’ as is done in standard quantum theory.60

If QT is capable of describing such a rich set of behaviour then we mightask what is to be gained by a QFT as opposed to the simpler QT. The an-swer to this question lies at the heart of complex emergent behaviour. Someclues are supplied by early work performed by pioneers such as Primas whonoted that it is possible to perform a perturbation expansion over multipletime scales,61 by Froolich who proposed that coherent phase correlationswill play “a decisive role in the description of biological materials and theiractivity”,62 and Davydov who proposed the concept of a biological solitonthat resists thermal fluctuations.63 Taken together these results suggestthat QFT has many applications in nontraditional fields, but despite thisinitial promise there has been a tendency for the larger physics communityto ignore these possible extra applications.

The full power of QFT becomes evident when symmetry breaking isincorporated into the formalism. If a system falls into a situation where itsground state does not have the same continuous symmetry as its dynamicsthen it is considered to be exhibiting spontaneous symmetry breaking. Ac-cording to Goldstone’s theorem, a system in such a state will dynamicallygenerate massless bosons, termed Nambu–Goldstone modes (NG-modes),in response to this break in symmetry. The number of NG-modes gener-ated will equal the number of broken symmetries in the system.30 Due totheir massless nature, NG-modes are long-range, they can move throughan entire system with no loss of energy, providing it with long range co-herence.54 Because of their boson status many NG-modes may occupy thesame ground state without changing the energy of the system. In such sys-tems there can be a number of structurally different ground states, eachin a lowest energy configuration. This is not possible in a classical system;there is only one lowest energy state in the classical realm, which means


that there is only one way in which a system can exhibit stable behaviour.On the other hand systems described by a QFT can exist in many differentstable configurations, which allows them to change state from one stableground state to another and in the process generate new behaviour of avery rich form. Because of this phenomenon, QFT is the only quantum for-malism capable of generating truly emergent behaviour rather than merelymodeling a set of components and their interactions.2,54,64 This is to be ex-pected since its basis could be argued to lie in the necessity of modeling thecreation and annihilation of particles in modern high energy experiments;QFT was invented in order to model the emergence of new particles.

Thus, the origin of Process Physics in QFT and, in particular its de-pendence upon NG phenomena via the extraction of (1) from the GCMsuggests that such a model can, not only exhibit contextual responses, butcan truly generate complex emergent behaviour.

6. Generating Complex Emergent Behaviour — a NewModeling Approach?

We shall conclude with a brief summary of some of the more recent devel-opments and applications that have followed from this more process drivenapproach to modeling complex emergent behavior.

The historical basis of Process Physics in QFT suggests that such the-ories are capable of describing far more systems than has generally beenconsidered the case. As described above, the process of action sequenc-ing applied in the extraction of hadronic structure from the GCM reliesstrongly upon the use of symmetry breaking and the associated emergenceof Nambu–Goldstone modes which are chosen as the dynamically extractednew variables. It is expected that this process will provide the key to thediscovery of a general mechanism whereby complex emergent behaviour canbe generated2 . This idea has been explored2,12 using a very simple modelof sympatric speciation which is based upon quantum field theory. Themodel utilises dynamical symmetry breaking and the associated emergenceof NG-modes to model the emergence of new sets of species within a systemundergoing dynamical change. This model of speciation can be consideredas a model of the differentiation of a species, and it is expected that suchmodels can be generalised to a model of the process of differentiation asit occurs in many systems exhibiting this form of behavior. For example,a key future goal is to use these concepts to create a model of the processof biological development. This process can be understood as consisting oftwo sub-processes, differentiation (the specification of different cell fates)

102 Kirsty Kitto

and morphogenesis (the development of shape), and leads to the formationof an organism. It is expected that far more applications are possible. Thisis because differentiation can be understood as the process responsible forforming new word meanings in language, new religions, new cultures andsocieties etc. Hence, a dynamical model of differentiation could be broadlyused in modeling processes of generation, evolution and the creation ofnovelty. A general class of such models would be very important in themodeling of systems exhibiting high end complexity.

Interestingly, some models along these lines already exist. For example,a quantum field theoretic model of the brain has been developed over anumber of years, which makes use of the infinite number of unitarily in-equivalent ground states required by QFT to model processes such as theformation of long term memories as situations where new stable states de-scribing the brain are created by the exposure of a person to new ideas etc.Short term memories are described as excitations of the ground state, andthe process of recall is modeled using the excitation of NG-modes from thecurrent ground state of the brain. This model explains the serial nature ofassociation as a situation where an excitation gradually decays via inter-mediate states back into a ground state. The book by Vitiello54 containsa good overview of this work, as well as an exhaustive list of referencesto the more detailed papers describing these results. Thus, there is everyreason to suppose that such models could be very useful in the dynamicalmodeling of emergent stable structures in general. Indeed, in the pioneeringpaper on the QFT brain model,65 Ricciardi and Umezawa suggest that asimilar model might possibly be used to explain how a stable DNA code isdynamically generated.

Another promising avenue for the quantum modeling of complexity liesin the modeling of human interactions with social, computational and eco-logical systems providing services to those humans. With the generalisedunderstanding of quantum theories summarised in Section 5, we begin tounderstand something that has been appreciated by the so-called ‘soft’sciences almost since their inception; that contradictory results can be ob-tained from the assumption that different choices of experiment are inde-pendent from one another. However, with an acceptance that contextualoutcomes are the norm of the quantum formalism we acquire a new, lessarbitrary, theory with which we can incorporate choice into scientific dis-course. For example, the concept of a “service ecosystem” has recently beendeveloped66 to describe the connection of different web services which aredeployed, published, discovered and delivered to different business channels


through specialist intermediaries. In such cases a richer range of semanticsin service descriptions and support of fuzzier search goals becomes necessaryin order to create adequate descriptions and results for the end users. Suchsystems are highly contextual, and as such, their modeling by the standardreductionist methodology is inappropriate. However, with a formalism thatincorporates contextual outcomes into its very foundation, quantum theoryprovides a number of possibilities for a set of more appropriate modelingtechniques. Instead of hardwiring a plethora of different services into anyattempt to describe such systems, a more malleable and hence long termsolution presents itself along the lines discussed in Section 5.60 Namely theeffect of the user upon a system can be modeled through finding a setof basis states to represent the system that are compatible with the ba-sis states available to functions performed by a user and then performing‘measurements’ as is done in standard quantum theory Thus, we see a gen-eral possibility that quantum theories can provide a new class of modelscapable of incorporating more fully the choices made by an observer whotakes a more active role in the behaviour of a system than is traditionallyconsidered to be the case, i.e., is interacting with the system in choosingwhich measurements will be performed upon it.

Finally, the simple relational model discussed in section 1.1 has beenparticularly successful as a pregeometric theory. It has been used as the ba-sis of a higher level theory of quantum gravity1 which is proving remarkablysuccessful in the explanation of a number of so far unexplained phenom-ena. However this theory is remarkably complex when compared to themore traditional object based approaches. This leads to the final questionof this paper and indeed to our conclusions.

7. What has been Gained by a Shift to Process Physics?(In Conclusion)

The shift to a process oriented physics is remarkably difficult to imple-ment. All of our standard modes of analysis are based upon reductive,object based techniques that assume a passive observer unable to interactwith the system of interest in any way other than via controllable, identifiedchannels. In attempting to construct a more complex understanding of theworld (which is in fact more complex) we lose many of the long held, almostsacrosanct, assumptions of the scientific method. The accompanying diffi-culty of analysis is to be expected. However, many of the phenomena andproblems currently unexplainable, and even impossible to correctly framein more standard approaches become at least approachable from this new

104 Kirsty Kitto

perspective. Many mysteries of physics become less mysterious, and as thispaper has attempted to show, some of the problems outside of standardphysics also appear to be amenable to this new methodology. It has beenargued in this paper that the success of such models will lie with the generalclass of quantum theories, and in particular, in the class of QFT’s.

In conclusion I would like to point out that I still teach the standardobject-driven and static physics to undergraduates, in much the same waythat I begin a discussion of mechanics with Newton’s laws rather than withGeneral Relativity. There is a very good reason for this, and it is not merelyhistorical. Taking account of complexity is a very difficult process, and inmany scenarios it is not necessary. Even in situations where context mustbe incorporated it is not always necessary, or even desirable to go to afully-fledged QFT, and it is highly likely that there will be a large numberof complex systems that will not be describable by any form of quantumformalism. Systems exhibiting high end complexity cannot be fully modeledby one language of description alone;2,12 we require more tools and moreapproaches before we can ever hope to understand this very important classof system, and the approach outlined above is merely one proposal in thisdirection.

Acknowledgement

The author would like to acknowledge the support of her PhD supervi-sor R.T. Cahill, without whose support she could have taken a lifetime togenerate these concepts.

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109

A Cross-disciplinary Framework for the Description ofContextually Mediated Change

Liane Gaboraa and Diederik Aertsb

aDepartment of Psychology, University of British ColumbiaOkanagan Campus, 3333 University Way, Kelowna BC, V1V 1V7, Canada

bLeo Apostel Centre for Interdisciplinary Studies andDepartment of Mathematics, Brussels Free University, Brussel, Belgium

[email protected]

We present a mathematical framework (referred to as Context-driven Actual-ization of Potential, or CAP) for describing how entities change over time underthe influence of a context. The approach facilitates comparison of change ofstate of entities studied in different disciplines. Processes are seen to differ ac-cording to the degree of nondeterminism, and the degree to which they aresensitive to, internalize, and depend upon a particular context. Our analysissuggests that the dynamical evolution of a quantum entity described by theSchrodinger equation is not fundamentally different from change provoked bya measurement often referred to as collapse but a limiting case, with only oneway to collapse. The biological transition to coded replication is seen as ameans of preserving structure in the face of context, and sexual replication asa means of increasing potentiality thus enhancing diversity through interactionwith context. The framework sheds light on concepts like selection and fitness,reveals how exceptional Darwinian evolution is as a means of ‘change of state’,and clarifies in what sense culture (and the creative process underlying it) areDarwinian.

Keywords: Change of State, Collapse; Context; Dynamical Evolution; Evolu-tion; Natural Selection; Nondeterminism; Unifying TheoryPACS(2006): 01.70.+w; 01.90.+g; 87.10.+e; 87.15.He; 89.75.Fb

1. Introduction

This paper elaborates a general theory of change of state (a nascent, non-mathematical draft of which is presented elsewhere1) with the goal of unit-ing physical, biological, and cultural evolution, not reductively, but througha process that may be referred to as inter-level theorizing2 . Other unifyingtheories have been put forward, such as that of Treur3 which takes tempo-ral factorization as its unifying principle. Our scheme focuses on the role of

110 Liane Gabora and Diederik Aerts

context ; i.e. on the fact that what entities of different kinds have in com-mon is that they change through a reiterated process of interaction with acontext. We refer to this fundamental process of change of state under theinfluence of a context as context-driven actualization of potential, or CAP.By “potential” we do not mean determined or preordained; indeed, differ-ent forms of evolution differ according to the degree of nondeterminism,as well as degree of contextuality and retention of context-driven change.We conclude with several examples of how the scheme has already bornfruit: it suggests a unifying scheme for the two kinds of change in quantummechanics4 , illustrates why the earliest forms of life could not have evolvedthrough natural selection5 , and helps clarify how the concept of evolutionapplies to culture6,7 and creative thought8,9 .

2. Deterministic and Nondeterministic Evolution under theInfluence of a Context

In this section we discuss the kinds of change that must be incoroporated ina general scheme for the description of the evolution of an entity under theinfluence of a changing or unchanging context. We use the term evolutionto mean simply ‘change of state’. Thus our neither implies natural selectionnor change in the absence of a measurement; the term is thus used as itwas prior to both Darwin and Schrodinger.

Since we do not always have perfect knowledge of the state of the entity,the context, and the interaction between them, a general description of anevolutionary process must be able to cope with nondeterminism. Processesdiffer with respect to the degree of determinism involved in the changes ofstate that the entity undergoes. Consider an entity—whether it be physical,biological, mental, or some other sort—in a state p(ti) at an instant oftime ti. If it is under the influence of a context e(ti), and we know withcertainty that p(ti)changes to state p(ti+1) at time ti+1, we refer to thechange of state as deterministic. Newtonian physics provides the classicexample of deterministic change of state. Knowing the speed and positionof a material object, one can predict its speed and position at any time inthe future. In many situations, however, an entity in a state p(ti) at timetiunder the influence of a context e(ti)may change to any state in the setp1(ti+1), p2(ti+1), ..., pn(ti+1), .... When more than one change of state ispossible, the process is nondeterministic.

Nondeterministic change can be divided into two kinds. In the first,the nondeterminism originates from a lack of knowledge concerning thestate of the entity p(ti) itself. This means that deep down the change is

Cross-disciplinary Framework for Contextually Mediated Change 111

deterministic, but since we lack knowledge about what happens at thisdeeper level, and since we want to make a model of what we know, the modelwe make is nondeterministic. This kind of nondeterminism is modeled bya stochastic theory that makes use of a probability structure that satisfiesKolmogorov’s axioms.

Another possibility is that the nondeterminism arises through lack ofknowledge concerning the context e(ti), or how that context interacts withthe entity. Yet another possibility is that the nondeterminism is ontologi-cal i.e. the universe is at bottom intrinsically nondeterministic. It has beenproven that in these cases, the stochastic model to describe this situation ne-cessitates a non-Kolmogorovian probability model. A Kolmogorovian prob-ability model (such as is used in population genetics) cannot be used10−15.It is only possible to ignore the problem of incomplete knowledge of contextif all contexts are equally likely, or if context has a temporary or limitedeffect. Because the entity has the potential to change to many differentstates (given the various possible states the context could be in, since welack precise knowledge of it), we say that it is in a potentiality state withrespect to context. This is schematically depicted in Figure 1.

We stress that a potentiality state is not predetermined, just waiting forits time to come along, at least not insofar as our models can discern, pos-sibly because we cannot precisely specify the context that will come alongand actualize it. Note also that a state is only a potentiality state in relationto a certain (incompletely specified) context. It is possible for a state to bea potentiality state with respect to one context, and a deterministic statewith respect to another. More precisely, a state that is deterministic withrespect to a context can be considered a limit case of a potentiality state,with zero potentiality.

In reality the universe is so complex we can never describe with com-plete certainty and accuracy the context to which an entity is exposed, andhow it interacts with the entity. There is always some possibility of evenvery unlikely outcomes. However, there are situations in which we can pre-dict the values of relevant variables with sufficient accuracy that we mayconsider the entity to be in a particular state, and other situations in whichthere is enough uncertainty to necessitate the concept of potentiality. Thusa formalism for describing the evolution of these entities must take intoaccount the degree of knowledge we as observers have about the context.


Fig. 1. Graphical representation of a general evolution process. Contexts e(t0), e(t1),e(t2) and e(t3) at times t0, t1, t2, and t3, are represented by vertical lines. States ofthe entity are represented by circles on vertical lines. At time t0 the entity is in statep(t0). Under the influence of context e(t0), its state can change to one of the states inthe set p1(t2), p2(t2), p3(t2), ..., pn(t2), .... These potential changes are represented bythin lines. Only one change actually takes place, the one represented by a thick line,

i.e. p(t0) changes to p4(t1). At time t1 the entity in state p4(t1) is under the influenceof another context e(t1), and can change to one of p1(t2), p2(t2), p3(t2), ..., pn(t2), ....Again only one change occurs, i.e. p4(t1) changes to p3(t2). The process then starts allover again. Under the influence of a new context e(t2), the entity can change to oneof p1(t3), p2(t3), p3(t3), p4(t3), ..., pn(t3), .... Again only one change happens: p3(t2)changes to p5(t3). The dashed lines from states that have not been actualized at acertain instant indicate that much more potentiality is present at time t0 than explicitlyshown. For example, if p(t0) had changed to p2(t1) instead of p4(t2) , which was possibleat time t0, then context e(t1) would have exerted a different effect on the entity at timet1, such that a new vertical line at time t1 would have to be drawn, representing anotherpattern of change.

3. SCOP: A Mathematics that IncorporatesNondeterministic Change

We have seen that a description of the evolutionary trajectory of an entitymay involve nondeterminism with respect to the state of the entity, the


context, or how they interact. An important step toward the developmentof a general framework for how entities evolve is to find a mathematicalstructure that can incorporate all these possibilities. There exists an elab-orate mathematical framework for describing the change and actualizationof potentiality through contextual interaction that was developed for quan-tum mechanics. However it has several limitations, including the linearityof the Hilbert space, and the fact that one can only describe the extremecase where change of state is maximally contextual. Other mathematicaltheories, such as State COntext Property (SCOP) systems, lift the quan-tum formalism out of its specific structural limitations, making it possibleto describe nondeterministic effects of context in other domains 16−29. Theoriginal motivation for these generalized formalisms was theoretical (as op-posed to the need to describe the reality revealed by experiments). In theSCOP formalism, for instance, an entity is described by three sets Σ, Mand L and two functions µ and ν. Σ represents that set of states of theentity, M the set of contexts, and L the set of properties of the entity.

The function µ is a probability function that describes how state p ∈ Σunder the influence of context e ∈ Mchanges to state q ∈ Σ. Mathemati-cally, this means that µ is a function defined as follows

µ : Σ ×M× Σ → [0, 1](q, e, p) → µ(q, e, p)

(1)

where µ(q, e, p) is the probability that the entity in state p changes to stateq under the influence of context e. Hence µ describes the structure of thecontextual interaction of the entity under study.

3.1. Sensitivity to Context

A parameter that differentiates evolving entities is the degree of sensitivityto context, or more precisely, the degree to which a change of state of con-text evokes a change of state of the entity. Degree of sensitivity to contextdepends on both the state of the entity and the state of the context. Thedegree of sensitivity to context is expressed by the probability of collapse.If the probability is close to 1, it means that this context almost with cer-tainty (deterministically) causes the state of the entity to change to thecollapsed state.


If the probability is close to zero, it means that this context is unlikely tocauses the state of the entity to change to the collapsed state. Specifically,if µ(q, e, p) = 0, then e has no influence on the entity in state p, and ifµ(q, e, p) = 1, then e has a strong, deterministic influence on the entity instate p. A value of µ(q, e, p) between 0 and 1, which is the general situation,quantifies the influence of context e on the entity in state p.

3.2. Degree of Nondeterminism

Suppose we consider the set

µ(q, e, p)|e ∈ M ⊂ [0, 1] (2)

which is a set of points between 0 and 1. Suppose this set equals the sin-gleton 1. This would mean that for all context e ∈ M the entity in statep changed to the entity in state q. Hence this would indicate a situation of‘deterministic change independent of context’ for the entity, between state p

and state q. On the contrary, if this set equals the singleton 0, this wouldmean that the entity in state p never changes to state q, again independentof context. The general situation of the set being a subset of the interval[0, 1] hence models in a detailed way the overal contextual way two statesp and q are dynamically connected.

3.3. Weights of Properties

We need a means of expressing that certain properties are more applicableto some entities than others, or more applicable to entities when they arein one state than when they are in another state. The function ν describesthe weight (which is the renormalization of the applicability) of a certainproperty given a specific state. This means that ν is a function from theset Σ×L to the interval [0, 1], where ν(p, a) is the weight of property a forthe entity in state p. We write

ν : Σ × L → [0, 1](p, a) → ν(p, a).

(3)

The function ν, contrary to µ, describes the internal structure of theentity under investigation. Again we can look to some special situations toclarify how ν models the internal structure. Suppose that we have ν(a, p) =1. This means that for the entity in state p the property ahas weight equalto 1, which means that it is actual with certainty (probability equal to 1).Hence, this represents a situation where the entity, in state p, ‘has’ the


property ‘in acto’. On the contrary, if ν(a, p) = 0, it means that for theentity in state p is not at all actual, but completely potential. A value ofν(a, p)between 0 and 1 describes in a refined way ‘how property a is’ whenthe entity is in state p. Hence the set

ν(a, p)|a ∈ L ⊂ [0, 1] (4)

gives a good description of the internal structure of the entity, i.e. howproperties are depending on the different states the entity can be in.

4. Expressing Dynamics: Context-driven Actualization ofPotential (CAP)

Context-driven Actualization of Potential, or CAP, is the dynamics of en-tities modeled by SCOP. This which means that the mathematical modelfor CAP will be as follows: at moment ti we have a SCOP

(Σ, M, L, µ, ν)(ti) = (Σ(ti), M(ti), L, µ, ν) (5)

where Σ(ti) is the set of states of the entity at time ti and M(ti) is theset of relevant contexts at time ti. L is independent of time, since it isthe collection of properties of the entity under consideration. Propertiesthemselves do not change over time, but their status of ‘actual’ or ‘potential’can change with the change of the state of the entity. One should in factlook to it the other way around: a property is an element of the entityidependent of time, or, it is exactly the elements independent of time thatgive rise to properties. That is also the reason that L and ν describe theinternal context independent structure of the entity under consideration.

In Figure 1 a typical dynamical pattern of CAP is presented. Four con-secutive moments of time t0, t1, t2 and t3 are considered. It can be seen inthe figure how, for example context e(t0) has an influence on the entity instate p(t0) which is such that the entity can change into one of the statesof the set p1(t1), p2(t1), p3(t1), ..., pn(t1). This is an example of a generalnondeterministic type of change. Similar types of changes are representedin the figure, under influence of contexts e(t1), e(t2) and e(t3). What is im-portant to remark is that the actual change taking place is a path throughthe graph of the figure, but the states not touched by this path remainof influence for the overall pattern of change, since they are potentialitystates.

The states p0(t0), pi1(t1), pi2(t2), ..., pin(tn), ... constitute the trajectoryof the entity through state space, and describe its evolution in time. Thus,the general evolution process is broadly construed as the incremental change


that results from recursive, context-driven actualization of potential, orCAP. A model of an evolutionary process may consist of both deterministicsegments, where the entity changes state in a way that follows predictablygiven its previous state and/or the context to which it is exposed, and/ornondeterministic segments, where this is not the case. With a generalizedquantum formalism such as SCOP it is possible to describe situations withany degree of contextuality. In fact, classical and quantum come out as spe-cial cases: quantum at one extreme of complete contextuality, and classicalat the other extreme, complete lack of contextuality30,31,32 . This is why itlends itself to the description of context-driven evolution.

4.1. Degree to which Context-driven Change is Retained

We saw that if µ(q, e, p) = 1, then e has a strong, deterministic influence onthe entity in state p. However, an entity may be sensitive—readily undergochange of state due to context—but through regulatory mechanisms or self-replication have a tendency to return to a previous state. Thus although it issusceptible to undergoing a change of state from p to q, its internal dymaicsare such that it goes back to state p. A simple example of a situation wherethis is not the case, i.e. context-driven change is retained is a rock breakingin two. An example where it is the case, i.e. context-driven change is notretained is the healing of an injury or the birth of offspring with none ofthe characteristics their parents acquired over their lifetimes. In the caseof biological organisms, we are considering two interrelated entities, oneembedded in the other: the organism itself, and its lineage.

4.2. Context Independence

The extent to which a change of context threatens the survival of the entitycan be referred to as context dependence. The degree to which an entity isable to withstand, not just its particular environment, but any environ-ment, can be referred to as context independence. Sensitivity to and reten-tion of context can lead, in the long run, to either context dependence orcontext independence. This can depend on the variability of the contextsto which an entity is exposed. A static, impoverished environment mayprovide contexts that foster specializations tailored to that particular envi-ronment, whereas a dynamic, rich, diverse environment may foster generalcoping mechanisms.


Thus for example, a species that develops an intestine specialized forthe absorption of nutrients from a certain plant that is abundant in itsenvironment exhibits context dependence, whereas a species that becomesincreasingly more able to consume any sort of vegetation exhibits contextindependence.

Whether an entity exhibits context dependence or independence maysimply reflect what one chooses to define as the entity of interest. If anentity splits into multiple ‘versions’ of itself (as through reproduction) eachof which adapts to a different context and thus becomes more contextdependent, when all versions are considered different lineages of one jointentity, that joint entity is becoming more context-independent. Thus forexample, while different mammalian species are becoming more contextdependent, the kindom as a whole is becoming more context independent.

5. Ways Found by Universe to Actualize Potential

We now look at how different kinds of evolution fit into the above frame-work, and how their trajectories differ with respect to the parameters intro-duced in the previous section. They are all means of actualizing potentialthat existed due to the state of the entity, the context, and the nature oftheir interaction, but differ widely with respect to these parameters.

5.1. The Evolution of Physical Objects and Particles

We begin by examining three kinds of change undergone by physical enti-ties. The first is the collapse of quantum particles under the influence of ameasurement. The second is the evolution of quantum particles when theyare not measured. The third is the change of state of macroscopic physicalobjects.

Nondeterministic Collapse of a Quantum ParticleLet us begin with change of state in the most micro of all domains,

quantum mechanics. The central mathematical object in quantum mechan-ics is a complex Hilbert space H , which is a vector space over the fieldC of complex numbers. Vectors |p(ti)〉 of this Hilbert space of unit lengthrepresent states p(ti) of a quantum particle. This is expressed in Hilbertspace by the bra-ket of the vector with itself being equal to 1, hence

〈p(ti) | p(ti)〉 = 1. (6)

A measurement e(ti) on a quantum particle is described by a self-adjointoperator E(ti), which is a linear function on the Hilbert space, hence

E(ti) : H → H (7)


such that

E(ti)(a |p1(ti)〉 + b |p2(ti)〉) = aE(ti) |p1(ti)〉 + bE(ti) |p2(ti)〉 (8)

A self-adjoint operator always has a set of special states associated with it,the eigenstates. A state p(ti) described in the Hilbert space Hby means ofthe vector |p(ti)〉 is an eigenstate of the measurement e(ti) represented bythe self-adjoint operator E(ti) if and only if we have

E(ti) |p(ti)〉 = a |p(ti)〉 (9)

where a ∈ C is a real number, and a is called the eigenvalue of themeasurement e(ti) the quantum particle being in state p(ti). Let us de-note the set of eigenstates corresponding to the measurement e(ti) byp1(ti), p2(ti), ..., pn(ti), ..., and the set of their corresponding eigenvaluesby a1, a2, ..., an, ....

Let us mention for mathematical completeness, that only when the self-adjoint operator has a point spectrum do we encounter the above situationin quantum mechanics. Measurements described by operators with a con-tinuous spectrum must be treated in a more sophisticated way. However,this is of no relevance to the points made here.

An eigenstate pj(ti) does not change under the influence of the mea-surement e(ti) described by the self-adjoint operator E(ti), and the corre-sponding eigenvalue aj is the value obtained deterministically by the mea-surement. However the set of eigenstates corresponding to a measuremente(ti) is only a subset of the total set of states of the quantum particle.States that are not eigenstates of the measurement e(ti) are called super-position states with respect to the measurement e(ti). Suppose that q(ti)is such a superposition state of the quantum particle. The vector |q(ti)〉 ofthe Hilbert space representing this superposition state can then alwaysbe written as a linear combination, i.e. a superposition, of the vectors|p1(ti)〉 , |p2(ti)〉 , ..., |pn(ti)〉 , ...representing the eigenstates of the quan-tum particle. More specifically

|q(ti)〉 =∑

j

λj |pj(ti)〉 (10)

where λj ∈ C are complex numbers such that∑

j

|λj |2 = 1. (11)

The physical meaning of the coefficients λj is the following: |λj |2 is theprobability that the measurement e(ti)will yield the value aj and as a conse-quence of performing the measurement e(ti), the superposition state |q(ti)〉


will change (collapse) to the eigenstate |pj(ti)〉with probability equal to|λj |2. This change of state from a superposition state to an eigenstate isreferred to as collapse. In general (depending on how many |λj |2 are non-zero), many eigenstates are possible states to collapse to under the influenceof this measurement. In other words, the collapse is non-deterministic. Thismeans that a genuine superposition state is a state of potentiality with re-spect to the measurement. This suggests that what we refer to as a contextis the same thing as what in the standard quantum case is referred to as ameasurement.

Thus a quantum entity exists in general in a superposition state, and ameasurement causes it to collapse nondeterministically to an eigenstate ofthat measurement. The specifics of the measurement constitute the contextthat elicit one of the states that were previously potential. Its evolution can-not be examined without performing measurements—that is, introducingcontexts—but the contexts unavoidably affect its evolution. The evolutionof a quantum particle is an extreme case of nondeterministic change, aswell as of context sensitivity, because its state at any point in time reflectsthe context to which it is exposed.

Evolution of Quantum Particles

The other mode of change in standard quantum mechanics is the dy-namical change of state when no measurement is executed, referred to as‘evolution’. This evolution is described by the Schrodinger equation, andit is considered a fundamentally different kind of change from ‘collapse’under the influence of a measurement. The Schrodinger evolution is de-scribed by the Hamiltonian H which is the self-adjoint operator correspond-ing to the measurement h(ti) of the energy of the quantum particle, andthe Schrodinger equation

|p(t + ti)〉 = e−iHt

|p(ti)〉 (12)

where |p(t + ti)〉 is the vector representing the state p(t+ti) of the quantumparticle at time t + ti and |p(ti)〉 is the vector representing the state p(ti)of the quantum particle at time ti, while e

−iHt is the unitary operator

describing the time translation of state p(ti) to state p(t + ti), and isPlanck’s constant. It is possible to interpret the change of the state of aquantum entity as described by the Schrodinger equation as an effect ofcontext, namely a context that is the rest of the universe. All the fieldsand matter present in the rest of the universe contribute to the effect. Thechange is deterministic. Specifically, if the quantum entity at a certain time


ti is in state p(ti), and the only change that takes place is this dynamicalchange governed by the Schrodinger equation, then state p(t + ti) at anytime t + ti later than ti is determined.

For historical reasons, physicists think of a measurement not as a con-text, but as a process that gives rise to outcomes that are read off a mea-surement apparatus. In this scheme of thought, the simplest measurementsare assumed to be those with two possible outcomes. A measurement withone outcome is rightly not thought of as a measurement, because if thesame outcome always occurs, nothing has been compared and/or measured.However, when measurements are construed as contexts, we see that themeasurement with two possible outcomes is not the simplest change possi-ble. It is the deterministic evolution process—which can be conceived as ameasurement with one outcome—that is the simplest kind of change. Thismeans that in quantum mechanics the effect of context on change is asfollows:

• When the context is the rest of the universe, its influence on thestate of a quantum entity is deterministic, as described by theSchrodinger equation.

• When the context is a measurement, its influence on a genuinesuperposition state is nondeterministic, described as a process ofcollapse.

• When the context is a measurement, its influence on an eigenstateis deterministic, the eigenstate is not changed by the measurement.

Thus, under the CAP framework, the two basic processes of change in quan-tum mechanics are united; what has been referred to as dynamical evolutionis not fundamentally different from collapse. They are both processes of ac-tualization of potentiality under the influence of a context. In dynamicalevolution there is only one possible outcome, thus it is deterministic. Incollapse, until the state of the entity becomes an eigenstate, there is morethan one possible outcome, thus it is nondeterministic.

It was mentioned that the standard quantum formalism contains somefundamental structural restrictions. For example, the state space, i.e. thecomplex Hilbert space, is a linear space. It was also mentioned that moregeneral axiomatic approaches to quantum mechanica have been developedwhere these structural restrictions are not present. More specifically, thestate space in such a quantum axiomatic approach does not need to bea linear space. The SCOP approach which is the underlying mathemat-ical framework for the ‘context driven actualization of potential’ change


envisaged here, is such an axiomatic quantum ‘and’ classical approach, i.e.both standard quantum mechanics and classical mechanics are special casesof this SCOP formalism33−41.

There is an even more specific structural restriction of standard quan-tum mechanics, namely the fact that the Schrodinger equation is a linearequation, and can only represent changes of states described by a uni-tary transformation. For this reason, even staying within the standardquantum mechanical Hilbert space formalism for the state space, non-linearand hence non-unitary evolution more general than the Schrodinger onehave been considered and studied on many occasions42−45. All these varia-tions on standard quantum mechanics are special cases of the SCOP formal-ism, and hence CAP incorporates the types of changes modeled by them.

Evolution of Classical Physical Entities

Classical physical entities are the paradigmatic example of lack of sen-sitivity to and internalization of context, and deterministic change of state.However, theorists are continually expanding their models to include moreof the context surrounding an entity in order to better predict its behavior,which suggests that things are not so tidy in the world of classical physicalobjects as Newtonian physics suggests. Macro-level physical entities mayexhibit structure similar to the entanglement of quantum mechanics46 .This is the case when change of state of the entity cannot be predicted dueto lack of knowledge of how it interacts with its context47,48 .

5.2. Biological Evolution

Some theorists seeking to develop an integrative framework for the phys-ical and life sciences (Ref. 49) focus on tools or phenomena that are ap-plicable to or appear in both such as power laws and adaptive landscapes.The real challenge, however, is to develop an adaptive framework that en-compasses what is unique about biological life, what makes some matteralive, and what give rise to lineages that undergo adaptive modification.This is closely tied with the origins of structure with the capacity to self-replicate, and the refinement of this capacity through the emergence of thegenetic code50,51 . Thus, this section is divided into four parts. The firstconcerns autocatalytic sets of polymers, which replicated themselves (slop-pily) without genetic information, and are therefore are widely believed tobe the first forms that can be considered ‘alive’. The second concerns organ-isms after genetically coded replication was established but prior to sexual


reproduction. The third concerns sexually reproducing organisms. Thefourth part is a general comment on the structure of biochemical change.

The Earliest Life Forms

Early life forms were more sensitive to context and prone to internal-ize context than present-day life because their replication took place notaccording to instructions (such as a genetic code), but through happen-stance interactions. In Kauffman’s model of the origin of life52 , polymerscatalyze reactions that generate other polymers, increasing their joint com-plexity, until together as a whole they form something that can more orless replicate itself53 . The set is autocatalytically closed because althoughno polymer catalyzes its own replication, each catalyzes the replication ofanother member of the set. So long as each polymer is getting duplicatedsomewhere in the set, eventually multiple copies of all polymers exist. Thusit can self-replicate, and continue to do so indefinitely, or at least until itchanges so drastically that its self-replicating structure breaks down. (No-tice that ‘death’ of such life forms is not a particularly noticeable event; theonly difference between a dead organism and an alive one is that the aliveone continues to spawn new replicants.) Replication is far from perfect, so‘offspring’ are unlikely to be identical to their ‘parent’. Different chanceencounters of polymers, or differences in their relative concentrations, orthe appearance of new polymers, could all result in different polymers cat-alyzing a given reaction, which in turn altered the set of reactions to becatalyzed. Context was readily internalized by incorporating elements ofthe environment, thus there was plenty of room for heritable variation.

Recent work has been shown that the dynamical structure of biochem-ical reactions is quantum-like, above and beyond their microscopic (andobvious) quantum structure54,55,56 . This stems from the fact that notonly in the micro-domain where standard quantum structures are knownto exist, but also in other domains where change-of-state depends on howan entity interacts with its context, the resulting probabilities can be non-Kolmogorovian, and the appropriate formalisms for describing them areeither the quantum formalisms or mathematical generalizations of them.Kolmogorovian probability models consider only actualized entities, andtheir actualized interactions, and assumes that all evolutionary change issteered by these actualized entities, and their actualized interactions.WithinCAP, potential states of entities, and potential interactions between them,can be described.


Genetic Code Impedes Retention of Context in Lineage

A significant transition in the history of life was the transition from un-coded, self-organized replication to replication as per instructions given bya genetic code. This has been beautifully described and modelled by Vet-sigian, Goldenfeld, and Woese57 They refer to the transition from sloppyself-replication of autocatalytic form to precise replication using a geneticcode as the Darwinian transition, since it is at this point that traits acquiredat the level of individuals were no longer inherited and natural selection,a population-level mechanism of change, becomes applicable. We saw thatprior to coded replication, a change to one polymer would still be presentin offspring after budding occurred, and this could cause other changesthat have a significant effect on the lineage further downstream. There wasnothing to prohibit inheritance of acquired characteristics. But with theadvent of explicit self-assembly instructions, acquired characteristics wereno longer passed on to the next generation. The reason for this stems fromthe fact that, as first noted by von Neumann58 , they are self-replicatingautomata, meaning that they contain a coded instruction set that is usedin two distinct ways: (1) as a self-assembly code that is interpreted duringdevelopment, and (2) as self-instructions that are passively copied during re-production. It is this separation of how the code is used to generate offspring,and how the code is used during development, that is at the foundation oftheir lack of inheritance of acquired traits.

Since acquired characteristics were no longer being passed on to thenext generation, the process became more constrained, robust, and shieldedfrom external influence. (Thus for example, if one cuts off the tail of amouse, its offspring will have tails of a normal length.) A context-drivenchange of state of an organism only affects its lineage if it impacts thegeneration and survival of progeny (such as by affecting the capacity toattract mates, or engage in parental care). Clearly, the transition from un-coded to coded replication, while ensuring fidelity of replication, decreasedlong-term sensitivity to and internalization of context, and thus capac-ity for context independence. Since one generation was almost certainlyidentical to the next, the evolution became more deterministic. As a re-sult, in comparison with entities of other sorts, biological entities are resis-tant to internalization and retention of context-driven change. Though theterm ‘adaptation’ is most closely associated with biology, biological formis resistant to adaptation. This explains why it has been possible to de-velop a theory of biological evolution that all but ignores the problem of


incomplete knowledge of context. As we saw earlier, it is possible to ignorethis problem if all contexts are equally likely, or if context has a limitedeffect on heritability. In biology, since acquired traits are not heritable, theonly contextual interactions that exert much of an effect are those thataffect the generation of offspring. Thus it is because classical stochasticmodels work fine when lack of knowledge concerns the state of the entityand not the context that natural selection has for so long been viewed asadequate for the description of biological evolution. In Aerts, Czachor, andD’Hooghe59 and Aerts et al.60 , Darwinian evolution is analyzed with re-spect to potentiality using a specific example, and various possible (andspeculative) consequences of this difference are put forward.

Sexual Reproduction

With the advent of sexual reproduction, the contextuality of biologicalevolution increased. Consider an organism that is heterozygous for trait Xwith two alleles A and a. The potential of this Aa organism gets actualizeddifferently depending on the context provided by the genotype of the or-ganism’s mate. In the context of an AA mate, the Aa organism’s potentialis constrained to include only AA or Aa offspring. In the context of an aamate, it has the potential for Aa or aa offspring, and once again some ofthis potential might get actualized. And so forth. But while the mate con-strains the organism’s potential, the mate is necessary to actualize some ofthis potential in the form of offspring. In other words, the genome of themate simultaneously makes some aspects of the Aa organism’s potentialitypossible, and others impossible. An organism exists in a state of poten-tiality with respect to the different offspring (variants of itself) it couldproduce with a particular mate. In other words, a mate constitutes a con-text for which an organism is a state of potentiality. One can get away withignoring this to the extent that one can assume mating is random. Notethat since a species is delineated according to the capacity of individuals tomate with one another, speciation can be viewed as the situation whereinone variant no longer has the potential to create a context for the other forwhich its state is a potentiality state with respect to offspring. A speciescan be said to be adapted to the extent that its previous states could havecollapsed to different outcomes in different contexts, and thus to the extentits form reflects the particular contexts to which it was exposed. Note alsothat although over time species becomes increasingly context dependent,collectively they are becoming more context independent. (For virtuallyany ecological niche there exists some branch of life that can cope with it.)


Some (Ref. 61)argue for expansion of the concept of selection to other hi-erarchical levels, e.g. group selection. We agree with Kitcher62 that ‘despitethe vast amount of ink lavished upon the idea of “higher-order” processes’,once we have the causal story, it’s a matter of convention whether we saythat selection is operating at the level of the species, the organism, the geno-type, or the gene. It is not the concept of selection that needs expansion,but the embedding of selection in a framework for how change can occur.The actual is but the realized fragment of the potential, and selection worksonly on this fragment, what is already actual. We can now return to ourquestion about what natural selection has to say about the fitness of theoffspring you might have with one mate as opposed to another. The answeris of course, nothing, but why? Because the situation involves actualizationof potential and nondeterminism with respect to context, and as we haveseen, a nonclassical formalism is necessary to describe the change of stateinvolved. The CAP perspective also clarifies why fitness has been so hardto nail down. We agree with Krimbas63 that fitness is a property of neitherorganism nor environment, but emerges at the interface between them, andchanges from case to case.

5.3. Change of State in Cognitive Processes

Campbell64−67 argues that a stream of creative thought is a Darwinianprocess. The basic idea is that we generate new ideas through variationand selection: ‘mutate’ the current thought in a multitude of different ways,select the variant that looks best, mutate it in various ways and select thebest, and so forth, until a satisfactory idea results.

Thus a stream of thought is treated as a series of tiny selections. Thisview has been extended, most notably by Simonton68−71. The problemswith this thesis are outlined in detail elsewhere72,73 , but one that is ev-ident following our preceding discussion is that thoughts simply are notvon Neumann self-replicating automata. Another is that selection theoryrequires multiple, distinct, simultaneously-actualized states. But in cogni-tion, each thought or cognitive state changes the ‘selection pressure’ againstwhich the next is evaluated; they are not simultaneously selected amongst.The mind can exist in a state of potentiality, and change through interac-tion with the context to a state that is genuinely new, not just an elementof a pre-existing set of states. Creative thought is a matter of honing inon an idea by redescribing successive iterations of it from different real orimagined perspectives74 ; i.e. actualizing potential through exposure to dif-ferent contexts. Once again, the description of contextual change of state


introduces a non-Kolmogorovian probability distribution, and a classicalformalism such as selection theory cannot be used. Thus an idea certainlychanges as it gets mulled over in a stream of thought, and indeed it appearsto evolve, but the process is not Darwinian. Campbell’s error is to treat aset of potential, contextually elicited states of one entity as if they wereactual states of a collection of entities, or possible states with no effect ofcontext, even though the mathematical structure of the two situations iscompletely different. In a stream of thought, neither are all contexts equallylikely, nor does context have a limited effect on future iterations. So the as-sumptions that make classical stochastic models useful approximations donot hold.

5.4. Cultural Evolution

Culture, even more often than creative thought, is interpretted in evolu-tionary terms. Some scientists view culture merely as a contributing to thebiological evolution of our species. Increasingly, however, culture is viewdas an evolutionary process in and of its own (though one that still is inter-twined with, and influences, biological evolution). In this section we lookat how cultural evolution (as a second evolutionary process) fits into theCAP framework.

Culture Evolves Without a Self-Assembly Code

The basic unit of culture has been assumed to be the behavior or ar-tifact, or the mental representations or ideas that give rise to concretecultural forms. The meme notion further implies that these cultural unitsare replicators75 , misleadingly, because it does not consist of self-assemblyinstructions76 . (An idea may retain structure as it passes from one indi-vidual to another, but does not replicate it.) Looking at cultural evolutionfrom the CAP framework we ask: what is really changing through cul-tural processes? Because of the distributed nature of human memory, itis never is just one discrete ‘meme’affected by a cultural experience; it isones view of how the world hangs together, ones’ model of reality, or world-view. A worldview is not merely a collection of discrete ideas or memes(nor do ideas or memes form an interlocking set like puzzle pieces) be-cause each context impacts it differently; concepts and ideas are alwayscolored by the situation in which they are evoked77−79. Indeed it has beenargued that a worldview is a replicator.80 We saw that living organismsprior to the genetic code—a pre-RNA set of autocatalytic polymers—were


primitive replicators because they generate self-similar structure, butin a self-organized, emergent, piecemeal manner, eventually, for eachpolymer, there existed another that catalyzed its formation. Sincethere was no self-assembly instructions to copy from, there was noexplicit copying going on. The presence of a given catalytic poly-mer, say X, simply speeded up the rate at which certain reac-tions took place, while another polymer, say Y, influenced the re-action that generated X. Just as polymers catalyze reactions thatgenerate other polymers, retrieval of an item from memory can trigger an-other, which triggers yet another and so forth, thereby cross-linking mem-ories, ideas, and so forth into a conceptual web. Like the autocatalytic setsof poymers considered earlier,81 the result can be described as a connectedclosure structure82,83 . Elements of a worldview are regenerated throughsocial learning. Since as with Kauffman’s origin of life scenario the pro-cess occurrs in a self-organized, piecemeal autocatalytic manner, throughbottom-up interactions rather than a top-down code, worldviews like theearliest life forms replicate with low fidelity, and their evolution is highlynondeterministic.

Inheritance of Acquired Traits in Culture

As with the earliest pre-DNA forms of life, characteristics of a worldviewacquired over a lifetime are heritable. We hear a joke and, in sharing itwith others, give it our own slant. We create a disco version of Beethoven’sFifth Symphony and a rap version of that. The evolutionary trajectory ofa worldview makes itself known indirectly via the behavior and artifacts itmanifests under the influence of the contexts it encounters. (For example,when you explain to a child how to brush ones’ teeth, certain facets of yourworldview are revealed, while your writing of a poem reveals other facets.)

Because acquired traits are heritable in culture, the probability of split-ting into multiple variants is high. These variants can range from virtuallyidentical to virtually impossible to trace back to the same ‘parent’ idea.They affect, and are affected by, the minds that encounter them. For exam-ple, books can affect all the individuals who read them, and these individ-uals subsequently provide new contexts for the possible further evolutionof the ideas they described and stories they told.


6. Conclusions

This paper introduced a general framework for characterizing how entitiesevolve through context-driven actualization of potential (CAP). By this wemean an entity has the potential to change in different ways under differentcontexts. Some aspects of this potentiality are actualized when the entityundergoes a change of state through interaction with the particular contextit encounters. The interaction between entity and context may also changethe context, and the constraints and affordances it offers the entity. Thusthe entity undergoes another change of state, and so forth, recursively.

When evolution is construed as the incremental change that results fromrecursive, context-driven actualization of potential, the domains throughwhich we have carved up reality can be united under one umbrella. Quan-tum, classical, biological, cognitive, and cultural evolution appear as differ-ent ways in which potential that is present due to the state of an entity,its context, and the nature of their interaction. They differ according tothe degree of sensitivity to context, internalization of context, dependenceupon a particular context. nondeterminism due to lack of knowledge con-cerning the state of the entity, and nondeterminism due to lack of knowledgeconcerning the state of the context.

The reason potentiality and contextuality are so important stems fromthe fact that we inevitably have incomplete knowledge of the universein which an entity is operating. When the state of the entity of interestand/or context are in constant flux, or undergoing change at a resolutionbelow that which we can detect but nevertheless affect what emerges atthe entity-context interface, this gives rise in a natural way to nondeter-ministic change. Nondeterminism that arises through lack of knowledgeconcerning the state of the entity can be described by classical stochas-tic models (Markov processes) because the probability structure is Kol-mogorovian. However, nondeterminism that arises through lack of knowl-edge concerning the interaction between entity and context introduces anon-Kolmogorovian probability model84 on the state space, necessitating anonclassical formalism. Historically, the first nonclassical formalism was thequantum formalism. This formalism has since been generalized to describesituations involving nonlinearity, and varying degrees of contextuality.

Let us sum up a few of the more interesting results to come out of thisframework. It has been thought that the two modes of change in quan-tum mechanics—dynamical evolution of the quantum entity as per theSchrodinger equation, and the collapse that takes place when the quan-tum entity is measured—were fundamentally different. However, when the


measurement is seen to be a context, we notice that it is always a contextthat could actualize the potential of the entity in different ways. Indeed, ifone knows the outcome with certainty one does not perform a measurement;it is only when there is more than one possible value that a measurement isperformed. Thus the two modes of change in quantum mechanics are united;the dynamical evolution of a quantum entity as per the Schrodinger equa-tion reduces to a collapse for which there was only one way to collapse (i.e.only one possible outcome), hence deterministic collapse. This also holdsfor the deterministic evolution of classical entities. This constitutes a trueparadigm shift, for evolution and collapse have been thought to be twofundamentally different processes.

Looking at biological evolution from the CAP perspective, self-replication appears as a means of testing the integrity of an entity—orrather different versions of an entity—against different contexts. While in-dividuals and even species become increasingly context-dependent, the jointentity of living organisms becomes increasingly context-independent.Thegenetic code afforded primitive life protection against contextually-induceddisintegration of self-replication capacity, at the cost of decreased diversity.The onset of sexual reproduction increased potentiality, and thus possibletrajectories for biological form. The CAP framework supports the notionthat fitness is a property of neither organism nor environment, but emergesat the interface between them. The concept of potential fitness includesall possible evolutionary trajectories under all possible contexts. Since itinvolves nondeterminism with respect to context, unless context has a lim-ited effect or all possible contexts are equally likely, a nonclassical formalismis necessary to describe the novel form that results when an organism inter-acts with its environment in a way that makes some of its potential becameactual (where actual fitness refers only to the realized segment of its poten-tiality). It now becomes clear why natural selection has been able to tell usmuch about changes in frequencies of existing forms, but little about hownew forms emerge in the first place! The CAP framework provides a per-spective from which we can see why the neo-Darwinian view of evolutionhas been satisfactory for so long, and it wasn’t until after other processesbecome prominently viewed in evolutionary terms that the time was ripe forpotentiality and contextuality to be taken seriously. We also see how uniquethe genetic code is, and the consequent lack of retention of context-drivenchange. The effects of contextual interaction in biology are in the long-runlargely invisible; context affects lineages only by influencing the numberand nature of offspring. Natural selection is such an exceptional means of


change, it is no wonder it does not transfer readily to other domains. Notethat it is often said that because acquired traits are inherited in culture,culture should not be viewed in evolutionary terms. It is ironic that thiscritique also applies to the earliest stage of biological evolution itself. Whatwas true of early life is also true of the replication of worldviews: acquiredcharacteristics can be inherited. Modern life is unique in this sense.

The same argument holds for what happens in a stream of creativethought. The mathematical formulation of the theory of natural selec-tion requires that in any given iteration there be multiple distinct, actual-ized states. In cognition however, each successive mental state changes thecontext in which the next is evaluated; they are not simultaneously selectedamongst. Creative thought is a matter of honing in on an idea by redescrib-ing successive iterations of it from different real or imagined perspectives;actualizing potential through exposure to different contexts. Thus selectiontheory is not applicable to the formal description of a stream of thought,and to the extent that creative thought powers cultural change, it is oflimited applicability there as well. Once again, a nonclassical formalism isnecessary.

The notion of culture as a Darwinian process probably derives from thefact that the means through which a creative mind manifests itself in theworld—language, art, and so forth—exist as discrete entities such as sto-ries and paintings. This can lead to the assumption that discrete creativeartifacts in the world spring forth from corresponding discrete, pre-formedentities in the brain. This in turn leads to the assumption that novelty getsgenerated through that most celebrated of all change-generating mecha-nisms, Darwinian selection, and that ideas and artifacts must therefore bereplicators. However, an idea or artifact is not a replicator because it doesnot consist of coded self-assembly instructions, and thus does not makecopies of itself. Moreover, ideas and artifacts do not arise out of separate,distinct compartments in the brain, but emerge from a dynamically andcontextually modifiable, web-like memory structure, a melting pot in whichdifferent components continually merge and blend, get experienced in newways as they arise in new contexts and combinations. The CAP perspec-tive suggests instead that the basic unit and the replicator of culture is anintegrated network of knowledge, attitudes, ideas, and so forth; that is, aninternal model of the world, or worldview, and that ideas and artifacts arehow a worldview reveals itself under a particular context.


Acknowledgments

We would like to thank Ping Ao for comments on the manuscript. This re-search was supported by Grant G.0339.02 of the Flemish Fund for ScientificResearch and a research grant from Foundation for the Future.

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84. For example Bayes’ formula for conditional probability is not satisfied.

135

Quantum-like Probabilistic Models Outside Physics

Andrei Khrennikov

International Center for Mathematical Modelingin Physics and Cognitive Sciences

University of Vaxjo, S-35195, [email protected]

We present a quantum-like (QL) model in that contexts (complexes of e.g. men-tal, social, biological, economic or even political conditions) are represented bycomplex probability amplitudes. This approach gives the possibility to applythe mathematical quantum formalism to probabilities induced in any domainof science. In our model quantum randomness appears not as irreducible ran-domness (as it is commonly accepted in conventional quantum mechanics, e.g.by von Neumann and Dirac), but as a consequence of obtaining incompleteinformation about a system. We pay main attention to the QL description ofprocessing of incomplete information. Our QL model can be useful in cognitive,social and political sciences as well as economics and artificial intelligence. Inthis paper we consider in a more detail one special application — QL modelingof brain’s functioning. The brain is modeled as a QL-computer.

Keywords: Incompleteness of quantum mechanics; Quantum-like Representa-tion of Information; Quantum-like Models in Biology; Psychology; Cognitiveand Social Sciences and Economy; Context; Complex Probabilistic AmplitudePACS(2006): 03.65.Fd; 03.67.a; 03.65.w; 05.30.d; 03.65.Ta; 03.65.Ca; 87.10.+e;89.75.k; 89.75.Fb; 89.65.s; 89.65.Gh

1. Introduction: Quantum Mechanics as Operation withIncomplete Information

Let us assume that, in spite of a rather common opinion, quantum me-chanics is not a complete theory. Thus the wave function does not providea complete description of the state of a physical system. Hence we as-sume that the viewpoint of Einstein, De Broglie, Schrodinger, Bohm, Bell,Lamb, Lande, ‘t Hooft and other believers in the possibility to provide amore detailed description of quantum phenomena is correct, but the view-point of Bohr, Heisenberg, Pauli, Fock, Landau and other believers in thecompleteness of quantum mechanics (and impossibility to go beyond it) iswrong. We remark that the discussion on completeness/incompleteness of

136 Andrei Khrennikov

quantum mechanics is also known as the discussion about hidden variables— additional parameters which are not encoded in the wave function. Inthis paper we would not like to be involved into this great discussion, seeRef. 12 for recent debates. We proceed in a very pragmatic way by takingadvantages of the incompleteness viewpoint on quantum mechanics. Thuswe would not like to be waiting for until the end of the Einstein–Bohrdebate. This debate may take a few hundred years more.∗

What are advantages of the Einstein’s interpretation of the QM-formalism?

The essence of this formalism is not the description of a special classof physical systems, so called quantum systems, having rather exotic andeven mystical properties, but the possibility to operate with incomplete in-formation about systems. Thus according to Einstein one may apply theformalism of quantum mechanics in any situation which can be character-ized by incomplete description of systems. This (mathematical) formalismcould be used in any domain of science, see Refs. 4, 7–9 : in cognitive andsocial sciences, economy, information theory. Systems under considerationneed not be exotic. However, the complete description of them should benot available or ignored (by some reasons).

We repeat again that in this paper it is not claimed that quantum me-chanics is really incomplete as a physical theory. It is not the problem underconsideration. We shall be totally satisfied by presenting an information QLmodel such that it will be possible to interpret the wave function as an in-complete description of a system. It might occur that our model could notbe applied to quantum mechanics as a physical theory. However, we shallsee that our model can be used at least outside physics. Therefore we shallspeak about a quantum-like (QL) and not quantum model.

We shall use Einstein’s interpretation of the formalism of quantum me-chanics. This is a special mathematical formalism for statistical decisionmaking in the absence of complete information about systems. By usingthis formalism one permanently ignore huge amount of information. How-ever, such an information processing does not induce chaos. It is extremelyconsistent. Thus the QL information cut off is done in a clever way. This isthe main advantage of the QL processing of information.

We also remark that (may be unfortunately) the mathematical for-malism for operating with probabilistic data represented by complex

∗Even if quantum mechanics is really incomplete it may be that hidden parameters wouldbe found only on scales of time and space that would be approached not very soon, cf.3


amplitudes was originally discovered in a rather special domain of physics,so called quantum physics. This historical fact is the main barrier on theway of applications of QL methods in other domains of science, e.g. biol-ogy, psychology, sociology, economics. If one wants to apply somewhere themathematical methods of quantum mechanics (as e.g. Hameroff10,11 andPenrose12,13 did in study of brain’s functioning) he would be typically con-strained by the conventional interpretation of this formalism, namely, theorthodox Copenhagen interpretation. By this interpretation quantum me-chanics is complete. There are no additional (hidden or ignored) parameterscompleting the wave function description. Quantum randomness is not re-ducible to classical ensemble randomness, see von Neumann14Birkhoff andvon Neumann15 .† It is not easy to proceed with this interpretation tomacroscopic applications.

In this paper we present a contextualist statistical realistic model forQL representation of incomplete information for any kind of systems: bi-ological, cognitive, social, political, economical. Then we concentrate ourconsiderations to cognitive science and psychology7,16 . In particular, weshall describe cognitive experiments to check the QL structure of mentalprocesses.

The crucial role is played by the interference of probabilities for men-tal observables. Recently one such experiment based on recognition of im-ages was performed, see17,7 . This experiment confirmed our prediction onthe QL behavior of mind. In our approach “quantumness of mind” hasno direct relation to the fact that the brain (as any physical body) is com-posed of quantum particles, see Refs. 10−13. We invented a new terminology“quantum-like (QL) mind.”

Cognitive QL-behavior is characterized by a nonzero coefficient of inter-ference λ (“coefficient of supplementarity”). This coefficient can be found onthe basis of statistical data. There are predicted not only cos θ-interferenceof probabilities, but also hyperbolic cosh θ-interference. The latter interfer-ence was never observed for physical systems, but we could not exclude thispossibility for cognitive systems. We propose a model of brain functioningas a QL-computer. We shall discuss the difference between quantum andQL-computers.

From the very beginning we emphasize that our approach has nothing

†The founders of the orthodox Copenhagen interpretation, Bohr and Heisenberg as wellas e.g. Pauli and Landau, did not discuss so much probabilistic aspects of this interpreta-tion. This is an astonishing historical fact, since quantum mechanics is a fundamentallyprobabilistic theory.


to do with quantum reductionism. Of course, we do not claim that ourapproach implies that quantum physical reduction of mind is totally im-possible. But our approach could explain the main QL-feature of mind —interference of minds — without reduction of mental processes to quantumphysical processes. Regarding the quantum logic approach we can say thatour contextual statistical model is close mathematically to some models ofquantum logic18 , but interpretations of mathematical formalisms are quitedifferent. The crucial point is that in our probabilistic model it is possi-ble to combine realism with the main distinguishing features of quantumprobabilistic formalism such as interference of probabilities, Born’s rule,complex probabilistic amplitudes, Hilbert state space, and representation of(realistic) observables by operators.

2. Levels of Organization of Matter and Information andtheir Formal Representations

This issue is devoted to the principles and mechanisms which let matterto build structures at different levels, and their formal representations. Wewould like to extend this issue by considering information as a more gen-eral structure than matter, see Ref. 19. Thus material structures are justinformation structures of a special type. In our approach it is more naturalto speak about principles and mechanisms which let information to buildstructures at different levels, and their formal representations. From thispoint of view quantum mechanics is simply a special formalism for opera-tion in a consistent way with incomplete information about a system. Herea system need not be a material one. It can be a social or political system,a cognitive system, a system of economic or financial relations.

The presence of OBSERVER collecting information about systems isalways assumed in our QL model. Such an observer can be of any kind:cognitive or not, biological or mechanical. The essence of our approach isthat such an observer is able to obtain some information about a systemunder observation. As was emphasized, in general this information is notcomplete. An observer may collect incomplete information not only becauseit is really impossible to obtain complete information. It may occur that itwould be convenient for an observer or a class of observers to ignore a partof information, e.g. about social or political processes.‡

‡We mention that according to Freud’s psychoanalysis human brain can even represssome ideas, so called hidden forbidden wishes and desires, and send them into the un-consciousness.


Any system of observers with internal or external structure of self-organization should operate even with incomplete information in a con-sistent way. The QL formalism provides such a possibility.

We speculate that observers might evolutionary develop the ability tooperate with incomplete information in the QL way — for example, brains.In the latter case we should consider even self-observations: each brainperforms measurements on itself. We formulate the hypothesis on the QLstructure of processing of mental information. Consequently, the ability ofQL operation might be developed by social systems and hence in economicsand finances.

If we consider Universe as an evolving information system, then we shallevidently see a possibility that Universe might evolve in such a way thatdifferent levels of its organization are coupled in the QL setting. In particu-lar, in this model quantum physical reality is simply a level of organization(of information structures, in particular, particles and fields) which is basedon a consistent ignorance of information (signals, interactions) from a pre-quantum level. In its turn the latter may also play the role of the quantumlevel for a pre-prequantum one and so on. We obtain the model of Universehaving many (infinitely many) levels of organization which are based onQL representations of information coming from the previous levels.

In same way the brain might have a few levels of transitions from theclassical-like (CL) to the QL description.

At each level of representation of information so called physical, bio-logical, mental, social or economic laws are obtained only through partialignorance of information coming from previous levels. In particular, in ourmodel the quantum dynamical equation, the Schrodinger’s equation, can beobtained as a special (incomplete) representation of dynamics of classicalprobabilities, see Section 12.

3. Vaxjo Model: Probabilistic Model for Results ofObservations

A general statistical realistic model for observables based on the contextualviewpoint to probability will be presented. It will be shown that classicalas well as quantum probabilistic models can be obtained as particular casesof our general contextual model, the Vaxjo model.

This model is not reduced to the conventional, classical and quantummodels. In particular, it contains a new statistical model: a model with hy-perbolic cosh-interference that induces ”hyperbolic quantum mechanics”.7

A physical, biological, social, mental, genetic, economic, or financial


context C is a complex of corresponding conditions. Contexts are funda-mental elements of any contextual statistical model. Thus construction ofany model M should be started with fixing the collection of contexts of thismodel. Denote the collection of contexts by the symbol C (so the familyof contexts C is determined by the model M under consideration). In themathematical formalism C is an abstract set (of “labels” of contexts).

We remark that in some models it is possible to construct a set-theoreticrepresentation of contexts — as some family of subsets of a set Ω. For ex-ample, Ω can be the set of all possible parameters (e.g. physical, or mental,or economic) of the model. However, in general we do not assume the pos-sibility to construct a set-theoretic representation of contexts.

Another fundamental element of any contextual statistical model M isa set of observables O : each observable a ∈ O can be measured under eachcomplex of conditions C ∈ C. For an observable a ∈ O, we denote the setof its possible values (“spectrum”) by the symbol Xa.

We do not assume that all these observables can be measured simultane-ously. To simplify considerations, we shall consider only discrete observablesand, moreover, all concrete investigations will be performed for dichotomousobservables.

Axiom 1: For any observable a ∈ O and its value x ∈ Xa, there aredefined contexts, say Cx, corresponding to x-selections: if we perform ameasurement of the observable a under the complex of physical conditionsCx, then we obtain the value a = x with probability 1. We assume that theset of contexts C contains Cx-selection contexts for all observables a ∈ Oand x ∈ Xa.

For example, let a be the observable corresponding to some question:a = + (the answer “yes”) and a = − (the answer “no”). Then the C+-selection context is the selection of those participants of the experimentwho answering “yes” to this question; in the same way we define the C−-selection context. By Axiom 1 these contexts are well defined. We pointout that in principle a participant of this experiment might not want toreply at all to this question. By Axiom 1 such a possibility is excluded. Bythe same axiom both C+ and C−-contexts belong to the system of contextsunder consideration.

Axiom 2: There are defined contextual (conditional) probabilities P(a =x|C) for any context C ∈ C and any observable a ∈ O .

Thus, for any context C ∈ C and any observable a ∈ O , there is definedthe probability to observe the fixed value a = x under the complex ofconditions C.


Especially important role will be played by probabilities:

pa|b(x|y) ≡ P(a = x|Cy), a, b ∈ O, x ∈ Xa, y ∈ Xb,

where Cy is the [b = y]-selection context. By axiom 2 for any context C ∈ C,there is defined the set of probabilities:

P(a = x|C) : a ∈ O.We complete this probabilistic data for the context C by contextual prob-abilities with respect to the contexts Cy corresponding to the selections[b = y] for all observables b ∈ O. The corresponding collection of dataD(O, C) consists of contextual probabilities:

P(a = x|C),P(b = y|C),P(a = x|Cy),P(b = y|Cx), ...,

where a, b, ... ∈ O. Finally, we denote the family of probabilistic dataD(O, C) for all contexts C ∈ C by the symbol D(O, C)(≡ ∪C∈CD(O, C)).

Definition 1. (Vaxjo Model) An observational contextual statisticalmodel of reality is a triple

M = (C,O,D(O, C)) (1)

where C is a set of contexts and O is a set of observables which satisfy toaxioms 1,2, and D(O, C) is probabilistic data about contexts C obtained withthe aid of observables belonging O.

We call observables belonging the set O ≡ O(M) reference of observ-ables. Inside of a model M observables belonging to the set O give the onlypossible references about a context C ∈ C.

4. Representation of Incomplete Information

Probabilities P(b = y|C) are interpreted as contextual (conditional) prob-abilities. We emphasize that we consider conditioning not with respect toevents as it is typically done in classical probability,20 but conditioning withrespect to contexts — complexes of (e.g. physical, biological, social, mental,genetic, economic, or financial) conditions. This is the crucial point.

On the set of all events one can always introduce the structure of theBoolean algebra (or more general σ-algebra). In particular, for any twoevents A and B their set-theoretic intersection A∩B is well defined and itdetermines a new event: the simultaneous occurrence of the events A andB.


In contract to such an event-conditioning picture, if one have two con-texts, e.g. complexes of physical conditions C1 and C2 and if even it is possi-ble to create the set-theoretic representation of contexts (as some collectionsof physical parameters), then, nevertheless, their set-theoretic intersectionC1 ∩ C2 (although it is well defined mathematically) need not correspondto any physically meaningful context. Physical contexts were taken just asexamples. The same is valid for social, mental, economic, genetic and anyother type of contexts.

Therefore even if for some model M we can describe contexts in theset-theoretic framework, there are no reasons to assume that the collectionof all contexts C should form a σ-algebra (Boolean algebra). This is themain difference from the classical (noncontextual) probability theory.20

One can consider the same problem from another perspective. Supposethat we have some set of parameters Ω (e.g. physical, or social, or mental).We also assume that contexts are represented by some subsets of Ω. Weconsider two levels of description. At the first level a lot of information isavailable. There is a large set of contexts, we can even assume that theyform a σ-algebra of subsets F . We call them the first level contexts. Thereis a large number of observables at the first level, say the set of all possiblerandom variables ξ : Ω → R (here R is the real line). By introducing onF a probability measure P we obtain the classical Kolmogorov probabilitymodel (Ω,F ,P), see Ref. 20. This is the end of the classical story aboutthe probabilistic description of reality. Such a model is used e.g. in classicalstatistical physics.

We point our that any Kolmogorov probability model induces a Vaxjomodel in such a way: a) contexts are given by all sets C ∈ F such thatP(C) 6= 0; b) the set of observables coincides with the set of all possiblerandom variables; c) contextual probabilities are defined as Kolmogorovianconditional probabilities, i.e. by the Bayes formula: P(a = x|C) = P(ω ∈C : a(ω) = x)/P(C). This is the Vaxjo model for the first level of descrip-tion.

Consider now the second level of description. Here we can obtain a non-Kolmogorovian Vaxjo model. At this level only a part of information aboutthe first level Kolmogorovian model (Ω,F ,P) can be obtained through aspecial family of observables O which correspond to a special subset of theset of all random variables of the Kolmogorov model (Ω,F ,P) at the firstlevel of description. Roughly speaking not all contexts of the first level, Fcan be “visible” at the second level. There is no sufficiently many observ-ables “to see” all contexts of the first level — elements of the Kolmogorov


σ-algebra F . Thus we should cut off this σ-algebra F and obtain a smallerfamily, say C, of visible contexts. Thus some Vaxjo models (those permittinga set-theoretic representation) can appear starting with the purely classicalKolmogorov probabilistic framework, as a consequence of ignorance of in-formation. If not all information is available, so we cannot use the first level(classical) description, then we, nevertheless, can proceed with the secondlevel contextual description.

We shall see that starting with some Vaxjo models we can obtain thequantum-like calculus of probabilities in the complex Hilbert space. Thusin the opposition to a rather common opinion, we can derive a quantum-like description for ordinary macroscopic systems as the results of using ofan incomplete representation. This opens great possibilities in applicationof quantum-like models outside the micro-world. In particular, in cognitivescience we need not consider composing of the brain from quantum particlesto come to the quantum-like model.

Example 1. (Firefly in the box) Let us consider a box which is dividedinto four sub-boxes. These small boxes which are denoted by ω1, ω2, ω3, ω4

provides the the first level of description. We consider a Kolmogorov prob-ability space: Ω = ω1, ω2, ω3, ω4, the algebra of all finite subsets F of Ωand a probability measure determined by probabilities P(ωj) = pj , where0 < pj < 1, p1 + ... + p4 = 1. We remark that in our interpretation it ismore natural to consider elements of Ω as elementary parameters, and notas elementary events (as it was done by Kolmogorov).

We now consider two different disjoint partitions of the set Ω :

A1 = ω1, ω3, A2 = ω2, ω4,

B1 = ω1, ω2, B2 = ω3, ω4.We can obtain such partitions by dividing the box: a) into two equal partsby the vertical line: the left-hand part gives A1 and the right-hand part A2;b) into two equal parts by the horizontal line: the top part gives B1 andthe bottom part B2.

We introduce two random variables corresponding to these partitions:ξa(ω) = xi, if ω ∈ Ai and ξb(ω) = yi ∈ if ω ∈ Bi. Suppose nowthat we are able to measure only these two variables, denote the corre-sponding observables by the symbols a and b. We project the Kolmogorovmodel under consideration to a non-Kolmogorovian Vaxjo model by us-ing the observables a and b — the second level of description. At this


Fig. 1. The first level (complete) description.

level the set of observables O = a, b and the natural set of contextsC : Ω, A1, A2, B1, B2, C1 = ω1, ω4, C2 = ω2, ω3 and all unions of thesesets. Here “natural” has the meaning permitting a quantum-like representa-tion (see further considerations). Roughly speaking contexts of the secondlevel of description should be large enough to “be visible” with the aid ofobservables a and b.

Intersections of these sets need not belong to the system of contexts(nor complements of these sets). Thus the Boolean structure of the originalfirst level description disappeared, but, nevertheless, it is present in thelatent form. Point-sets ωj are not “visible” at this level of description.For example, the random variable

η(ωj) = γj , j = 1, ..., 4, γi 6= γj , i 6= j,

is not an observable at the second level.Such a model was discussed from positions of quantum logic, see Ref. 6.

There can be provided a nice interpretation of these two levels of descrip-tion. Let us consider a firefly in the box. It can fly everywhere in this box.Its locations are described by the uniform probability distribution P (onthe σ-algebra of Borel subsets of the box). This is the first level of descrip-tion. Such a description can be realized if the box were done from glass or


Fig. 2. The second level description: the a-observable.

Fig. 3. The second level description: the b-observable

if at every point of the box there were a light detector. All Kolmogorovrandom variables can be considered as observables.

Now we consider the situation when there are only two possibilities toobserve the firefly in the box:

1) to open a small window at a point a which is located in such a way(the bold dot in the middle of the bottom side of the box) that it is possible


to determine only either the firefly is in the section A1 or in the section A2

of the box;2) to open a small window at a point b which is located in such a way

(the bold dot in the middle of the right-hand side of the box) that it ispossible to determine only either the firefly is in the section B1 or in thesection B2 of the box.

In the first case I can determine in which part, A1 or A2, the firefly islocated. In the second case I also can only determine in which part, B1 orB2, the firefly is located. But I am not able to look into both windows simul-taneously. In such a situation the observables a and b are the only sourceof information about the firefly (reference observables). The Kolmogorovdescription is meaningless (although it is incorporated in the model in thelatent form). Can one apply a quantum-like description, namely, representcontexts by complex probability amplitudes? The answer is to be positive.The set of contexts that permit the quantum-like representation consistsof all subsets C such that P(Ai|C) > 0 and P(Bi|C) > 0, i = 1, 2 (i.e.for sufficiently large contexts). We have seen that the Boolean structuredisappeared as a consequence of ignorance of information.

Finally, we emphasize again that the Vaxjo model is essentially moregeneral. The set-theoretic representation need not exist at all.

5. Quantum Projection of Boolean Logic

Typically the absence of the Boolean structure on the set of quantumpropositions is considered as the violation of laws of classical logic, e.g.in quantum mechanics.15 In our approach classical logic is not violated, itis present in the latent form. However, we are not able to use it, becausewe do not have complete information. Thus quantum-like logic is a kind ofprojection of classical logic. The impossibility of operation with completeinformation about a system is not always a disadvantages. Processing ofincomplete set of information has the evident advantage comparing with“classical Boolean” complete information processing — the great saving ofcomputing resources and increasing of the speed of computation. However,the Boolean structure cannot be violated in an arbitrary way, because insuch a case we shall get a chaotic computational process. There should bedeveloped some calculus of consistent ignorance by information. Quantumformalism provides one of such calculi.

Of course, there are no reasons to assume that processing of informationthrough ignoring of its essential part should be rigidly coupled to a special


class of physical systems, so called quantum systems. Therefore we prefer tospeak about quantum-like processing of information that may be performedby various kinds of physical and biological systems. In our approach quan-tum computer has advantages not because it is based on a special class ofphysical systems (e.g. electrons or ions), but because there is realized theconsistent processing of incomplete information. We prefer to use the ter-minology QL-computer by reserving the “quantum computer” for a specialclass of QL-computers which are based on quantum physical systems.

One may speculate that some biological systems could develop in theprocess of evolution the possibility to operate in a consistent way withincomplete information. Such a QL-processing of information implies evi-dent advantages. Hence, it might play an important role in the process ofthe natural selection. It might be that consciousness is a form of the QL-presentation of information. In such a way we really came back to White-head’s analogy between quantum and conscious systems.22

6. Contextual Interpretation of ‘Incompatible’ Observable

Nowadays the notion of incompatible (complementary) observables isrigidly coupled to noncommutativity. In the conventional quantum formal-ism observables are incompatible iff they are represented by noncommutingself-adjoint operators a and b : [a, b] 6= 0. As we see, the Vaxjo model isnot from the very beginning coupled to a representation of information ina Hilbert space. Our aim is to generate an analogue (may be not direct) ofthe notion of incompatible (complementary) observables starting not fromthe mathematical formalism of quantum mechanics, but on the basis of theVaxjo model, i.e. directly from statistical data.

Why do I dislike the conventional identification of incompatibility withnoncommutativity? The main reason is that typically the mathematicalformalism of quantum mechanics is identified with it as a physical theory.Therefore the quantum incompatibility represented through noncommuta-tivity is rigidly coupled to the micro-world. (The only possibility to transferquantum behavior to the macro-world is to consider physical states of theBose–Einstein condensate type.) We shall see that some Vaxjo models canbe represented as the conventional quantum model in the complex Hilbertspace. However, the Vaxjo model is essentially more general than the quan-tum model. In particular, some Vaxjo models can be represented not in thecomplex, but in hyperbolic Hilbert space (the Hilbert module over the twodimensional Clifford algebra with the generator j : j2 = +1).


Another point is that the terminology — incompatibility — is mislead-ing in our approach. The quantum mechanical meaning of compatibility isthe possibility to measure two observables, a and b simultaneously. In sucha case they are represented by commuting operators. Consequently incom-patibility implies the impossibility of simultaneous measurement of a andb. In the Vaxjo model there is no such a thing as fundamental impossibil-ity of simultaneous measurement. We present the viewpoint that quantumincompatibility is just a consequence of information supplementarity of ob-servables a and b. The information which is obtained via a measurementof, e.g. b can be non trivially updated by additional information which iscontained in the result of a measurement of a. Roughly speaking if oneknows a value of b, say b = y, this does not imply knowing the fixed valueof a and vice versa, see Ref. 16 for details.

We remark that it might be better to use the notion “complementary,”instead of “supplementary.” However, the first one was already reserved byNils Bohr for the notion which very close to “incompatibility.” In any eventBohr’s complementarity implies mutual exclusivity that was not the pointof our considerations.

Supplementary processes take place not only in physical micro-systems.For example, in the brain there are present supplementary mental pro-cesses. Therefore the brain is a (macroscopic) QL-system. Similar supple-mentary processes take place in economy and in particular at financial mar-ket. There one could also use quantum-like descriptions.8 But the essenceof the quantum-like descriptions is not the representation of informationin the complex Hilbert space, but incomplete (projection-type) represen-tations of information. It seems that the Vaxjo model provides a rathergeneral description of such representations.

We introduce a notion of supplementary which will produce in somecases the quantum-like representation of observables by noncommuting op-erators, but which is not identical to incompatibility (in the sense of im-possibility of simultaneous observations) nor complementarity (in the senseof mutual exclusivity).

Definition 2. Let a, b ∈ O. The observable a is said to be supplementaryto the observable b if

pa|b(x|y) 6= 0, (2)

for all x ∈ Xa, y ∈ Xb.

Let a = x1, x2 and b = y1, y2 be two dichotomous observables. In this


case (2) is equivalent to the condition:

pa|b(x|y) 6= 1, (3)

for all x ∈ Xa, y ∈ Xb. Thus by knowing the result b = y of the b-observationwe are not able to make the definite prediction about the result of the a-observation.

Suppose now that (3) is violated (i.e., a is not supplementary to b), forexample:

pa|b(x1|y1) = 1, (4)

and, hence, pa|b(x2|y1) = 0. Here the result b = y1 determines the resulta = x1.

In future we shall consider a special class of Vaxjo models in that thematrix of transition probabilities Pa|b = (pa|b(xi|yj))2i,j=1 is double stochas-tic: pa|b(x1|y1)+pa|b(x1|y2) = 1; pa|b(x2|y1)+pa|b(x2|y2) = 1. In such a casethe condition (4) implies that

pa|b(x2|y2) = 1, (5)

and, hence, pa|b(x1|y2) = 0. Thus also the result b = y2 determines theresult a = x2.

We point out that for models with double stochastic matrix

Pa|b = (pa|b(xi|yj))2i,j=1

the relation of supplementary is symmetric! In general it is not the case.It can happen that a is supplementary to b : each a-measurement givesus additional information updating information obtained in a precedingmeasurement of b (for any result b = y). But b can be non-supplementaryto a.

Let us now come back to Example 1. The observables a and b are sup-plementary in our meaning. Consider now the classical Kolmogorov modeland suppose that we are able to measure not only the random variables ξa

and ξb — observables a and b, but also the random variable η. We denotethe corresponding observable by d. The pairs of observables (d, a) and (d, b)are non-supplementary:

pa|d(x1|γi) = 0, i = 3, 4; pa|d(x2|γi) = 0, i = 1, 2,

and, hence,

pa|d(x1|γi) = 1, i = 1, 2; pa|d(x2|γi) = 1, i = 3, 4.

Thus if one knows , e.g. that d = γ1 then it is definitely that a = x1 and soon.


7. A Statistical Test to Find Quantum-like Structure

We consider examples of cognitive contexts:1) C can be some selection procedure that is used to select a special

group SC of people or animals. Such a context is represented by this groupSC (so this is an ensemble of cognitive systems). For example, we select agroup Sprof.math. of professors of mathematics (and then ask questions a or(and) b or give corresponding tasks). We can select a group of people ofsome age. We can select a group of people having a “special mental state”:for example, people in love or hungry people (and then ask questions orgive tasks).

2) C can be a learning procedure that is used to create some specialgroup of people or animals. For example, rats can be trained to react tospecial stimulus.

We can also consider social contexts. For example, social classes:proletariat-context, bourgeois-context; or war-context, revolution-context,context of economic depression, poverty-context, and so on. Thus our modelcan be used in social and political sciences (and even in history). We cantry to find quantum-like statistical data in these sciences.

We describe a mental interference experiment.Let a = x1, x2 and b = y1, y2 be two dichotomous mental observables:

x1=yes, x2=no, y1=yes, y2=no. We set X ≡ Xa = x1, x2, Y ≡ Xb =y1, y2 (“spectra” of observables a and b). Observables can be two differentquestions or two different types of cognitive tasks. We use these two fixedreference observables for probabilistic representation of cognitive contextualreality given by C.

We perform observations of a under the complex of cognitive conditionsC :

pa(x) =the number of results a = x

the total number of observations.

So pa(x) is the probability to get the result x for observation of the a underthe complex of cognitive conditions C. In the same way we find probabilitiespb(y) for the b-observation under the same cognitive context C.

As was supposed in axiom 1, cognitive contexts Cy can be created corre-sponding to selections with respect to fixed values of the b-observable. Thecontext Cy (for fixed y ∈ Y ) can be characterized in the following way. Bymeasuring the b-observable under the cognitive context Cy we shall obtainthe answer b = y with probability one. We perform now the a-measurements


under cognitive contexts Cy for y = y1, y2, and find the probabilities:

pa|b(x|y) =number of the result a = x for context Cy

number of all observations for context Cy

where x ∈ X, y ∈ Y. For example, by using the ensemble approach to prob-ability we have that the probability pa|b(x1|y2) is obtained as the frequencyof the answer a = x1 = yes in the ensemble of cognitive system that havealready answered b = y2 = no. Thus we first select a sub-ensemble of cogni-tive systems who replies no to the b-question, Cb=no. Then we ask systemsbelonging to Cb=no the a-question.

It is assumed (and this is a very natural assumption) that a cognitivesystem is “responsible for her (his) answers.” Suppose that a system τ hasanswered b = y2 = no. If we ask τ again the same question b we shall getthe same answer b = y2 = no. This is nothing else than the mental formof the von Neumann projection postulate: the second measurement of thesame observable, performed immediately after the first one, will yield thesame value of the observable).

Classical probability theory tells us that all these probabilities have tobe connected by the so called formula of total probability:

pa(x) = pb(y1)pa|b(x|y1) + pb(y2)pa|b(x|y2), x ∈ X.

However, if the theory is quantum-like, then we should obtain7 the formulaof total probability with an interference term:

pa(x) = pb(y1)pa|b(x|y1) + pb(y2)pa|b(x|y2) (6)

+2λ(a = x|b, C)√

pb(y1)pa|b(x|y1)pb(y2)pa|b(x|y2),

where the coefficient of supplementarity (the coefficient of interference) isgiven by

λ(a = x|b, C) =pa(x)− pb(y1)pa|b(x|y1)− pb(y2)pa|b(x|y2)

2√

pb(y1)pa|b(x|y1)pb(y2)pa|b(x|y2). (7)

This formula holds true for supplementary observables. To prove its validity,it is sufficient to put the expression for λ(a = x|b, C), see (7), into (6). Inthe quantum-like statistical test for a cognitive context C we calculate

λ(a = x|b, C) =pa(x)− pb(y1)pa|b(x|y1)− pb(y2)pa|b(x|y2)

2√

pb(y1)pa|b(x|y1)pb(y2)pa|b(x|y2),


where the symbol p is used for empirical probabilities – frequencies. An em-pirical situation with λ(a = x|b, C) 6= 0 would yield evidence for quantum-like behaviour of cognitive systems. In this case, starting with (experimen-tally calculated) coefficient of interference λ(a = x|b, C) we can proceedeither to the conventional Hilbert space formalism (if this coefficient isbounded by 1) or to so called hyperbolic Hilbert space formalism (if thiscoefficient is larger than 1). In the first case the coefficient of interferencecan be represented in the trigonometric form λ(a = x|b, C) = cos θ(x), Hereθ(x) ≡ θ(a = x|b, C) is the phase of the a-interference between cognitivecontexts C and Cy, y ∈ Y. In this case we have the conventional formula oftotal probability with the interference term:

pa(x) = pb(y1)pa|b(x|y1) + pb(y2)pa|b(x|y2) (8)

+2 cos θ(x)√

pb(y1)pa|b(x|y1)pb(y2)pa|b(x|y2).

In principle, it could be derived in the conventional Hilbert space formal-ism. But we chosen the inverse way. Starting with (8) we could introducea “mental wave function” ψ ≡ ψC (or pure quantum-like mental state)belonging to this Hilbert space. We recall that in our approach a mentalwave function ψ is just a representation of a cognitive context C by a com-plex probability amplitude. The latter provides a Hilbert representation ofstatistical data about context which can be obtained with the help of twofixed observables (reference observables).

8. The Wave Function Representation of Contexts

In this section we shall present an algorithm for representation of a con-text (in fact, probabilistic data on this context) by a complex probabilityamplitude. This QL representation algorithm (QLRA) was created by theauthor.7 It can be interpreted as a consistent projection of classical prob-abilistic description (the complete one) onto QL probabilistic description(the incomplete one).

Let C be a context. We consider only contexts with trigonometric inter-ference for supplementary observables a and b — the reference observablesfor coming QL representation of contexts. The collection of all trigonomet-ric contexts, i.e., contexts having the coefficients of supplementarity (withrespect to two fixed observables a and b) bounded by one, is denoted by thesymbol C tr ≡ C tr

a|b. We again point out to the dependence of the notion ofa trigonometric context on the choice of reference observables.


We now point directly to dependence of probabilities on contexts by us-ing the context lower index C. The interference formula of total probability(6) can be written in the following form:

paC(x) =

∑

y∈Xb

pbC(y)pa|b(x|y) + 2 cos θC(x)

√Πy∈Y pb

C(y)pa|b(x|y) . (9)

By using the elementary formula:

D = A + B + 2√

AB cos θ = |√

A + eiθ√

B|2,for A,B > 0, θ ∈ [0, 2π], we can represent the probability pa

C(x) as thesquare of the complex amplitude (Born’s rule):

paC(x) = |ϕC(x)|2, (10)

where a complex probability amplitude is defined by

ϕ(x) ≡ ϕC(x) =√

pbC(y1)pa|b(x|y1) + eiθC(x)

√pb

C(y2)pa|b(x|y2) . (11)

We denote the space of functions: ϕ : Xa → C, where C is the field ofcomplex numbers, by the symbol Φ = Φ(Xa,C). Since Xa = x1, x2, theΦ is the two dimensional complex linear space. By using the representation(11) we construct the map

Ja|b : C tr → Φ(X,C)

which maps contexts (complexes of, e.g. physical or social conditions) intocomplex amplitudes. This map realizes QLRA.

The representation (10) of probability is nothing other than the famousBorn rule. The complex amplitude ϕC(x) can be called a wave functionof the complex of physical conditions (context) C or a (pure) state. Weset ea

x(·) = δ(x − ·). The Born’s rule for complex amplitudes (10) can berewritten in the following form:

paC(x) = |(ϕC , ea

x)|2, (12)

where the scalar product in the space Φ(X, C) is defined by the standardformula:

(ϕ,ψ) =∑

x∈Xa

ϕ(x)ψ(x). (13)

The system of functions eaxx∈Xa is an orthonormal basis in the Hilbert

space

H = (Φ, (·, ·)).


Let Xa be a subset of the real line. By using the Hilbert space represen-tation of the Born’s rule we obtain the Hilbert space representation of theexpectation of the reference observable a:

E(a|C) =∑

x∈Xa

x|ϕC(x)|2 =∑

x∈Xa

x(ϕC , eax)(ϕC , ea

x) = (aϕC , ϕC),

where the (self-adjoint) operator a : H → H is determined by its eigenvec-tors: aea

x = xeax, x ∈ Xa. This is the multiplication operator in the space of

complex functions Φ(X,C) :

aϕ(x) = xϕ(x).

It is natural to represent this reference observable (in the Hilbert spacemodel) by the operator a.

We would like to have Born’s rule not only for the a-observable, butalso for the b-observable.

pbC(y) = |(ϕ, eb

y)|2 , y ∈ Xb.

Thus both reference observables would be represented by self-adjoint oper-ators determined by bases ea

x, eby, respectively.

How can we define the basis eby corresponding to the b-observable?

Such a basis can be found starting with interference of probabilities. We

set ubj =

√pb

C(yj), pij = pa|b(xj |yi), uij = √pij , θj = θC(xj). We have:

ϕ = ub1e

b1 + ub

2eb2, (14)

where

eb1 = (u11, u12), eb

2 = (eiθ1u21, eiθ2u22). (15)

We consider the matrix of transition probabilities Pa|b = (pij). It is always astochastic matrix: pi1 +pi2 = 1, i = 1, 2). We remind that a matrix is calleddouble stochastic if it is stochastic and, moreover, p1j+p2j = 1, j = 1, 2. Thesystem eb

i is an orthonormal basis iff the matrix Pa|b is double stochasticand probabilistic phases satisfy the constraint: θ2−θ1 = π mod 2π, see Ref.7 for details.

It will be always supposed that the matrix of transition probabilitiesPa|b is double stochastic. In this case the b-observable is represented by theoperator b which is diagonal (with eigenvalues yi) in the basis eb

i. TheKolmogorovian conditional average of the random variable b coincides withthe quantum Hilbert space average:

E(b|C) =∑

y∈Xb

ypbC(y) = (bφC , φC), C ∈ C tr.


9. Brain as a System Performing a Quantum-likeProcessing of Information

The brain is a huge information system that contains millions of elementarymental states . It could not “recognize” (or “feel”) all those states at eachinstant of time t. Our fundamental hypothesis is that the brain is ableto create the QL-representations of mind. At each instant of time t thebrain creates the QL-representation of its mental context C based on twosupplementary mental (self-)observables a and b. Here a = (a1, ..., an) andb = (b1, ..., bn) can be very long vectors of compatible (non-supplementary)dichotomous observables. The reference observables a and b can be chosen(by the brain) in different ways at different instances of time. Such a changeof the reference observables is known in cognitive sciences as a change ofrepresentation.

A mental context C in the a|b− representation is described by the mentalwave function ψC . We can speculate that the brain has the ability to feelthis mental field as a distribution on the space X. This distribution is givenby the norm-squared of the mental wave function: |ψC(x)|2.

In such a model it might be supposed that the state of our consciousnessis represented by the mental wave function ψC . The crucial point is that inthis model consciousness is created through neglecting an essential volumeof information contained in subconsciousness. Of course, this is not just arandom loss of information. Information is selected through QLRA, see (11):a mental context C is projected onto the complex probability amplitude ψC .

The (classical) mental state of sub-consciousness evolves with time C →C(t). This dynamics induces dynamics of the mental wave function ψ(t) =ψC(t) in the complex Hilbert space.

Further development of our approach (which we are not able to presenthere) induces the following model of brain’s functioning9 :

The brain is able to create the QL-representation of mental contexts,C → ψC (by using the algorithm based on the formula of total probabilitywith interference).

10. Brain as Quantum-like Computer

The ability of the brain to create the QL-representation of mental contextsinduces functioning of the brain as a quantum-like computer.

The brain performs computation-thinking by using algorithms of quan-tum computing in the complex Hilbert space of mental QL-states.

We emphasize that in our approach the brain is not quantum computer,


but a QL-computer. On one hand, a QL-computer works totally in accor-dance with the mathematical theory of quantum computations (so by usingquantum algorithms). On the other hand, it is not based on superpositionof individual mental states. The complex amplitude ψC representing a men-tal context C is a special probabilistic representation of information statesof the huge neuronal ensemble. In particular, the brain is a macroscopicQL-computer. Thus the QL-parallelism (in the opposite to conventionalquantum parallelism) has a natural realistic base. This is real parallelismin the working of millions of neurons. The crucial point is the way in whichthis classical parallelism is projected onto dynamics of QL-states. The QL-brain is able to solve NP -problems. But there is nothing mysterious inthis ability: an exponentially increasing number of operations is performedthrough involving of an exponentially increasing number of neurons.

We point out that by coupling QL-parallelism to working of neurons westarted to present a particular ontic model for QL-computations. We shalldiscuss it in more detail. Observables a and b are self-observations of thebrain. They can be represented as functions of the internal state of brainω. Here ω is a parameter of huge dimension describing states of all neuronsin the brain: ω = (ω1, ω2, ..., ωN ) :

a = a(ω), b = b(ω).

The brain is not interested in concrete values of the reference observablesat fixed instances of time. The brain finds the contextual probability dis-tributions pa

C(x) and pbC(y) and creates the mental QL-state ψC(x), see

QLRA – (11). Then it works with the mental wave function ψC(x) by usingalgorithms of quantum computing.

11. Two Time Scales as the Basis of the QL-representationof Information

The crucial problem is to find a mechanism for producing contextual prob-abilities. We think that they are frequency probabilities that are createdin the brain in the following way.There are two scales of time: a) internalscale, τ -time; b) QL-scale, t-time. The internal scale is finer than the QL-scale. Each instant of QL-time t corresponds to an interval ∆ of internaltime τ. We might identify the QL-time with mental (psychological) timeand the internal time with physical time. We shall also use the terminology:pre-cognitive time-scale - τ and cognitive time-scale - t.

During the interval ∆ of internal time the brain collects statistical data


for self-observations of a and b. The internal state ω of the brain evolves as

ω = ω(τ, ω0).

This is a classical dynamics (which can be described by a stochastic differ-ential equation).

At each instance of internal time τ there are performed nondisturbativeself-measurements of a and b. These are realistic measurements: the braingets values a(ω(τ, ω0)), b(ω(τ, ω0)). By finding frequencies of realization offixed values for a(ω(τ, ω0)) and b(ω(τ, ω0)) during the interval ∆ of internaltime, the brain obtains the frequency probabilities pa

C(x) and pbC(y). These

probabilities are related to the instant of QL-time time t correspondingto the interval of internal time ∆ : pa

C(t, x) and pbC(t, y). We remark that

in these probabilities the brain encodes huge amount of information —millions of mental “micro-events” which happen during the interval ∆. Butthe brain is not interested in all those individual events. (It would be toodisturbing and too irrational to take into account all those fluctuations ofmind.) It takes into account only the integral result of such a pre-cognitiveactivity (which was performed at the pre-cognitive time scale).

For example, the mental observables a and b can be measurements overdifferent domains of brain. It is supposed that the brain can “feel” probabil-ities(frequencies) pa

C(x) and pbC(y), but not able to “feel” the simultaneous

probability distribution pC(x, y) = P (a = x, b = y|C).This is not the problem of mathematical existence of such a distribution.

This is the problem of integration of statistics of observations from differentdomains of the brain. By using the QL-representation based only on prob-abilities pa

C(x) and pbC(y) the brain could be able to escape integration of

information about individual self-observations of variables a and b relatedto spatially separated domains of brain. The brain need not couple thesedomains at each instant of internal (pre-cognitive time) time τ. It couplesthem only once in the interval ∆ through the contextual probabilities pa

C(x)and pb

C(y). This induces the huge saving of time and increasing of speed ofprocessing of mental information.

One of fundamental consequences for cognitive science is that our mentalimages have the probabilistic structure. They are products of transitionfrom an extremely fine pre-cognitive time scale to a rather rough cognitivetime scale.

Finally, we remark that a similar time scaling approach was developedin Ref. 3 for ordinary quantum mechanics. In Ref. 3 quantum expectationsappear as results of averaging with respect to a prequantum time scale.


There was presented an extended discussion of possible choices of quantumand prequantum time scales.

We can discuss the same problem in the cognitive framework. We maytry to estimate the time scale parameter ∆ of the neural QL-coding. Thereare strong experimental evidences, see, Ref 21 that a moment in psycho-logical time correlates with ≈ 100 ms of physical time for neural activity.In such a model the basic assumption is that the physical time requiredfor the transmission of information over synapses is somehow neglected inthe psychological time. The time (≈ 100 ms) required for the transmissionof information from retina to the inferiotemporal cortex (IT) through theprimary visual cortex (V1) is mapped to a moment of psychological time.It might be that by using

∆ ≈ 100ms

we shall get the right scale of the QL-coding.However, it seems that the situation is essentially more complicated.

There are experimental evidences that the temporial structure of neuralfunctioning is not homogeneous. The time required for completion of colorinformation in V4 (≈ 60 ms) is shorter that the time for the completion ofshape analysis in IT (≈ 100 ms). In particular it is predicted that there willbe under certain conditions a rivalry between color and form perception.This rivalry in time is one of manifestations of complex level temporialstructure of brain. There may exist various pairs of scales inducing theQL-representations of information.

12. The Hilbert Space Projection of ContextualProbabilistic Dynamics

Let us assume that the reference observables a and b evolve with time: x =x(t), y = y(t), where x(t0) = a and y(t0) = b. To simplify considerations,we consider evolutions which do not change ranges of values of the referenceobservables: Xa = x1, x2 and Xb = y1, y2 do not depend on time. Thus,for any t, x(t) ∈ Xa and y = y(t) ∈ Xb.

In particular, we can consider the very special case when the dy-namical reference observables correspond to classical stochastic processes:x(t, ω)), y(t, ω)), where x(t0, ω) = a(ω) and y(t0, ω) = b(ω). Under the pre-vious assumption these are random walks with two-points state spaces Xa

and Xb. However, we recall that in general we do not assume the existenceof Kolmogorov measure-theoretic representation.


Since our main aim is the contextual probabilistic realistic reconstruc-tion of QM, we should restrict our considerations to evolutions with thetrigonometric interference. We proceed under the following assumption:

(CTRB) (Conservation of trigonometric behavior) The set of trigono-metric contexts does not depend on time: C tr

x(t)|y(t) = C trx(t0)|y(t0)

= C tra|b.

By (CTRB) if a context C ∈ C trx(t0)|y(t0)

, i.e. at the initial instant oftime the coefficients of statistical disturbance |λ(x(t0) = x|y(t0), C)| ≤ 1,

then the coefficients λ(x(t) = x|y(t), C) will always fluctuate in the segment[0, 1]. §

For each instant of time t, we can use QLRA, see (11): a context C canbe represented by a complex probability amplitude:

ϕ(t, x) ≡ ϕx(t)|y(t)C (x) =

√p

y(t)C (y1)px(t)|y(t)(x|y1)

+eiθx(t)|y(t)C (x)

√p

y(t)C (y2)px(t)|y(t)(x|y2).

We remark that the observable y(t) is represented by the self-adjoint oper-ator y(t) defined by its with eigenvectors:

eb1t =

(√pt(x1|y1)√pt(x2|y1)

)eb2t = eiθC(t)

(√pt(x1|y2)

−√

pt(x2|y1)

)

where pt(x|y) = px(t)|y(t)(x|y), θC(t) = θx(t)|y(t)C (x1) and where we set eb

jt ≡ey(t)j . We recall that θ

x(t)|y(t)C (x2) = θ

x(t)|y(t)C (x1) + π, since the matrix of

transition probabilities is assumed to be double stochastic for all instancesof time.

We shall describe dynamics of the wave function ϕ(t, x) starting withfollowing assumptions (CP) and (CTP). Then these assumptions will becompleted by the set (a)-(b) of mathematical assumptions which will implythe conventional Schrodinger evolution.

(CP)(Conservation of b-probabilities) The probability distribution of theb-observable is preserved in process of evolution: p

y(t)C (y) = p

b(t0)C (y), y ∈

Xb, for any context C ∈ C tra(t0)|b(t0). This statistical conservation of the b-

quantity will have very important dynamical consequences. We also assumethat the law of conservation of transition probabilities holds:

(CTP) (Conservation of transition probabilities) Probabilities pt(x|y)are conserved in the process of evolution: pt(x|y) = pt0(x|y) ≡ p(x|y).

§Of course, there can be considered more general dynamics in which the trigonometricprobabilistic behaviour can be transformed into the hyperbolic one and vice versa.


Under the latter assumption we have:

eb1t ≡ eb

1t0 , eb2t = ei[θC(t)−θC(t0)]eb

2t0 .

For such an evolution of the y(t)-basis b(t) = b(t0) = b. Hence the wholestochastic process y(t, ω) is represented by one fixed self-adjoint operator b.

This is a good illustration of incomplete QL representation of information.Thus under assumptions (CTRB), (CP) and (CTP) we have:

ϕ(t) = ub1e

b1t + ub

2eb2t = ub

1eb1t0 + eiξC(t,t0)ub

2eb2t0 ,

where ubj =

√p

b(t0)C (yj), j = 1, 2, and ξC(t, t0) = θC(t) − θC(t0). Let us

consider the unitary operator U(t, t0) : H → H defined by this transfor-mation of basis: eb

t0 → ebt . In the basis eb

t0 = eb1t0 , e

b2t0 the U(t, t0) can be

represented by the matrix:

U(t, t0) =(

1 00 eiξC(t,t0)

).

We obtained the following dynamics in the Hilbert space H:

ϕ(t) = U(t, t0)ϕ(t0). (16)

This dynamics looks very similar to the Schrodinger dynamics in theHilbert space. However, the dynamics (16) is essentially more general thanSchrodinger’s dynamics. In fact, the unitary operator U(t, t0) = U(t, t0, C)depends on the context C, i.e., on the initial state ϕ(t0) : U(t, t0) ≡U(t, t0, ϕ(t0)). So, in fact, we derived the following dynamical equation:ϕ(t) = U(t, t0, ϕ0)ϕ0, where, for any ϕ0, U(t, t0, ϕ0) is a family of unitaryoperators.

The conditions (CTRB), (CP) and (CTP) are natural from the physi-cal viewpoint (if the b-observable is considered as an analog of energy, seefurther considerations). But these conditions do not imply that the Hilbertspace image of the contextual realistic dynamics is a linear unitary dy-namics. In general the Hilbert space projection of the realistic prequantumdynamics is nonlinear. To obtain a linear dynamics, we should make thefollowing assumption:

(CI) (Context independence of the increment of the probabilistic phase)The ξC(t, t0) = θC(t)− θC(t0) does not depend on C.

Under this assumption the unitary operator U(t, t0) = diag(1, eiξ(t,t0))does not depend on C. Thus the equation (16) is the equation of the lin-ear unitary evolution. The linear unitary evolution (16) is still essentiallymore general than the conventional Schrodinger dynamics. To obtain theSchrodinger evolution, we need a few standard mathematical assumptions:


(a) Dynamics is continuous: the map (t, t0) → U(t, t0) is continuous¶; (b)Dynamics is deterministic; (c) Dynamics is invariant with respect to time-shifts; U(t, t0) depends only on t− t0 : U(t, t0) ≡ U(t− t0).

The assumption of determinism can be described by the following re-lation: φ(t; t0, φ0) = φ(t; t1, φ(t1; t0, φ0)), t0 ≤ t1 ≤ t, where φ(t; t0, φ0) =U(t, t0)φ0.

It is well known that under the assumptions (a), (b), (c) the family of(linear) unitary operators U(t, t0) corresponds to the one parametric groupof unitary operators: U(t) = e−

ih Ht, where H : H → H is a self-adjoint

operator. Here h > 0 is a scaling factor (e.g. the Planck constant). We have:H = diag(0, E), where

E = −h[θC(t)− θC(t0)

t− t0

].

Hence the Schrodinger evolution in the complex Hilbert space correspondsto the contextual probabilistic dynamics with the linear evolution of theprobabilistic phase:

θC(t) = θC(t0)− E

h(t− t0).

We, finally, study the very special case when the dynamical referenceobservables correspond to classical stochastic processes: x(t, ω)), y(t, ω)).This is a special case of the Vaxjo model: there exist the Kolmogorovrepresentation of contexts and the reference observables. Let us consid-xer a stochastic process (rescaling of the process y(t, ω)) : H(t, ω) = 0if y(t, ω) = y1 and H(t, ω) = E if y(t, ω) = y2. Since the probabil-ity distributions of the processes y(t, ω)) and H(t, ω) coincide, we havep

H(t)C (0) = p

H(t0)C (0); pH(t)

C (E) = pH(t0)C (E).

If E > 0 we can interpret H(t, ω) as the energy observable and theoperator H as its Hilbert space image. We emphasize that the whole “energyprocess” H(t, ω) is represented by a single self-adjoint nonnegative operatorH in the Hilbert space. This is again a good illustration of incomplete QLrepresentation of information. This operator, “quantum Hamiltonian”, isthe Hilbert space projection of the energy process which is defined on the“prespace” Ω. In principle, random variables H(t1, ω),H(t2, ω), t1 6= t2,

can be very different (as functions of ω). We have only the law of statisticalconservation of energy: p

H(t)C (z) ≡ p

H(t0)C (z), z = 0, E.

¶We recall that there is considered the finite dimensional case. Thus there is no problemof the choice of topology.


13. Concluding Remarks

We created an algorithm – QLRA – for representation of a context (in fact,probabilitic data on this context) by a complex probability amplitude.

QLRA can be applied consciously by e.g. scientists to provide a consis-tent theory which is based on incomplete statistical data. By our interpreta-tion quantum physics works in this way. However, we can guess that a com-plex system could perform such a self-organization that QLRA would workautomatically creating in this system the two levels of organization: CL-leveland QL-level. Our hypothesis is that the brain is one of such systems withQLRA-functioning. We emphasize that in the brain QLRA-representationis performed on the unconscious level (by using Freud’s terminology: in theunconsciousness). But the final result of application of QLRA is presentedin the conscious domain in the form of feelings, associations and ideas, seeRef. 23. and Ref. 24. We guess that some complex social systems are ableto work in the QLRA-regime. As well as for the brain for such a socialsystem, the main advantage of working in the QLRA-regime is neglectingby huge amount of information (which is not considered as important).Since QLRA is based on choosing a fixed class of observables (for a brainor a social system they are self-observables), importance of information isdetermined by those observables. The same brain or social system can useparallely a number of different QL representations based on applications ofQLRA for different reference observables.

We can even speculate that physical systems can be (self-) organizedthrough application of QLRA. As was already pointed out, Universe mightbe the greates user of QLRA.

Conclusion

The mathematical formalism of quantum mechanics can be applied out-side of physics, e.g. in cognitive, social, and political sciences, psychology,economics and finances.

References

1. A. Yu. Khrennikov (2002) editor, Quantum Theory: Reconsideration ofFoundations. Vaxjo Univ. Press, Vaxjo.

2. G. Adenier and A.Yu. Khrennikov (2005) editors, Foundations of Probabilityand Physics-3. American Institute of Physics, Melville, NY, 750.

3. A. Yu. Khrennikov (2006) To quantum mechanics through random fluctua-tions at the planck time scale. http://www.arxiv.org/abs/hep-th/0604011.

4. D. Aerts and S. Aerts (1994) Foundations of Science 1, (1) pp 85–97.


5. Accardi L (1997) Urne e Camaleoni: Dialogo sulla realta, le leggi del caso ela teoria quantistica, Il Saggiatore, Rome.

6. K. Svozil (1998) Quantum Logic. Springer, Berlin.7. A. Yu. Khrennikov (2004) Information Dynamics in Cognitive, Psychologi-

cal and Anomalous Phenomena. Kluwer Academic, Dordreht.8. O. Choustova (2006) Quantum bohmian model for financial market. Physica

A 374, 304–314 .9. A. Yu. Khrennikov (2006) Quantum-like brain: Interference of minds.

BioSystems 84, 225–241.10. S. Hameroff (1994) Quantum coherence in microtubules. A neural basis for

emergent consciousness? J. of Consciousness Studies 1, pp 91–118.11. S. Hameroff (1994) Quantum computing in brain microtubules? The

Penrose-Hameroff Orch Or model of consciousness. Phil. Tr. Royal Sc., Lon-don A pp 1–28.

12. R. Penrose (1989)The emperor’s new mind. Oxford Univ. Press, New-York.13. R. Penrose (1994) Shadows of the mind. Oxford Univ. Press, Oxford.14. J. von Neumann (1955) Mathematical foundations of quantum mechanics,

Princeton Univ. Press, Princeton, N.J.15. G. Birkhoff and J. von Neumann (1936) The logic of quantum mechanics,

Ann. Math. 37, pp 823–643.16. A. Yu. Khrennikov (2005) The principle of supplementarity: A contextual

probabilistic viewpoint to complementarity, the interference of probabilities,and the incompatibility of variables in quantum mechanics, Foundations ofPhysics 35(10), pp 1655–1693.

17. E. Conte, O. Todarello, A. Federici, F. Vitiello, M. Lopane, A. Khrennikovand J. P. Zbilut (2006) Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantummechanical formalism to describe cognitive entity and its dynamics, Chaos,Solitons and Fractals 31, pp 1076–1088.

18. G. W. Mackey (1963) Mathematical Foundations of Quantum Mechanics,W. A. Benjamin Inc., New York.

19. A. Yu. Khrennikov (1999) Classical and quantum mechanics on informationspaces with applications to cognitive, psychological, social and anomalousphenomena, Foundations of Physics 29, pp 1065–1098.

20. A. N. Kolmogoroff (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung,Springer Verlag, Berlin.

21. K. Mori (2002) On the relation between physical and psychological time.In Toward a Science of Consciousness, Tucson University Press, Tucson,Arizona, p. 102.

22. A. N. Whitehead, (1929) Process and Reality: An Essay in Cosmology,Macmillan Publishing Company, New York.

23. A. Yu. Khrennikov (1998) Human subconscious as the p-adic dynamicalsystem. J. of Theor. Biology 193, pp 179–196.

24. A. Yu. Khrennikov (2002) Classical and quantum mental models and Freud’stheory of unconscious mind, Vaxjo Univ. Press, Vaxjo.

165

Phase Transitions in Biological Matter

Eliano Pessa

Centro Interdipartimentale di Scienze Cognitive, Universita di Paviaand

Dipartimento di Psicologia, Universita di PaviaPiazza Botta, 6 , 27100 Pavia, Italy

[email protected]

In this paper we will deal with usefulness of physical theory of phase transitionsin order to describe phenomena of change occurring in the biological world. Inparticular, we will assess the role of quantum theory in accounting for theemergence of different forms of coherence seemingly characterizing a numberof biological behaviours. In this regard we will introduce some arguments that,while supporting the convenience (as well as the unavoidability) of resortingto a quantum-theoretical framework to describe biological emergence, point tothe need for a suitable generalization of actual quantum theory. Some possibleways to achieve such a generalization will be shortly discussed.

Keywords: Biophysics; Phase Transition; Biological EmergencePACS(2006): 87.16.Ac; 87.10.+e; 05.70.a; 05.70.Fh; 64.60.i; 89.75.k; 89.75.Fb

1. Introduction

The world of biological phenomena (here the attribute ‘biological’ will beused in the widest sense, including even psychological, social, and economicphenomena) is characterized by a number of phenomena of change, whoseunderstanding, forecasting, and (when possible) control is the main goal ofmany scientific disciplines, such as Biology, Neuroscience, Psychology, So-ciology, Economics. Most phenomena of this kind seem to consist in someform of emergence (this word is here used in an intuitive sense) of a new,coherent, form of organization. Surely the appearance of life itself, underthe form of a new individual or of a new species, belongs to this category.But the latter includes even the formation of a flock of birds flying to-gether, the synchronization between electric activities of spatially distantbrain neurons, evidenced by the electroencephalographic recordings, thedeep understanding of a new concept by a student (the so-called insight),

166 Eliano Pessa

the establishment of a new social habit. It is obvious that scientists deal-ing with phenomena of this kind have been attracted by the conceptualstructure of physical theory of phase transitions. The latter, after all, hasbeen dealing with similar problems since more than one century, achievinga remarkable success in explaining and controlling a number of phenom-ena, even if in a restricted domain. This circumstance has triggered theexplosive growth of models trying to use the methods of physical theory ofphase transitions (hereafter shortly denoted as TPT) to describe biologicalchanges. Quantum Biology, Quantum Brain Theory, Dynamical Approachin Psychology, Econophysics are all examples of this new way of thinking.

Faced with this situation, the basic question is: can TPT be directlyapplied, as it stands (that is, without modifications or generalizations),to describe biological changes? Would the answer to the basic questionbe yes, then a further question would arise: what general methodology(if any) should be used to apply TPT to biological problems? Would theanswer be no, then the next question would be: how should we modify orgeneralize TPT? Within this paper we will introduce a number of argumentssupporting a partially negative answer to the basic question. The expression‘partially negative’ means that our conclusion is, not in favor of a reject ofTPT, but rather of a suitable (and unavoidable) generalization of it whenapplied to biological realm. As a consequence, in the final part of this paperwe will introduce a number of proposals in order to generalize TPT so asto allow its application to a wider range of phenomena. We feel, however,that these generalizations will turn out to be useful, sooner or later, evenin applications concerning strictly physical domains.

The organization of the paper can be described as follows. In section twowe will summarize the features of TPT which are relevant for our discussion.In section three there will be a synthetic description of main mechanismsunderlying biological emergence phenomena so far hypothesized within theactual context of TPT. As these mechanisms are grounded on quantum the-ory, section four will be devoted to a discussion of the possible role of thislatter within biological phenomena, with particular reference to the prob-lem of decoherence. As most arguments based on Quantum Field Theoryare heavily dependent on infinite volume limits, section five will containa discussion of finite volume effects (domains, defects, vortices, and like)arising in phase transitions, as these effects can be of utmost importancewhen trying to apply TPT to biological realm. Section six will contain adiscussion about a number of proposals of generalization of TPT, in or-der to make easier its application to biological phenomena, while section


seven will be devoted to a synthetic conclusion ensuing from the argumentspresented within the paper.

2. The Main Features of TPT

Within this exposition we will assume that the reader is familiar with stan-dard formalism of Thermodynamics, as well as with the one of StatisticalMechanics, and of usual TPT. In this way we can focus only on main con-ceptual aspects, neglecting the explanation of concepts and formulae whichcan be easily found on standard textbooks. We start by remarking that,roughly speaking, TPT adopts a framework based on a distinction betweenmicroscopic and macroscopic descriptions. The former deal with a worldof elementary constituents whose interactions are described through (hope-fully) simple laws, while the latter refer to the appearance of phenomenaon our normal observational scale. The science of macroscopic descriptionscoincides with Thermodynamics and phase transitions belong to phenom-ena dealt with by this discipline. Namely phase transitions are consideredas existing and observable only on a macroscopic scale, even if producedby phenomena occurring at the microscopic level. The connection betweenmacroscopic and microscopic level should be assured by Statistical Mechan-ics, allowing, in principle, a deduction of features of macroscopic phenomenastarting only from a knowledge of laws ruling the behaviour of microscopicconstituents.

Within this context it is possible to introduce a suitable definition of theconcept of phase, and characterize in a precise way the conditions associatedwith a phase transition. We will not enter into details about the compli-cated history of this subject, and we will limit ourselves to mention that,according to actual views, phase transitions are associated to the presenceof discontinuities of derivatives of thermodynamical Gibbs potential of thesystem. When these discontinuities affect first-order derivatives, we speakof first-order phase transitions, while, in presence of continuous first-orderderivatives and of discontinuities in derivatives whose order is greater thanthe first, we speak of second-order phase transitions. We follow here theclassification scheme of phase transitions first introduced by Landau, andthen adopted by most modern authors, neglecting the existence of otherclassification schemata (like the older one proposed by Ehrenfest) whichdon’t matter for our purposes.

Leaving aside technical details (for a summary of which we refer totextbooks on this subjects; see, for instance, Sewell 1986, Chap. 4), westress that the connection mentioned above between phase transitions and

168 Eliano Pessa

discontinuities of derivatives of Gibbs potential is nothing but a conse-quence of a number of hypotheses, the most important of which can bethus summarized:

A) the microscopic dynamics can be described in terms of a suitableset of extensive conserved quantities, including the energy (whence we arein presence of a Hamiltonian dynamics), which are bounded and transformcovariantly with respect to space translations;

B) it is possible to define in an unambiguous way, starting from thesemicroscopic quantities, their global macroscopic densities, which act as in-tensive variables at the macroscopic level;

C) the macroscopic densities, defined in B), can be used to characterizein an unambiguous way the macroscopic states in the sense that, every timethey are constrained to take well defined values, the entropy density of thesystem is maximized by precisely one translationally invariant state.

Already at this level, we can easily realize that these hypotheses can bedevoid of any meaning in most biological phenomena. In first place, mostobservations about these phenomena occur only on a macroscopic scale, intotal absence of any hypothesis about the existence of some microscopic en-tities producing the observed effects. For instance, a psychologist looking atthe behaviour of a person troubled by problems will usually try to accountfor what has been observed in terms of hypothetical (macroscopic) men-tal processes, without resorting to some unknown microscopic entities (ofcourse, it would be possible to explain the behaviour in terms of brain neu-rons electrical activity, but the gap between the level of neuronal dischargesand the one of human behaviours is still too great to allow this kind of ex-planation). In second place, even when we are in presence of a microscopicdescription dealing with interacting elementary components responsible forthe behaviours observed on a macroscopic scale, almost never this descrip-tion can be cast under a Hamiltonian form. Namely in the biological worldalmost all components are driven by a dissipative dynamics, owing to theirinteraction with external environment, so that it is virtually impossible tofind conserved quantities. Biological cells or individuals belonging to a so-ciety are typical examples of open systems, for which there is no form ofenergy conservation (and perhaps the concept itself of ‘equilibrium’ cannotbe defined).

However, even when the hypotheses A), B), C) are satisfied, they arenot enough for building a theory of phase transitions. The latter has beenbuilt by resorting to a suitable combination of phenomenological observa-tions, of phenomenological theories describing them, of suitable theoretical


frameworks into which the latter have been embedded, and of statisticalarguments supporting all this machinery. TPT can be thus seen as a sortof complicated device used to accounting for experimental data and, whenpossible, forecasting them. The operation of this device is not easy to under-stand and often unsatisfactory. We will shortly summarize the main aspectsof this operation, by listing some essentials of actual TPT.

Phenomenological observationsThere are two kinds of observations which played a major role in shap-

ing the actual TPT. The first kind includes the behaviours observed whentemperature approaches critical temperature, that is the divergence of (gen-eralized) susceptibility, the divergence of amplitude of fluctuations of orderparameter, the critical slowing down, and the discontinuity in the curvegiving specific heat as a function of temperature. The second kind refers toexistence of universality classes, evidenced by the fact that different phasetransitions are associated to (almost) the same set of critical exponentsappearing in the laws describing macroscopic behaviours near the criticalpoint.

It is to be stressed that generally we lack similar observations in bio-logical contexts. In most domains, each phenomenon of change is studiedin its own right, without any comparison with other change phenomena, inabsence of any search for similarities between different contexts of change.On the other hand, it is to be acknowledged that many models of biologicalchanges include an explicit computation of critical exponents, thus postu-lating the existence, close to the critical point, of yet unobserved phenomenawhich should parallel the ones observed in the physical domain. Besides,it is to be taken into account that, within biological realm, observations ofthe kinds quoted above would be very difficult to make. However we mustremark that perhaps, in some cases, observations of this nature have beenalready made, but get unnoticed. This is particularly the case of electricalrecordings of neural activity. The latter are associated to an apparently‘noisy’ character which makes difficult to extract from the raw signal thetypical behaviours which could occur close to a critical point, even becauseit is very difficult to identify the correct ‘order parameter’ to be used inthese contexts. In recent times (see Beggs and Plentz 2003, 2004) it has beenreported the observation of neuronal avalanches close to a critical networkstate, whose EEG was characterized by a characteristic activity pattern.However, other authors (see, for instance, Bedard et al., 2006) questionedthe connection between EEG power spectra and the occurrence of criticalstates.

170 Eliano Pessa

Phenomenological theoriesIn this regard, as it is well known, two main approaches constitute the

skeleton of macroscopic TPT: the Ginzburg–Landau theory and the Renor-malization Group. While neglecting any technical description of these top-ics (which can be easily found on standard textbooks; see, for instance,Patashinskij and Pokrovskij 1979; Amit 1984; Huang 1987; Toledano andToledano 1987; Goldenfeld 1992; Benfatto and Gallavotti 1995; Cardy 1996;Domb 1996; Minati and Pessa 2006, Chap. 5), we will limit ourselves tomention the outstanding importance of two main ideas: on one hand, theidentification of phase transitions with symmetry breaking phenomena (andin practice with bifurcation phenomena in Ginzburg–Landau functional),and, on the other hand, the scaling hypothesis, allowing the elimination ofall irrelevant parameters close to critical point. While both ideas are correctonly in a limited number of cases, however they underlie a huge numberof models of phase transitions, so that every concrete computation, donewithin this context, is, in a direct or indirect way, based on them.

Theoretical frameworksIn many cases the framework adopted in building models within TPT

is the one of classical physics. This doesn’t mean that quantum theoryis neglected. In most contexts the interactions allowing the occurrence ofphase transitions have a quantum origin (the typical case being the one offerromagnetism). However in all these cases quantum theory (most oftenquantum mechanics) is used only to deduce effective potentials which, infurther computations, can be considered as entirely classical. The typicalcase is given by London dispersion forces whose quantum description endsin effective potentials like the celebrated one of Lennard–Jones. The latterare then used in subsequent computations as if they were classical entities,completely neglecting their quantum origin.

However, even the approach based on classical physics has its pitfalls,the main one being that this approach entails the belief in classical thermo-dynamics and in classical statistical physics. The latter, as it is well known(cfr. Rumer and Ryvkin 1980; Akhiezer and Peletminski 1980; De Boerand Uhlenbeck 1962), is based on the so-called correlation weakening prin-ciple stating that, during a relaxation process, all long-range correlationsbetween the individual motions of single particles tend to vanish when thevolume tends to infinity. This, in turn, implies that, while spontaneous sym-metry breaking phenomena, like the ones described by Landau theory, areallowed by classical physics without any problem (see, for instance, Green-berger 1978; Sivardiere 1983; Drugowich de Felıcio and Hipolito 1985), they


give rise to new equilibrium states which are unstable with respect to ther-mal perturbations, even of small amplitude (one of the first proofs of thisfact was given in Stein 1980).

At first sight, such a circumstance could not appear as a problem. Af-ter all, nobody pretends to build models granting for structures absolutelyinsensitive to any perturbation. However, a deeper analysis shows that in-stability with respect to thermal perturbations is equivalent to instabilitywith respect to long-wavelength disturbances, and whence entails the im-possibility of taking infinite volume limits. The latter is a serious flaw, asone of the main pillars of TPT, that is the divergence of correlation lengthat the critical point, due to the occurrence of infinite-range correlations,has a meaning if and only if we go at the infinite volume limit. Thereforethe aforementioned results imply that classical physics is a framework un-tenable if we are searching for a wholly coherent formulation of TPT, inwhich phenomenological models are in agreement with statistical physics.

How to find an alternative framework? The most obvious candidate isgiven by quantum theories. However, in this regard, it is immediate to rec-ognize that ordinary quantum mechanics must be ruled out, mostly becausethe celebrated Von Neumann theorem shows that all possible representa-tions of the state vector of a given quantum system are unitarily equiva-lent. This means that different representations give rise to the same valuesof probabilities of occurrence of the results of all possible measurementsrelated to the physical system under consideration, independently of theparticular representation chosen (for a more recent discussion on this topicsee Halvorson 2001; Redei and Stolzner 2001; Halvorson 2004). In this wayit is clearly impossible to describe the different phases of a same quantumsystem (obviously deeply differing as regards their physical properties), and,a fortiori, phase transitions.

At this point the only remaining possibility is to make resort to Quan-tum Field Theory (QFT). The attractiveness of QFT stems from the factthat within QFT, and only within it, there is the possibility of having differ-ent, non-equivalent, representations of the same physical system (cfr. Haag1961; Hepp 1972; a more recent discussion on the consequences arising fromthis result, often denoted as ‘Haag Theorem’, can be found in Bain 2000;Arageorgis et al. 2002; Ruetsche 2002). As each representation is associatedwith a particular class of macroscopic states of the system (via quantumstatistical mechanics) and this class, in turn, can be identified with a par-ticular thermodynamical phase of the system (for a proof of the correctnessof such an identification, see Sewell, 1986), we are forced to conclude that

172 Eliano Pessa

only QFT allows for the existence of different phases of the system itself.Within this approach all seems to work very well. Namely the occur-

rence of a phase transition through the mechanism of spontaneous symme-try breaking (SSB) implies the appearance of collective excitations, whichcan be viewed as zero-mass particles carrying long-range interactions. Theyare generally called Goldstone bosons (cfr. Goldstone et al. 1962; for a moregeneral approach see Umezawa 1993). Such a circumstance endows thesesystems with a sort of generalized rigidity, in the sense that, acting uponone side of the system with an external perturbation, such a perturbationcan be transmitted to a very distant location, essentially unaltered. Thereason for the appearance of Goldstone bosons is that they act as order-preserving messengers, preventing the system from changing the particu-lar ground state chosen at the moment of the SSB transition. Moreover,the interaction between Goldstone bosons explains the existence of macro-scopic quantum objects (Umezawa 1993; Leutwyler 1997). Here, it mustbe stressed that the long-range correlations associated with a SSB ariseas a consequence of a disentanglement between the condensate mode andthe rest of system (Shi 2003). This finding, related to the fact that withinQFT we have a stronger form of entanglement than within Quantum Me-chanics (Clifton and Halvorson 2001), explains how the structures arisingfrom a SSB in QFT have a very different origin from those arising from aSSB in classical physics. Namely, while in the latter case we need an exact,and delicate, balance between short-range activation (due to non-linearity)and long-range inhibition (due, for instance, to diffusion), a balance whichcan be broken even by a small perturbation, in the former case we havesystems which, already from the starting, lie in an entangled state, withstrong correlations which cannot be altered as much by the introduction ofperturbations.

Despite these advantages, however, the situation within QFT-based ap-proach is not so idyllic at is could appear at a first sight. Namely themathematical machinery of QFT is essentially based on propagators (orGreen functions) which allow to compute only transition probabilities be-tween asymptotic states, without any possibility of describing the transientdynamics occurring during the transitions themselves. And, what is worse,it is easy to prove that, in most cases, such a dynamics must be described byclassical physics. In this regard we recall that quantum effects are negligiblewhen, at a given temperature T , the following relationship is satisfied:

kT/(~ωc) >> 1 (1)


where ωc denotes a suitable characteristic frequency and, as customary, k

and ~ denote, respectively, the Boltzmann constant and the Planck con-stant divided by 2π. Now, close to transition point, we can identify thecharacteristic frequency with the inverse of spontaneous relaxation time τ

which, owing to the phenomenon of critical slowing down, can be written,by resorting to scaling arguments, as:

τ ≈ ξz (2)

in which ξ is the correlation length and z denotes the so-called dynamicalcritical exponent. In turn, the correlation length in proximity of the criticalpoint follows the scaling relationship:

ξ ≈ δν . (3)

Here ν is another critical exponent and δ denotes a suitable measureof the distance with respect to critical point. For instance, if the orderedphase corresponds to T < Tc , where Tc is the critical temperature, we willchose δ = (Tc− T )/Tc. By substituting (2) and (3) into (1) and identifyingkT with kTc , we will find that (1) becomes:

kTc/(~δ−ν z) >> 1. (4)

Now, as experimental data evidenced that the value of ν is not very farfrom – 1 (close to – 0.7), we can simplify (4) under the form:

kTc >> ~δz. (5)

It is immediate to see that (5) is always satisfied, provided Tc be greaterthan zero (when Tc = 0 the possibility is left open for quantum phasetransitions; see in this regard, for a short review of some examples, Brandes2005, pp. 449–455). Namely this occurs not only for the trivial case in whichδ = 0, but even for nonzero values of δ. For instance, if we deal with a3-dimensional system and we accept for z the value 3/2 (on which mostphysicists seem to agree), it is easy to see that, in correspondence to thecritical point of superfluid transition of liquid Helium, that is Tc = 2.18oK,if we choose δ = 10− 6 then the left hand member of the inequality (5) hasa value approximately of 3.008 × 10− 2 3 J, while the right hand memberhas a value of 1.054× 10− 4 3 J.

In short: the framework for describing phase transitions is based onQFT, but the description of what occurs in correspondence to a phase

174 Eliano Pessa

transition can be done only in classical terms. No doubt, a very strangesituation! This raises a number of further problems, because the transitionfrom the quantum regime, existing far from critical point, and the classicalregime, holding close to critical point, is nothing but a phenomenon ofdecoherence, due to the interaction with the external environment. But,what are the physical features of the latter process (and of the symmetricprocess of recoherence, taking place after the phase transition and restatingthe quantum regime)? How to describe the external environment? Howthese phenomena can influence the formation of structures (like defects)surviving even after the completion of phase transition and signalling itspast occurrence? All these questions cannot, unfortunately, be answeredwithin the traditional framework of QFT. Namely these problems have beendealt with by resorting to suitable generalizations of it. We will postponea discussion of these fundamental topics until the following sections fourand five, being here enough to mention the main problems connected tothe choice of a framework for TPT.

Statistical argumentsThe main feature related to the use of statistical arguments (and of

statistical models) within TPT is that often the latter rely on canonicalensembles and deal therefore with systems in thermal equilibrium with theenvironment. All that remains to be done, within such a context, is tocompute the partition function (admittedly a very difficult task). However,processes far from equilibrium (which are of interest when dealing withbiological phenomena) are out of reach by these methods.

3. Attempts to Use TPT to Model Biological Emergence

Before dealing with the topic of this section, we must take into accountthat biological phenomena (and their models) need to be subdivided intodifferent classes as a function of their spatial and temporal scales. The lattercan be arranged within a sort of hierarchy, which can roughly summarizedas follows:

(1) phenomena occurring on atomic or molecular scales, on times of theorder of nanoseconds, such as, for instance, the ones related to interac-tions between electrical dipoles in water solutions, or to conformationalchanges of macromolecules;

(2) phenomena occurring on a cell scale, on times of the order of millisec-onds, or dozens of milliseconds, such as the discharge of a neuron;

(3) phenomena occurring on the macroscopic scale of a single organ, or


of a whole living being, on times of the order of seconds, or tenthsof seconds, such as, for instance, the visual recognition of a perceivedimage;

(4) phenomena occurring on the macroscopic scale of a whole organism, ontimes of the order of minutes, or hours, such as filling a questionnaire,solving a problem, understanding a concept;

(5) phenomena occurring on the macroscopic scale of a whole organism, buton times of the order of weeks, months, years, such as the growth ofa tree, the acquisition of mathematical competence, the reorganizationof personality;

(6) phenomena occurring on the macroscopic scale of a set of organisms, ontimes which often are very long, such as the evolution of a species, thechange of economic relationships, the acquisition of new social habits.

At this point, two remarks are in order. The first is that it appears ashighly implausible that changes occurring at different levels of this hierar-chy can be dealt with by the same theory of phase transitions. The secondis that it seems very difficult that phase transitions occurring at the lowestlevels of this hierarchy can have a direct influence on phenomena takingplace at the highest levels. As a consequence, it seems unlikely that actualTPT can account for all change phenomena occurring at any level, from a)to f). And it appears likewise unlikely that phase transitions occurring, forinstance, at level a) can explain the occurrence of phenomena occurring atlevel e), such as the emergence of consciousness. These considerations let usunderstand why the applications of traditional TPT have been, so far, lim-ited to level a), with little incursions at level b). However we could justifythe need for applying a single TPT to all levels of the previous hierarchyby resorting to one of two different (but mutually compatible) strategies.The first consists in building a many-level TPT (obviously generalizing theactual TPT). This is a very difficult enterprise, as actual statistical physicsis typically a two-level theory. However the improvement of our knowledgeabout the interactions between defects and, more in general, about meso-scopic physics, could be a promising starting point for continuing on thisway. The second strategy is based on the assumption that changes occur-ring at level n be nothing but emergent phenomena related to interactionsoccurring at level n – 1, and that this emergence is based on mechanismswhich are universal (or belonging to universality classes) and independentfrom n. We could name this statement as hypothesis of universality of inter-level transitions. It has been, more or less implicitly adopted, for instance,by the followers of the so-called connectionist approach in Psychology (the

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approach dates back to the Eighties; for more recent reviews see Chris-tiansen and Chater 2001; Quinlan 2003; Houghton 2005). As a matter offact, a look at experimental data about human behaviour would seem, atfirst sight, to prove the existence of phase transitions endowed with featuresvery similar to the ones observed in physical world. Developmental curvesfor prehension in infants (Wimmer et al. 1998) or for analogical reasoningin children (Hosenfeld et al. 1997), and even for cognitive changes within apsychotherapy (Tang et al. 2005) look very similar to the curve character-izing the lambda transition for liquid helium. Therefore we could considerthe hypothesis of universality of interlevel transitions as a working hypoth-esis which, even if not proved, so far doesn’t seem absurd or contradictingexperimental findings.

In any case, all these considerations point to a theory which should gobeyond traditional TPT. The application of the latter to biological phe-nomena has been so far limited mostly to level a) and is based on twomain mechanisms, which here we call Davydov effect and Frohlich effect.Before shortly discussing them, we must remark that at this level the useof quantum framework is widely diffused and accepted since long time,as most phenomena regarding biological macromolecules involve entitiessuch as electrons or ions for which quantum effects can be important (seeHelms 2002). Even most chemical reactions of biological relevance, such asenzyme-catalyzed reactions, are dominated by tunneling phenomena whichare typically of quantum nature (see, for a review, Kohen and Klinman1999, and, for a recent example, Masgrau et al. 2006). Even neurons couldbehave as quantum systems (Bershadskii et al. 2003), and odor recognitionmechanism appears to be based on quantum effects (Brookes et al. 2007).

Let us now take first into consideration the Davydov effect. It consistsin the production of solitons moving on long biomolecular chains, whenthe metabolic energy inflow is able to produce a localized deformation onthe chains themselves. The original context in which Davydov introducedhis ideas was the one of protein α-helices in which different neighbouringmonomers interact through hydrogen bonds (the typical case is given bymuscle myosin). The energy liberated in the hydrolysis of ATP in a site cor-responding to a particular monomer gives rise to a vibration excitation ofthe latter, and this excitation propagates from one monomer to the other,owing to the dipole-dipole interaction between them. However this excita-tion interacts even with the hydrogen bonds, producing a deformation ofspatial lattice structure of the polymer. Such a deformation, in presenceof suitable conditions, propagates along the protein under the form of a


soliton. The origin of Davydov effect, therefore, lies in the non-linear cou-pling between the vibrational excitation and lattice displacements.

Since the appearance of first papers by Davydov on this subject (seeDavydov 1973; Davydov and Kislukha 1973; Davydov 1979, 1982) the lit-erature about it has grown in such a way that it is impossible even to quoteonly the most relevant papers (here we will limit ourselves to mention onlypapers that, in a way or in another, contain reviews of this topic, such asScott 1992; Forner 1997; Cruzeiro–Hansson and Takeno 1997; Brizhik etal. 2004; Georgiev 2006). Before shortly discussing the Davydov model, wemust point out that, strictly speaking, it doesn’t describe a phase transi-tion, but deals, rather, with the state of affairs which could occur after thephase transition has taken place. Namely the goal of this model is only toargument in favour of a possible collective effect whose occurrence is al-lowed by the fact that model parameter values are confined within suitableranges. While being obvious that such a confinement results from a previ-ous phase transition in which parameter values crossed some critical point,both the mechanism of this transition and the phase eventually precedingit are not investigated.

As regards the explicit formulation of Davydov model, it can be done inmany different ways. Davydov himself started from a general Hamiltonian,describing the dynamics of the excitations, of phonons, and their interac-tions within a 1-dimensional lattice with N sites and constant spacing a, inwhich each site contains a monomer of mass M . The Hamiltonian operatorof this system is written, in second-quantized form, as a sum of three terms:

H = Hex + Hph + Hint (6)

where Hex is the exciton Hamiltonian:

Hex =∑

n[(ε−D)A+

n An − J(A+n An−1 −A+

n An+1)] (7a)

Hph is the phonon Hamiltonian:

Hph = (1/2)∑

n[κ(un − un−1)2 + (p2

n/M)] (7b)

while Hint is the interaction Hamiltonian:

Hint = χ∑

n[(un+1 − un−1)A+

n An]. (7c)

In the previous formulae A+n (An) denotes the fermionic creation (anni-

hilation) operator for a quantum excitation in the site n, while un is thedisplacement operator and pn the momentum operator in the same site (thelatter, however, have a different character, as phonons are bosons). More-over, ε is the excitation energy released by a single external input (like ATP

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hydrolysis), D the deformation excitation energy, J is the absolute value ofdipole-dipole interaction energy between neighbouring sites, κ the elasticityconstant of the lattice, and χ a parameter giving the strength of couplingbetween vibrational excitations and lattice displacements.

At this point many different routes can be followed. For instance onecould derive from the Hamiltonian the Heisenberg equations of motion forthe operators and then try to derive from the latter suitable consequences.However Davydov adopted a different approach. First of all, he focussed hisinterest on the stationary states of the Hamiltonian (6), hypothesizing thatthe latter were containing all information needed to describe the phenomenahe was searching for. In the second place, he assumed (the so-called Davydovansatz ) that these stationary states were of the form:

ψs(t) =∑

nan(t) exp[σ(t)]A+

n |0〉 (8)

where the complex function an(t) describes, through its squared modulus,the distribution of excitations within the molecular chain. This functionis normalized in a suitable way (we omit the normalization rule, to savespace). Besides, the operator σ(t) is given by:

σ(t) = −(i/~)∑

n[βn(t)pn − πn(t) un] (9)

It can be shown that βn(t) and πn(t) describe, respectively, the averagevalues of displacements and of momenta of molecules in the stationary state.When imposing the condition that, in correspondence to stationary states,the functional 〈ψs|H |ψs〉 has a minimum value, it is possible to obtain,by exploiting the commutation rules of the different operators appearing in(6), a set of equations which, if we go to a continuum limit in which thesite numbers n are replaced by a continuous spatial coordinate x, becomea set of partial differential equations for the unknown functions a(x,t),β(x, t) (as regards the function π(x, t) it is possible to show that it canbe expressed through β(x, t), as expected on intuitive grounds). A numberof manipulations (for details we refer to the literature, chiefly to Davydov1979, which, despite some mistakes in the formulae, is still one of the bestpapers on this subject) let to obtain an equation for the function a(x, t)having the form:

[(i~) ∂t − Λ + J ∂xx + G |a(x, t)|2] a(x, t) = 0 (10)

where:

Λ = ε−D +κ− 2 J, G = 4 χ2/[κ(1− s2)], s = ν/νac, νac = (κ/M)1/2

(11)


It is easy to recognize in (10) a particular version of Non-linear Schrodingerequation which, as it is well known, allows for the existence of solitonicsolutions. We will limit ourselves, here, to show the explicit form of thesolitonic solution for |a(x, t)|2, given by:

|a(x, t)|2 = 2 µ sec h2[µ (x− vt− x0)]. (12)

Here v denotes the velocity of propagation of soliton along the molecularchain, always lesser than the velocity of propagation of longitudinal soundwaves νac. Moreover, the symbol µ is defined by:

µ = χ2/[κ(1− s2)]. (13)

Leaving aside the question of biological plausibility of the mechanism pro-posed by Davydov, we remark that the approach described above leavesopen even the question of the stability of soliton with respect to externalperturbations, for instance of thermal nature. Without dealing with thisdifficult problem, here we will limit ourselves to stress that it could be bet-ter tackled by adopting, already from the starting, a QFT-based approach.Within the latter (for an extensive list of references we refer to Umezawa etal. 1982; Umezawa 1993; in the following we will partly follow the schemeadopted in Del Giudice et al. 1985; a useful reference is also Matsumoto etal. 1979) the Nonlinear Schrodinger equation (10) is no longer a c-numberequation arising as a consequence of the choice of the Hamiltonian (6), butrather a field equation for an excitation quantum field described by a suit-able field operator ψ(x, t). It is convenient, in order to save space, to rewritethis field equation in the shortened form:

Q (∂)ψ = I(ψ) (14)

where, in conformity with (10), the operator Q (∂) is given by:

Q (∂) = (i~) ∂t − Λ + J ∂xx (15)

while the ‘current’ operator is:

I(ψ) = − Gψ+ψψ (16)

Now, in order to endow the theory of this field with a physical meaning,we must connect it with the so-called asymptotic ‘free’ field, the only onewhich, in principle, is accessible to observation, do not being perturbed bythe interactions described by current operator. The asymptotic field ϕ istherefore described by the field equation:

Q (∂) ϕ = 0. (17)

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The complex boson field ϕ is to be conceived as a quasi-particle field, de-scribing the vacuum excitations. The vacuum itself, of course, is nothing butthe degenerate vacuum state arising after a symmetry-breaking transitionand, in our case of Davydov soliton, consists in the state of the molecu-lar chain in presence of Goldstone bosons (the quasi-particle excitations)granting for the stability of the particular choice of a ground state, occurredafter the transition.

It is to be taken into account that the interacting field ψ must be ex-pressed as a function of the asymptotic field ϕ, in order to have a physicallymeaningful theory. This implies that we must have a map of the form:

ψ = F [ϕ]. (18)

In practice such a map (in some contexts called dynamical map) can befound, for instance, by recasting (16) under the form of an integral equation(the so-called Yang–Feldman equation) which, by using (17), can be writtensynthetically as:

ψ = ϕ + Q−1(∂) ∗ I(ψ) (19)

where the asterisk denotes the convolution operator and Q−1(∂) is noth-ing but the Green function for the ϕ field. At first sight the equation (19)could be solved only by resorting to suitable iteration procedures, whichshould let express ψ in terms of suitable powers of ϕ, thus specifying theconcrete form of the map (18) up to the wanted degree of approximation.However, in order to implement such a computation, we need further infor-mation. Namely, as mentioned in the previous section, within a QFT-basedframework, we have the possibility of different, not unitarily equivalent,representations of the same system. In our case these representations corre-spond to the different vacua, arising as a consequence of vacuum degeneracyoccurring after the symmetry-breaking phase transition. In order to havephysical predictions, we must choose a particular kind of vacuum among themultiplicity of the possible ones. Otherwise the dynamical map is devoidof any physical significance. Namely both (18) and (19) allow the freedomfor a choice of the boundary conditions, essentially because the dynamicalmap holds only in a weak sense, that is relating expectation values and notoperators. In order to eliminate the ambiguity in the choice of boundaryconditions, let us now introduce a c-number function f(x, t) satisfying theequation:

Q (∂)f = 0 (20)


and a space-time dependent translation of the asymptotic quantum field ϕ

defined by:

ϕ → ϕ + f. (21)

The (21) is also called the boson transformation. It is to be remarked thatit is a canonical transformation, as it leaves unchanged the canonical formof commutation relations. Moreover, it can be associated to a generator ∆,allowing to write (21) under the form:

ϕ′ = exp(−i∆) ϕ exp(i∆) (22)

the explicit form of ∆ being given, in the case of Davydov soliton and withina reference frame comoving with it, by:

∆ = −∫

dx [g(x, t) ∂tϕ− ϕ ∂tg(x, t)], g(x, t) = θ(x) f0(x, t) (23)

where θ(x) is the Heaviside step function, while f0(x, t) denotes the solutionof (20) when the velocity of soliton is set to zero.

Let us now denote by |0〉 the vacuum state for the field ϕ. It is, then,convenient to introduce the state |f0〉 defined by:

|f0〉 = exp (i∆) |0〉 . (24)

Easy considerations (here, as usually, omitted), based on the fact that thegenerator ∆ induces a space-time displacement even of the annihilationoperator relative to the vacuum state |0〉, show that |f0〉 is a coherent state,as it is an eigenstate of the latter operator.

In order to exploit the consequences of the previous formalism, let usfirst denote by the symbol ψf the interacting quantum field, satisfying (14)or (19), when the asymptotic field, undergoing the boson transformation,has been defined by (21) or (22). It is then immediate to recognize from(19) that ψf fulfils the following form of Yang–Feldman equation:

ψf = ϕ + f + Q−1(∂) ∗ I(ψf ) (25)

Taking the vacuum expectation value of (25) we will obtain:

Φf = 〈0| ψf |0〉 = f + 〈0| Q−1(∂) ∗ I(ψf ) |0〉 (26)

Here we made use of the fact that the vacuum expectation value of the quasi-particle excitation field ϕ is vanishing. The macroscopic order parameterΦf , defined through (26), has clearly a quantum nature, implicitly containedin the developments in powers of ~ needed to obtain an explicit (thoughapproximate) form of the right hand member of this equation. In this regard

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the lowest level of approximation (the so-called Born approximation) is thatin which ~ tends to zero and 〈0| I(ψf ) |0〉 tends to I(Φf ). In this case (26)becomes:

Φf = f + Q−1(∂) ∗ I(Φf ). (27)

When applying to both members of (27) the operator Q(∂) we will imme-diately obtain, by taking into account (20), the equation:

Q(∂)Φf = I(Φf ) (28)

that is the classical equation describing the Davydov soliton. This time,however, differently from the approach adopted by Davydov, we can moreeasily understand the origin of the order parameter Φf . Namely from (22)and (24) we can immediately recognize that:

〈f0| ϕ |f0〉 = f (29)

so that f appears as a coherent condensation of quasi-particle excitationsdescribed by the quantum field ϕ. In absence of the self-interaction term(27) therefore tells us that even Φf is nothing but a coherent condensate ofquasi-particles. However, this condensate evolves dynamically as a soliton,owing to non-linear self-interaction contained in the right hand member of(28). We underline that the above picture is valid only within an approx-imation which is of zero order in ~. Nobody prevents, however, from go-ing to higher-order approximations, where more complicated effects shouldemerge. All these computations, as well as the stability analysis of the soli-ton, are allowed by the adoption of a QFT-based framework, in which Gold-stone bosons, arising as a consequence of a symmetry breaking, undergo acoherent condensation, giving rise to observable effects, whose detailed na-ture depends essentially on the interactions connecting the quantum fields.Within this approach all effects are from the beginning endowed with aquantum nature and we don’t need additional assumptions about the cou-pling between phonons and vibrational excitations. This occurs because inthis case phonons coincide just with the quasi-particle excitations acting asGoldstone bosons. As a proof for this interpretation we can add the factthat the expectation value of density probability of ϕ in the coherent state|f0〉 fulfils the equation of sound propagation (like phonons).

Before going further we remark that, despite the high inner coherenceand the conceptual economy allowed by this approach, it leaves unsolved anumber of problems, mainly connected to the study of transient processestaking place during the phase transition, not after nor before. Unfortunatelythese processes have a fundamental role in determining the outcome of a

184 Eliano Pessa

From the latter relationships it would be possible to sketch an hypotheti-cal, even if simplistic, path leading from the critical point of a symmetry-breaking phase transition to the stable propagation of solitons along themacromolecule. As it is possible to see, the critical point of the phase tran-sition can be identified with the condition G = 0. Very close to this criticalpoint we deal with a very small (even if positive) value of G, and thereforeof b, owing to (35). As the behaviour is classical, the equation (32) has asolution which, while being a soliton, is very flattened, as it follows from(34). However, such a first solitonic ‘hint’ is just a cue for the subsequentquantum regime, occurring when, as a consequence of the growth of G,the classical description is no longer valid and Goldstone bosons enter intoplay. Namely the latter give rise to a coherent condensate favoured justby the initial soliton solution, in such a way as to induce the QFT-baseddynamics previously described when discussing the boson transformation.In other words, the initial classical solitonic solution ‘forces’ the choice of aparticular vacuum state, among the ones present after the symmetry break-ing, a choice kept stable against perturbations by the essentially quantummechanism of coherent boson condensation.

Let us now shortly discuss the other mechanism quoted at the begin-ning of this section, that is the Frohlich effect. The latter, in its essence,consists in the excitation of a single (collective) vibrational mode withina system of electric dipoles interacting in a nonlinear way with a suitablesource of energy, or with a thermal bath. These dipoles are thought to bepresent both in water constituting the intracellular and extracellular liquidas well as in biological macromolecules, owing to ionization state connectedto the existence of high-intensity electric fields close to cellular membranes.This effect can be seen as a sort of Bose condensation of phonons describ-ing dipole vibrational modes, and therefore it appears as a truly quantumeffect. The original model proposed by Frohlich (see Frohlich 1968; theheuristic arguments introduced in the following have been adapted fromthe exposition given in Sewell 1986, pp. 209–214) was described in termof kinetic equations describing the rate of changes of occupation numbersni (i = 1, ..., N) of N phonon modes with frequencies ωi. The equationshave the explicit form:dni/dt = si − ai[ni − (1 + ni) exp(−βωi)]

−∑

kbik[ni(1 + nk) exp(−βωi)− nk(1 + ni) exp(−βωk)]. (36)

Here β = ~/kT while si is a constant pumping term. The term withcoefficient ai describes the emission and absorption of quanta due to


interactions with thermal bath, and the term with coefficient bik describesthe exchanges of quanta between different modes. These equations can befurther simplified by assuming rates which do not depend on mode indices.Then straightforward mathematical transformations reduce them to theform:

dni/dt = s− ni − gθ ni + gν (1 + ni) exp(−βωi) (37)

where s is the pump parameter, g a suitable coupling constant and ν, θ aredefined by:

ν =∑

knk/N, θ =

∑i

(1 + ni) exp (−β ωi)/N. (38)

When looking at the equilibrium solution of (37), that is the stationarydistribution of occupation numbers, it is possible to cast it under the form:

ni = [1 + (1/g) exp(−βωi)]/ exp[β(ωi − µ)]− 1 (39)

where the quantity µ is defined by:

µ = (1/β) ln [g s/(1 + g θ)]. (40)

It is immediate to recognize that (39) defines a Bose–Einstein distribution,apart from the multiplicative factor [1 + (1/g) exp (−β ωi)]. Within it µ

plays the role of a chemical potential. It is then possible to see that, whenthe pump parameter s tends to a suitable critical value, the chemical poten-tial µ tends to the lowest frequency ω1 so that n1 diverges: a Bose–Einsteincondensation of phonon quanta occurs (for a rigorous proof of this assertionwe address the reader to the quoted literature, as well as to Del Giudice etal. 1982). It is to be remarked that the critical value of pump parameterdefines a true phase transition.

Despite the fact that original Frohlich model was formulated withoutresorting to a microscopic Hamiltonian, it has been possible, however, torecast it within the language of QFT (see Del Giudice et al. 1985). TheQFT-based interpretation of Frohlich effect is, anyway, slightly more com-plicated than the one of Davydov effect, as dipoles need a (3+1)-dimensionalmodel, while for solitons on macromolecular systems (1+1)-dimensionalmodels were enough. In the case of Frohlich effect the coherent state canstill be interpreted as deriving from a condensation of quasi-particle Gold-stone modes, but the picture is made more complex because a 3-dimensionalspace forces to introduce a symmetry breaking of a SU(2)-invariant model,implying a richer mathematical structure and different kinds of Goldstonebosons. In any case, it is to be stressed that in the course of time a num-ber of Hamiltonian descriptions of Frohlich effect have been proposed, all

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evidencing how Davydov and Frohlich effects be strictly interrelated, so asto be nothing but different consequences of the same theoretical framework(see, in this regard, Bolterauer et al. 1991; Nestorovic et al. 1998; Mesquitaet al. 1998, 2004).

Before ending this short discussion of Frohlich effect, we remark that theadoption of QFT-based perspective allows a natural coupling between thelatter and Davydov effect, giving rise to a possible general theory about theorigin of biological coherence (see Del Giudice et al. 1985, 1988). Namely,as biological macromolecules are embedded within water, it is possible tohypothesize the existence of a general effect, arising from the coupling oftwo successive phases. In the first the random release of energy producedby metabolic chemical reactions is channelled (and, in a sense, stored) inan ordered way through solitons travelling along 1-dimensional molecu-lar chains, owing to Davydov effect. In the second phase the motion ofsolitons themselves induces an electric polarization in the water surround-ing the molecular chains themselves. The fact that this polarization hasbeen induced by a non-thermal phenomenon, such as the travelling of asoliton, breaks the rotational symmetry of water dipoles, and, as a conse-quence, triggers the occurrence of Goldstone bosons (the so-called dipolewave quanta). The condensation of these latter gives rise to a Frohlich-likeeffect, resulting in an ordered (electret) state of the system of water dipoles,associated to coherence and appearance of long-range correlations. The netresult of the two phases is that a disordered energetic input has produced acoherent state. It is important to remark that this effect is based on a singledynamical scheme, the one of QFT-based description of symmetry break-ing. Without entering into further details on this topic, we recall that theapproach based on QFT gave rise to a highly interesting theory of the oper-ation of one of most important biological systems: the brain. This QuantumBrain Theory (the huge amount of literature on this topic is summarizedin Jibu and Yasue 1995, 2004; Vitiello 2001; Del Giudice et al. 2005) hada number of important applications, such as the study of the role of cy-toskeleton in neural cells (Tuszynski et al. 1997, 1998; Hagan et al. 2002;Tuszynski 2006) the operation of memory (Ricciardi and Umezawa 1967;Stuart et al. 1978, 1979; Vitiello 1995; Alfinito and Vitiello 2000; Pessa andVitiello 2004; Freeman and Vitiello 2006), and the basis for consciousness(Penrose 1994; Hameroff and Penrose 1996; Hameroff et al. 2002; Tuszynski2006).

We can now summarize the discussion performed in this section by say-ing that, provided we adopt a QFT-based description of phase transitions


interpreted as symmetry breaking phenomena, traditional TPT is able toaccount for phase transitions in biological matter at the lowest, microscopiclevel. However, such a possibility exists only within a perspective in whichGoldstone bosons are considered as entities able to produce physical effectsthrough condensation phenomena, a view actually not shared by all physi-cists. In any case, it is to be stressed that the application itself of QFT tobiological phenomena evidenced a number of failures of traditional TPT.These can be listed as follows:

1) within the biological realm the most important process is the inter-action between a system and its environment; as it is well known the lattercould destroy quantum coherence, so as to transform the picture presentedin this section in nothing but a dream, a sort of illusion in disagreementwith the realm of experimental observations;

2) the formalism of QFT is based on an Hamiltonian structure, in turnrelated to a physics based on energy conservation principle; however, mostbiological phenomena appear as typically dissipative and whence not re-ducible to an Hamiltonian treatment;

3) within the biological world one must take into account finite volumeeffects, which destroy the picture presented above, as the lack of infinitevolume limits opens the way to the formation of defects, impurities, andsometimes of disorder;

4) the previous models are useful only to describe the emergence ofbiological coherence at the microscopic level; however they are unable toaccount for the plurality of levels existing in biological matter and to providethe basis for a multi-level approach in which we can describe the formationof a new level from a lower level, in a way which is partially independentfrom the level taken into consideration;

5) the solution of the problems described in the points 1) - 4) requiresthe introduction of a description of the environment and whence of a generaltheory of organism-environment interactions; such a theory, however, is outof reach of actual TPT.

In the next sections we will try to discuss the problems 1)-5), by intro-ducing some arguments which perhaps could be useful to the enterprise ofgeneralizing traditional TPT so as to account for biological changes.

4. Decoherence and Dissipation

The phenomenon of decoherence consists in the suppression of non-diagonal(interference) terms in the density operator of a quantum system, owing toa number of different possible influences, most of which include the action of

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external environment. This results in the destruction of interference effectsconnected to entangled states which, in turn, are at the basis of quantumcoherence (for reviews on this topic see, for instance, Giulini et al. 1996;Zurek 2003). The existence of this phenomenon induced a number of authorsto make claims about the uselessness of quantum theories in the biologicaldomain, as the decoherence time would be, in normal situations in whichbiological systems operate, so small as to make the classical description ofthese latter the only one feasible (see, for instance, Tegmark 2000; it is,however, to be taken into account the criticism by Alfinito et al. 2001).

A short discussion of decoherence phenomenon should start by listingthe possible mechanisms for decoherence so far taken into consideration.Following a tentative classification introduced by Anastopoulos (1999) wewill distinguish between two different kinds of mechanisms: the extrinsicand the intrinsic ones. Within the former the decoherence is produced asa consequence of a suitable coupling between the system and the externalenvironment, while the latter give rise to decoherence owing to the formitself of system dynamics, which selects in a very short time particularpointer states. Among the extrinsic mechanisms we can quote:

d.1) the interaction with a thermal bath;d.2) the interaction with external noise;d.3) the interaction with a dissipating environment;d.4) the interaction with a chaotic environment;d.5) the introduction of finite volume or of special boundary conditions.We first stress that this list is rather rough, as there can be a signifi-

cant overlap between different categories — such as, for instance, d.2) andd.3) — when the environment is described by detailed microscopic mod-els. In the second place we must remark that in some cases these mecha-nisms fail to achieve decoherence (see, for instance, Ford and O’Connell,2001, as well as the considerations contained in Pascazio 2004, concern-ing the role of noise in decoherence). In any case it is to be underlinedthat the assessment of the role of these mechanisms, as well as the esti-mates of decoherence time, are based mostly on a microscopic descriptionof the environment itself. In this regard, the mechanisms listed above dif-fer one from another in the possibility of giving an explicit description ofthis kind. While in the case d.1) the environment can be modelled througha system of oscillators in equilibrium at a given temperature, coupled tothe system in a suitable way, and in d.2) we can resort to some kind ofBrownian motion, in the case d.3) we have only a macroscopic descrip-tion of a dissipation, whose microscopic description is not fixed in advance.


A simple example is given by the elementary damped harmonic oscillator:

d2x/dt2 + γ dx/dt + ω20 x = 0 (41)

Here the friction coefficient γ could be related, via the traditionalfluctuation-dissipation theorem (in the Callen–Welton interpretation), tothe autocorrelation function of force fluctuations. However, this tells usnothing about the role of friction in inducing decoherence, for instance, ina system of coupled quantum oscillators. We should need a further infor-mation on a microscopic theory of forces acting between the undampedoscillator and the environment. And — what is worse — even this theorywould be useless if, as showed by many authors (see, for instance, Cuglian-dolo et al. 1997; Perez–Madrid et al. 2003), we were in a context in whichfluctuation-dissipation theorem no longer holds.

Unfortunately, it seems that such a form of description of dissipationis unavoidable, at least if we want to work within the framework of QFT.As shown by a number of authors (Calzetta and Hu 2000; Calzetta et al.2001, 2002), the dynamical content of QFT is completely given by theso-called Haag map, which is nothing but an explicit version of Yang–Feldman equation (19), containing an infinite series of approximations interms of asymptotic fields. It is to be remarked that each term of this seriesdepends on a suitable correlation function of asymptotic fields, whose orderdepends on the term taken into consideration. As the series is infinite, it willcontain correlation function of every order. Besides, we will have that, ingeneral, correlations of different orders are reciprocally connected, so that,for instance, 2-point correlation functions will depend on 3-point correlationfunctions, and so on. In most cases of interest correlations of higher ordercan be practically neglected, so that the theory can stop the expansionat a given correlation order. This is the case of traditional QFT, which islimited to 2-point correlation functions. However, in presence of an externalenvironment, the latter can influence higher-order correlations, so as tomake important their contribution to lower-order correlations. What occurs,at this point, is that the traditional theory must account for these influencesunder the form of dissipative terms, generally having a macroscopic form,owing to the practical impossibility of monitoring in a detailed way theinfluence of changes in higher-order correlations produced by the externalenvironment. This explains why, in principle, every QFT-based descriptionis automatically dissipative in presence of an environment.

These arguments support the claim that every application of QFT tobiological phenomena should necessary include, within model equations, a

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description of dissipative effects. At a first sight, such a claim seems to giverise to new problems. Namely QFT, like Quantum Mechanics, is based onan Hamiltonian description of dynamics, and, as it is well known, it is verydifficult, if not impossible, to cast even the simplest models of dissipativedynamics in an Hamiltonian form. In this regard, it suffices to deal withthe elementary example of damped harmonic oscillator described by (41).This case, as shown by a number of authors (see, for instance, Denmanand Buch 1973; Nagem et al. 1991) can be described by a (time-dependent)Hamiltonian having the form:

H(x, p) = (1/2)[p2 exp(−γ t) + ω20 x2 exp(γ t)] (42)

However this Hamiltonian does not solve the problems arising when wetry to quantize the damped harmonic oscillator through the well knownprocedures. Namely from the Schrodinger equation associated to (41) it ispossible to derive (see, for instance, Edwards 1979; Bolivar 1998) that theuncertainty principle assumes the form:

∆x∆p = (~ω0/2Ω) exp(−γ t), Ω = [ω20 − (γ2/4)]1/2 (43)

As it is easy to understand, this relationship leads to physically unaccept-able consequences when we let t → +∞.

Many different strategies have been proposed to deal in a correct waywith the problem of quantization of dissipative systems. Among them wequote the one based on suitable modifications of quantization rules (foran overview see Tarasov 2001), and the one, inspired by Caldeira–Leggettapproach (see, among the others, Caldeira and Leggett 1983), which, afterintroducing a suitable microscopic model of the interaction with the dissi-pative environment, derives a generalized master equation for the quantumdensity matrix (the so-called Lindblad form; see, for instance, Lindblad1976; for an overview see Rajagopal 1998). The latter strategy, while inagreement with general principles of Quantum Mechanics, entails a com-plex mathematical apparatus, and is heavily based on the choices madein describing the environment (examples of this strategy can be found inRajagopal and Rendell 2001; Mensky and Stenholm 2003).

A more convenient and simpler strategy has been introduced by Celegh-ini, Rasetti and Vitiello, who, developing earlier proposals by Bateman(1931) and Feshbach and Tikochinsky (1977), described the influence ofdissipating environment by doubling the original dissipative system throughthe introduction of a time-reversed version of it, which acts as an absorberof the energy dissipated by the original system (see Celeghini et al. 1992;


for an overview of both the principles underlying this method and its ap-plications it is highly recommended the reading of Vitiello 2001). In thisregard, a dissipative system like the damped harmonic oscillator describedby (41) is doubled by introducing its “mirror” (i.e. time-reversed) systemgiven by:

d2y/dt2 − γ dy/dt + ω20 y = 0. (44)

It is possible to show that the dynamics of the whole system including both(41) and (44) can be described by the following (time-invariant) Hamilto-nian:

H(x, y, px, py) = px py + (γ/2)(y py − x px) + [ω20 − (γ2/4)] x y. (45)

It has to be remarked that in this way the reduction to an Hamiltonianform has been obtained at the price of doubling the number of degrees offreedom. However, this lets us introduce the usual quantization procedureswithout additional troubles. In particular, when resorting to an occupationnumber representation we must introduce two different kinds of creationand annihilation operators:

a = g (px− i Ω x), a+ = g (px + i Ω x), b = g (py − i Ω y), b+ = g (py + i Ω y)(46)

where Ω has been defined by (43), while g denotes the quantity (1/2 ~Ω)1/2.The commutation relations fulfilled by these operators have the form:

[a, a+] = 1 = [b, b+], [a, b] = 0 = [a, b+]. (47)

In terms of these operators the Hamiltonian (45) becomes:

H = ~Ω(a+b+b+a)+(i ~Γ/2)[a+a+−b+b+−(a a−b b)], Γ = γ/2. (48)

As the form (48) is very far from a diagonal one, we can perform a partialdiagonalization through the canonical transformation to new operators A

and B:

A = (1/√

2)(a + b), B = (1/√

2)(a− b). (49)

The latter allows to write the Hamiltonian (49) under the more tractableform:

H = ~Ω(A+A + B+B) + (i ~Γ)(A+B+ −AB). (50)

A straightforward extension of this Hamiltonian to QFT consists in writing(50) under the form of an infinite collection of damped harmonic oscillators:

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H =∑

k~Ωk(A+

k Ak + B+k Bk) +

∑k(i~Γk)(A+

k B+k −AkBk) (51)

where k denotes the spatial momentum.The simple arguments presented above put into evidence the existence

of two different possible strategies in using the doubling mechanism. Thefirst one starts from a macroscopic classical description of dissipation, in-troduced by hand, like in (41), and resorts to doubling mechanism in orderto obtain a quantum or a QFT counterpart, shown in (50) or (51), so as todeal with a dissipative system without abandoning the traditional quantumformalism. The second one tries to take advantage of eventual (of course,not canonical) transformations leading from the traditional harmonic oscil-lator Hamiltonian to the damped oscillator Hamiltonian, in order to derive,from a whatsoever non-dissipative Hamiltonian, the corresponding dissipa-tive version, expressed in terms of damped oscillators of the form (41).Obviously, both strategies assume that Γ or Γk be known in advance or canbe derived by a suitable microscopic theory.

Here we will show a simple example of implementation of the secondstrategy, starting from the well known harmonic oscillator Hamiltonian:

H = ~ Ω a+a. (52)

Here the zero point energy was discarded to speed up the computations, byredefining the value of energy in the ground state. Now the problem is tointroduce a formal procedure in order to transform (52) into the form (48).This can be done through the following steps:

• t.1) introduction of new operators α, β through the canonical trans-formation:

a = (1/√

2)(α + β), a+ = (1/√

2)(α+ + β+) (53)

• t.2) discarding from the new Hamiltonian, written in terms of α, β, allcontributions having the form α+α + β+β, as they cancel one with theother when applied to the vacuum state (they have opposite eigenval-ues); this procedure leads to the Hamiltonian:

H = (1/2) ~ Ω (α+β + β+α) (54)

• t.3) introduction of new operators A, B through the (not canonical)formal transformation:

α+ = A+ + A + B+ + B (55a)


α = (iΓ/2Ω)A++[1+(iΓ/2Ω)]A−(iΓ/2Ω) B++[1−(iΓ/2Ω)] B (55b)

β+ = A+ −A + B+ −B (55c)

β = (iΓ/2Ω)A++[1−(iΓ/2Ω)]A−(iΓ/2Ω) B++[1+(iΓ/2Ω)] B (55d)

When we apply (55.a)–(55.d) to the Hamiltonian (54) we will obtain, bydiscarding the contributions of the form A+A + B+B, a new Hamiltonianwhich, in terms of the new operators A, B, has exactly the form (48). Itis to be remarked that, as expected on general grounds, the choice of thecoefficients of (55.a)–(55.d) is not unique. Namely they are solutions of a(nonlinear) system of algebraic equations coding the constraints allowing asatisfactory solution of the problem of transforming (52) into (48). Withoutreporting here, in order to save space, the complete form of this system,we will limit ourselves to mention that, among these constraints, there arethe ones of fulfilment, by the involved operators, of commutation relationshaving the form (47), as well as the one of granting that the terms A+A

and B+B have equal coefficients.Let us now apply the previous findings to a simple toy model, consisting

in a (1+1) self-interacting real scalar field described by the Hamiltonian:

H =∫

dx(1/2)[π2 + (∇ϕ)2] + V (ϕ)

,

V (ϕ) = (1/2) µ2ϕ2 + (1/4)λϕ4,

π = (1/c) ∂ϕ/∂ t. (56)

By resorting to the usual expansion of field operators in terms of creationand annihilation operators straightforward computations show that theHamiltonian (56), besides standard contributions of the form (52), containsfurther terms, coming from λ ϕ4 self-interaction, defined by products of cre-ation and annihilation operators having the form a+a+a+a+, or a a a a, ora+a+a a. If we apply to these operators the transformations (53)–(55) de-scribed above we will obtain that the contribution of self-interaction termwill contain a large number of different kinds of products of operators A+,A, B+, B. When expressing the latter in terms of momenta and coordinatesof damped oscillators and their mirrors through formulae like (46) we willeasily obtain that the leading order contributions will be given by termsof the form (y2p2

y − x2p2x), or (y3py − x3px), or (y p3

y − x p3x). In general

these terms produce a mixing of the dynamics of every damped oscillator

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with the one of its mirror. However, when we stop our developments to thezero order in px, it becomes possible to obtain a separation of the equationdescribing the dynamics of a damped oscillator from the one describing thedynamics of its mirror. Within this approximation (small momenta) it iseasy to derive that the field equation of the self-interacting scalar field inpresence of dissipation has (by using suitable measure units) the form:

∂ν∂νϕ + µ2ϕ + (γ + 3 λϕ2)(∂ϕ/∂t) + λϕ3 = 0. (57)

This equation can be used as a starting point for the study of dynamicalevolution of a self-interacting scalar field in presence of dissipation (a prob-lem dealt with by a number of authors by using equations similar to (57);see, for instance, Laguna and Zurek 1997; Dziarmaga et al. 1999).

As regards the role of doubling mechanism in QFT, a number of remarksare in order. First of all, it introduces, instead of a microscopic descriptionof the environment, a shortened version of the latter under the form of asuitable “mirror” quantum field. Of course, this field can be always con-nected to a microscopic description. In any case, however, it is always en-tangled with the “normal” quantum field describing the original dissipativesystem. This entanglement characterizes the dissipation as a “quantum dis-sipation”, very different from a classical one. Namely, within the context ofspecific models, such as the dissipative quantum brain model (Vitiello 1995;Pessa and Vitiello 2004), it can be shown that the quantum dissipation is acoherence-keeping, rather than a decoherence-producing, mechanism. As amatter of fact, the existence within these models of an infinite multiplicityof different ground states, each one of which is a coherent state resultingfrom the entanglement of the “normal” field with its “mirror”, is just aconsequence of quantum dissipation. Without entering into details for lackof space, it is to be recalled that this approach allows to prove that, withinthese models, all possible dynamical trajectories in the space of represen-tations of canonical commutation relations have a classical nature and arechaotic (Vitiello 2004; Pessa and Vitiello 2004; Vitiello 2005).

If the ground states quoted above are interpreted as memory states ofthe brain, this result implies that dissipative quantum brain model describesa chaotic memory. In the last years this kind of memory became popular fora number of different reasons. First of all, the investigations of Freeman andcoworkers on rabbit olfactory bulb evidenced the prominent role played bychaotic behaviour within the operation of biological systems, and, in par-ticular, of biological memories (Freeman 1987; 1992; 2000a; 2000b; 2005;Freeman and Vitiello 2006; Freeman et al. 2001; Kozma and Freeman 2002;


Skarda and Freeman 1987). In the second place, a number of researchersevidenced, through both computer simulations and experimental investiga-tions, the possibility of occurrence of chaotic behaviors in single neurons(see, e.g. Aihara et al. 1984; Nakagawa and Okabe 1992). In the third place,chaotic artificial neural networks gained much popularity, being consideredas the best models of human memory operation, without being affected bythe heavy shortcomings of traditional attractor neural networks. Namely,while in the latter the retrieval dynamics stops in a particular attractor,which most often is a spurious one, owing to the impossibility of controllingthe dynamics itself (as these models are nothing but special cases of spinglasses), in a chaotic neural network the retrieval dynamics never stops, asit consists in a never-ending wandering around the different chaotic attrac-tors, characterized by times (generally long enough) in which the systemsis in the neighbourhood of one of them, followed by shifts to the neighboursof other attractors. In other words, in a chaotic neural network the retrievaldynamics consists in a visit of the whole memory landscape, rather thanof a specific attractor. It is easy to understand that, if the network wouldbe endowed with a rule for controlling the retrieval dynamics in such away as to transform its chaotic character into an attractive character, weshould have realized a memory having just the features observed in humanmemory. Namely the latter acts on the basis of control strategies which letus know, for instance, if the retrieved pattern is the right or correct one,and, if this should not be the case, trigger a new retrieval attempt (see fora review Koriat 2000; Metcalfe 2000; Levy and Anderson 2002; Anderson2003).

In this regard, we recall that since the beginning of the Nineties it be-came evident (Ott et al. 1990) that chaotic processes can be controlled inmany different ways, most of which easier than expected. Such a circum-stance gave rise to a number of models of associative memories based onchaotic neural networks, differing one from another mostly on control rulesadopted (see, for instance, Kushibe et al. 1996; Sinha 1996; Bondarenko2002; Wagner and Stucki 2002; Crook et al. 2003; He et al. 2003; Yu etal. 2004). Unfortunately, so far no connection has been established betweenthese models and the model of memory based on dissipative quantum braintheory. In any case, these remarks make evident how the adoption of adissipative version of QFT relying on the doubling mechanism be the mostpromising approach for achieving a generalization of QFT able to accountfor change phenomena in the biological world.

To conclude this section, the previous arguments showed that traditional

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methods of describing decoherence, based on standard quantum mechan-ics, are useless when we deal with biological systems for two main reasons:1) these systems are constituted by components interacting through forcefields, a circumstance which need the resort to a theory of quantum fields(deeply different from quantum mechanics); 2) these systems interact withan external environment, in turn producing quantum dissipation phenom-ena which are outside the range of quantum mechanics. Therefore the argu-ments against the use of a quantum framework when modelling biologicalsystems are reliable only as regards standard quantum mechanics but failwhen we introduce QFT. Of course, it could be held that quantum fieldsare not important, as all interactions with environment in the biologicalcase have to do with macroscopic entities which are fully classical. In thisregard, first of all it is to be remarked that every classical feature disappearson very short distances and that these latter cannot be reached only owingto the existence of short-range repulsive interactions (like Lennard-Jonesones), which have a fully quantum origin. Then it is, possible to argue that,even in absence of this effect, the presence of a field-mediated interactioncould work against decoherence, provided the field be of a particular kind.Let us suppose, for instance, to have a simple quantum system lying in anentangled state (for example the Schrodinger cat state), interacting witha classical field inducing a dissipative dynamics. Then (we follow here theargument of Anglin et al. 1995), owing to the fact the different degrees offreedom of the system react in a different way to action of the field, theinterference which supported the entanglement disappears and the systemstate reduces to the product of the single states of its components. In otherwords, decoherence occurred. However, as the dynamics is dissipative, thesystem is forced to evolve, independently from its initial conditions, towardsan attractor whose dimensionality is lesser than the number of degrees ofthe system. Thus, after a suitable relaxation time, some degrees of free-dom (or even all of them, when the attractor is an equilibrium point) fallin the same state, just as if the system were in an entangled state. Deco-herence disappeared, and recoherence took place! This apparent paradox iseasily solved if we take into account that, as already stressed in this section,the dissipation induced another stronger kind of entanglement, the one be-tween the system and the environment. And just such entanglement wasresponsible for the relaxation towards the equilibrium state. We could addthat it has been shown that classical dissipation can be viewed as the originof quantum spectrum of harmonic oscillator. To save space, we will not dis-cuss here this topic, by referring the reader to the literature (Blasone et al.


2001; Blasone et al. 2005). We will go now to shortly discuss the problemof finite volume effects: what really happens during a phase transition?

5. Finite Volume Effects. The Realistic Dynamics of aPhase Transition

So far, QFT dealt with symmetry breaking phase transitions by resortingto infinite volume limits and focussing on instantaneous changes. While al-lowing to obtain a number of important results (such as the ones relatedto Goldstone bosons), such approach neglects what really occurs in practi-cal cases of biological interest, where spatial domains and transition timesare finite and nonzero. On the other hand, we need to know some detailsabout what occurs inside the critical region, because only in this way wecan assess the possibility of controlling and forecasting the eventual statesfollowing the transition itself. In this regard we remark that in biologicaldomain the time extent of the critical region could even be very large.Think, for instance, of the phase transitions which, in the past, gave riseto the birth of new biological species. The fossil remnants tell us that inthese cases the critical period lasted for times of the order of million years,surely enough to undertake some control action. On the other hand, in thecase of processes related to human behaviour, the phase transitions canbe associated to critical periods of the order of minutes, hours, days, andsometimes years, still very long if compared to the very short times charac-terizing phase transitions commonly dealt with by physicists. In short, theneed for information about the details of processes occurring in the criticalperiod is far greater in the biological case than in most cases of physicalinterest (where the critical region is often unobservable).

Of course, the study of dynamics inside the critical region is a very dif-ficult topic. Usually it is dealt with by resorting to advanced techniques ofQFT and suitable simplifying hypotheses. Obviously many different ques-tions must be answered, the first of which deals with the kind of dynamicscharacterizing the critical region. We recall, in this regard, that the latteris defined as the region in which the amplitude of fluctuations of physicalquantities around their equilibrium points is far greater than the equilib-rium values themselves. From a quantitative point of view it can be iden-tified, provided we use temperature as a control parameter, as the regionassociated to the temperature interval (Tc−∆T , Tc +∆T ), where Tc is thecritical temperature and ∆T = TG−Tc. As usual, TG denotes the Ginzburgtemperature. We already argued in Section 2 that within this region we havea classical dynamics, as quantum fluctuations can be neglected. In order to

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understand what could occur we can then resort to a simple example basedon a (1+1)-dimensional self-interacting scalar field of the kind described by(56). Owing to previous arguments, the field can be considered as classicaland we can study its behaviour by resorting to a test particle of unit masswhich moves under the influence of a potential given by the scalar fielditself. For reasons of computational convenience we will write this potentialunder the specific form:

U(x) = (x− 1)2[(x− 1)2 − α] (58)

where α is a control parameter which we can imagine as dependent ontemperature T , and whence on time t. While this potential is nothing but aparticular case of the more general form (56), it has the further advantagethat the critical value of control parameter is αc = 0. When α < αc thepotential has only one stable minimum (ground state), given by x = 1,while, when α > αc, there are two different stable minima (ground states),given by x = 1 ± √

α, and x = 1 becomes an unstable maximum. Wewill suppose that the absolute values of α be very small and that our testparticle be confined within a small neighbourhood of x = 1. If we introducean auxiliary variable ξ = x−1, the dynamical equation ruling the behaviourof test particle, in presence of dissipation, has the form:

d2ξ/dt2 + γ dξ/dt = −4 ξ3 + 2 α ξ. (59)

Here γ is the damping coefficient. This term has been introduced for threedifferent reasons, the first one being the need for taking into account thedissipation produced by the interaction with the environment, discussed inthe previous Section. The second reason is that, to be realistic, we mustsuppose that our system be confined within a finite and small volume. Inthis regard, to fulfil this requirement we could introduce explicit boundaryconditions or explicit forces acting only when the particle is close to theboundary. However the form itself of potential function guarantees that,whatever be the value of control parameter, a strong damping prevents theparticle for going too much away from the location x = 1. Thus we intro-duced a strong damping term as the easiest way for obtaining a confinementwithout resorting to explicit boundary constraints.

The third reason for introducing the damping term is connected toscaling hypothesis, asserting the divergence of correlation length and ofdynamical relaxation time when approaching the critical point. Thisamounts to introduce scaling laws of the form:

l = l0/δν , τ = τ0/δµ, δ = |Tc − T | /Tc (60)


phase transition. However, as we have seen before, they cannot be studiedby resorting to a QFT-based framework, as, very close to a phase tran-sition, all phenomena are ruled by classical physics. Within this context,however, we remark that a mechanism giving rise to solitons in macro-molecular systems exists even in classical physics. An example is given bythe model proposed by Bhaumik et al. (1982), which introduce, within a1-dimensional macromolecule spanned by a continuous spatial coordinatex, a polarization field P (x, t) and an acoustic field A(x, t), both of clas-sical nature. This (1 + 1)-dimensional model is described by the followingLagrangian density:

L = (1/2)[(∂tP )2 + (∂tA)2 − λ2(∂xP )2 − σ2(∂xA)2 − ω20P 2 − c (∂xA)2P 2

− (1/2) d2P 4] (30)

The coefficients entering in this Lagrangian are nothing but parameters, tobe specified on the basis of experimental data. Now, making use of Lagrangeequations, and introducing new variables defined by:ϕ = ∂xA, P = P + P∗, P = pk(ξ) exp [i(k x− ωk t)],

with ξ = x− v t, ωk = λ2 k/v (31)

some mathematical manipulations show that pk(ξ) fulfils the ordinary dif-ferential equation:

d2pk/dξ2 − b2pk + (2b2/a2) p3k = 0 (32)

where:

b2 = [(ω20 − cϕ0)/(λ2 − v2)]− (λ2 k2/v2) (33a)

2 b2/a2 = [3 d2/(λ2 − v2)(σ2 − v2)][v2 − σ2 + (c2/3 d2)]. (33b)

It can be shown that (32) allows for a solitonic solution:

pk = a sec h (bξ). (34)

It can be easily shown that the Davydov equation (10) can be reduced tothe form (32) if we introduce the hypothesis of dependence of a(x,t) onlyon the variable ξ = x − v t and we suppose that both the imaginary partof a(x,t) as well as the derivative of this latter with respect to ξ can beneglected in a first approximation. It is then possible to make the followingidentification between the parameters introduced by Davydov and the onesappearing in the model of Bhaumik et al.:

b2 = Λ/J , 2 b2/a2 = G/J. (35)


where l is the correlation length and τ the dynamical relaxation time. Fromthese relationships, we derive that, close to critical point, the first order timederivative scales as:

l/τ = (l0/τ0) δ µ− ν (61)

while the second order time derivative scales as:

l/τ2 = (l0/τ20 ) δ 2µ− ν . (62)

If we adopt for the critical exponents the customary values µ = 1, ν =1/2, it is immediate to see that the ratio between first order and secondorder time derivatives scales as 1/δ. Thus close to critical point the roleof first order time derivatives becomes more and more prominent. Thisimplies overdamping and the need to take it into account by introducingthe damping term.

Let us now write, for computational convenience, the control parameterα under the form:

α = α0 − z(ξ, dξ/dt, t). (63)

Here α0 is a constant parameter, which, for reasons of convenience, willbe supposed positive. By introducing a specific law ruling the dynamics ofthe function z(ξ, dξ/dt, t) we do nothing but specifying the law of controlexerted on the system by the external environment during the critical pe-riod. It is thus evident that the kind of dynamical behaviour ensuing from(59) is strongly depending on the form of adopted control law. In any case ashort look at the form of (59) suggests that, even in the simplest cases, suchdynamics is very complex. We remark that this complexity doesn’t dependon stochastic fluctuations induced by the environment, which we have nottaken into consideration in this simple model, but is already present per se.It is not difficult to obtain chaotic deterministic behaviours. For instance,if we introduce a simple law of control of the form:

z = β0 − (1/ξ) ε cos(Ω t) (64)

where β0 is a positive parameter whose value is greater than 2 α0, it is im-mediate to see that (59), (63) and (64) give rise to the following dynamicalequation:

d2ξ/dt2 + γ dξ/dt + 4 ξ3 + ω20 = ε cos(Ω t), ω2

0 = β0 − 2 α0, (65)

describing a particular case of the well known Duffing oscillator, which,for suitable values of ε, has a chaotic attractor reached through a period-doubling route (for an elementary introduction to Duffing oscillator see

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Davis 1962, pp.386-400; this equation is dealt with by a number of text-books, such as Tabor 1989; Ott 1993; Strogatz 1994).

Another interesting example is obtained if we introduce a control lawhaving the form of the following differential equation:

dz/dt = − [(ξ/r)2 + (ξ/σ r) ξ dξ/dt + b z] (66)

where r, σ, b are suitable positive parameters. If we make use of the auxiliarydependent variable given by η = dξ/dt, so as to transform the single secondorder equation (59) into a pair of first order equations, it is possible toshow, through suitable algebraic manipulations, that the transformation ofvariables:

ξ = r ξ′, η = σ (η′ − ξ′), z = z′ (67)

allows to cast the system constituted by (59), (63) and (66) in the form:

dξ′/dt = ρ (η′− ξ′), dη′/dt = −χη′+β ξ′− (2r/σ) ξ′ z′, dz′/dt = ξ′η′− b z′

(68)where the coefficients:

ρ = σ/r, χ = (γ/σ)− (σ/r), β = (2 α0 r/σ)− (σ/r) (69)

are all positive in the case of strong overdamping. It is easy to recognizethat (68) is nothing but a particular version of the celebrated Lorenz system,one of first examples of systems of differential equations having a chaoticattractor (see, for a complete information, Sparrow 1982).

However, despite these examples and other proofs of occurrence of de-terministic chaos in first-order phase transitions (see Bystrai et al. 2004),we cannot assert on solid grounds that they prove the existence of a chaoticphase of dynamics close to critical point. Namely our elementary model ne-glects most aspects of the influence exerted on the system by the stochas-tic fluctuations of environment. The only way in which this influence isdescribed in an implicit way is through damping coefficient. We should,however, add at least an explicit noise term, specifying its nature (whiteor coloured). In this regard we remark that, provided some form of deter-ministic chaos would be occurring in the critical region, the presence ofnoise would influence it, leading, in some cases, to its disappearing (an ef-fect known since long time; see, for instance, Matsumoto and Tsuda 1983;Herzel and Pompe 1987; Shibata et al. 1999; Shiino and Yoshida 2001; seealso Redaelli et al. 2002), and, in other cases, to a situation of stochasticchaos (Freeman et al., 2001).


As a consequence of the above the best way to describe the spatio-temporal (classical) dynamics of a field in the critical region is to makeuse of well known Ginzburg–Landau equation (GLE), eventually extendedto the so-called cubic-quintic form (see, for instance, Crasovan et al. 2001;Akhmediev and Soto–Crespo 2003), and supplemented by suitable noiseterms. Such a popular strategy has been adopted by a number of authors(among the others Laguna and Zurek 1997; Dziarmaga et al. 1999; Rivers2000; Lythe 2001), even if alternative approaches have been used, such asthe one of including the noise contributions within the definitions them-selves of GLE coefficients. In any case it is to be remarked that, in absenceof noise terms, GLE can have chaotic attractors as well as exhibit spatio-temporal chaos (Doering et al. 1987; Sirovich and Rodriguez 1987; Shraimanet al. 1992; Hernandez–Garcıa et al. 1999).

Once found a strategy for describing the critical dynamics, the next stepis to understand what are its outcomes and whether they could or not con-trolled through external actions. In this regard, we begin to recall that thisdynamics has two main effects: to settle the system in a suitable asymptoticstate (which, in general, could be a ground state or a metastable state, staticor dynamic), kept by the action of quantum mechanisms, such as Goldstonebosons, and to give rise, owing to finite volume, to the formation of suit-able defects. The latter process is of paramount importance for two mainreasons, the first one being that defects can be experimentally detected andthis lets us test the predictions of microscopic theories of phase transitions.In the second place defects, while accounting for the macroscopic featuresassociated to the new phase, often behave as autonomous entities whichcan be, in turn, viewed as the microscopic constituents of a further, andhigher, macroscopic level. In short, the study of defect formation, as wellas of their interactions, is the starting point for the building of a general-ization of TPT, able to describe the birth of a hierarchy of different levelsof organization of condensed matter. It is easy to understand that such ageneralization would be just the tool required to describe the organizationof biological systems.

Defect formation has been the subject of a theory proposed by Kibbleand Zurek (see, for instance, Kibble 1976; 1980; Zurek 1993; 1996). Thetheory has been refined by a number of authors who introduced a number ofdifferent microscopic models of processes taking place in the critical region(among them we quote Rivers 2000; Alfinito and Vitiello 2002; Alfinito et al.2002; Lombardo et al. 2002; Rivers et al. 2002; Almeida et al. 2004; Antuneset al. 2005; Blasone et al. 2006). Here, without discussing the Kibble-Zurek

202 Eliano Pessa

theory, we will focus on a numerical study of a simple (1+1)-dimensionaltoy model of the behaviour of a scalar field in the critical region, in order toevidence the possible scenarios. More precisely, we will resort to the model ofa self-interacting scalar field with dissipation, whose dynamics is describedby the equation (57) introduced in the previous Section. In order to derivea dynamical equation in the Ginzburg–Landau form we will simply neglectthe second order time derivatives, on the basis of the arguments describedabove. Moreover, we will rescale the variables and add an environmentalnoise in such a way as to obtain a dynamical equation in the Langevin form:

(1 + 3 ϕ2)(∂ϕ/∂ t) + ∂xxϕ− (α/8)ϕ + (1/2)ϕ3 = θ (70)

In this way (we adopted exactly the same form of potential used by Lagunaand Zurek 1997) the only control parameter is given by α. Elementaryarguments show that the critical value is αc = 0. When α < 0, there isonly one ground state ϕ = 0, while, when α > 0, we have a bifurcationto two stable ground states ϕ = ± (1/2)

√α. In all our simulations we will

suppose that the absolute value of α be very small, in such a way as to liewithin a close neighbourhood of the critical point. The symbol θ denotesthe environmental noise, modelled as a Gaussian white noise, with zeromean and variance given by 2 T , where T is the reservoir temperature. Asregard the time variation of control parameter (conceptually equivalent totime variation of temperature adopted in most models), we chose a simplelaw of the form:

α(t) = α0 + β0 t. (71)

To perform a numerical study of (70)-(71) we replaced the finite 1-dimensional space, in which the system is supposed to be contained, with afinite 1-dimensional lattice of points, endowed with a toroidal topology soas to have periodic boundary conditions. If we substitute the second orderspatial derivative with its discretized version, the equation (70) becomesa system of N ordinary differential equations (supplemented by additivenoise), where N is the number of lattice points. The numerical solution ofthis system was achieved through a traditional fourth-order Runge–Kuttamethod. In the Figure 1 below we show two snapshots of ϕ, taken at timesteps 50 and 980. In this simulation the adopted parameter values were:α0 = − 0.1, β0 = 0.02, T = 0.01, ∆ t = 0.01, N = 500. Moreover, the initialvalues of ϕ in the different lattice points were chosen at random, accordingto a uniform distribution between 0.1 and - 0.1. Of course, having onlyone spatial dimension, our model is nothing but a caricature of a phasetransition. Namely the only possible defects are given by kinks.


As it is possible to see from Figure 1, after an initial stage (when α < 0)in which we have only small fluctuations of ϕ around its ground state value,it is possible to observe (when α > 0) a stage characterized by the formationof kinks having a profile stable enough. This method would even let us testthe predictions of Kibble–Zurek theory but, as here we are interested onlyin proving the possibility that (70) could be a reliable model of defectformation, we will avoid a discussion of this topic. It is possible to ask

Fig. 1. Snapshots of ϕ in a dynamical evolution ruled by (70)-(71) in correspondenceto 50 and 980 time steps. The parameter values are listed within the text.

whether the form of damping coefficient, present within (70) and dependingon ϕ, be in some way responsible of the form of kinks shown in Figure1. To answer this question we repeated the same simulation previously

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described with the same parameters, but with a damping coefficient equalto 1 (exactly like in Laguna and Zurek 1997). The snapshots of ϕ at 50 and980 time steps are shown below in Figure 2. As it is possible to see from

Fig. 2. Snapshots of ϕ when damping is equal to 1 with the same conditions as inFigure 1.

Figure 2 the only difference consists in the fact that kink amplitudes areslightly greater than in Figure 1. This means that a damping coefficientgiven by (1 + 3 ϕ2) entails only a bounding of kink amplitudes, withoutchanging the overall dynamical picture.

We can now ask ourselves whether it is possible to introduce some formof control on critical dynamics. The answer is affirmative. In this regard,the simplest form of control consists in modifying the rate of time change


of control parameter, given by β0. We performed a numerical simulation of(70)-(71) with the same parameter values listed before, but with β0 = 0.2.In correspondence to this parameter value the number of time steps requiredto reach a value of α = 0.1 is only 100, instead of the 1000 required in thecase of simulation depicted in Figure 1. Thus, for a comparison with thelatter, in this case the snapshots are to be taken at time steps 50 and 98.They are shown below in the Figure 3.

Fig. 3. Snapshots of ϕ at time steps 50 and 98 when β0 = 0.2.

As we can see a high change rate of control parameter produces asuppression of kink formation. Of course, we could also interpret the snap-shot at time step 98 as showing the occurrence of a large number of very

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small kinks. Obviously this circumstance has an important influence on thenature of processes occurring when leaving the critical region. We can thussay that acting on the change rate of control parameter, we can exert somesort of control on the output of phase transition.

There is, however, another simple way for controlling the output of aphase transition, by adding a suitable external force field. The latter, forinstance, is responsible for the occurrence of an inner magnetization havingthe same direction of the external field in the paramagnetic-ferromagnetictransition. Besides, the presence of an external field changes the nature ofphase transition itself, which is no longer an equilibrium phase transitionbut becomes a nonequilibrium phase transition. As in the latter case theMermin–Wagner theorem no longer applies, true phase transitions becomepossible even in the one-dimensional case (see, for instance, Katz et al. 1983;Privman 1997).

To test the influence of external force fields in our toy model we added tothe left hand side of (70) a term given by −α2/8. In this case the equilibriumpoints of (70) are given by the roots of the cubic equation:

4 ϕ3 − α ϕ− α2 = 0. (72)

From elementary theory of cubic equations it is possible to obtain that,when α < 0, (72) has only one real root, while, when 0 < α < 4/27,there are three different real roots. Thus α = 0 is still a bifurcation point.However, as this time the two stable equilibrium points occurring afterthe bifurcation are associated to different values of ϕ, the dynamics of thesystem will be attracted by the absolute minimum.

This forecasting was confirmed by another numerical study, in which,while introducing the external force described above, we used the sameparameter values adopted for the case depicted in Figure 1. Even in thiscase the simulation was performed for 1000 time steps. The snapshot of ϕ

at time step 50 is reported below in Figure 4.As it is possible to see, the dynamical evolution up to this time step

doesn’t seem to differ from the one observed in other cases, as if the externalforce were not influential. However, as number of time steps was increasing,the value of ϕ in all lattice points was tending towards a common value,perturbed only by very small fluctuations. After 1000 time steps the value ofϕ, averaged on all lattice points, was 4.271, while the variance of fluctuationsaround this value was only 7.0645 × 10− 5. In other words, the systemsettled, as predicted, on a particular equilibrium point, determined by thechoice of external forcing term, without kink production.


Fig. 4. Snapshot of ϕ at time step 50 in presence of the external force α2/8.

Once taken into consideration the description of critical dynamics anddefect production, as well as the ways for controlling these processes, wewill deal with another important topic, the one of recoherence, that is ofmechanisms allowing, after the occurrence of phase transition, the shift fromthe classical regime to a new regime ruled again by QFT. We remark, in thisregard, that recoherence is not a popular topic, as few authors, so far, dealtwith this subject (among them we can quote Anglin et al. 1995; Angelo et al.2001; Mokarzel et al. 2002; the reading of Farini et al. 1996 is illuminatingon what occurs in presence of deterministic chaos). In any case, the originof recoherence is easy to understand: classical dynamics close to criticalpoint, ruled by some form of Ginzburg–Landau equation, gives rise, aftersuitable time, to the occurrence of a mean-field coupling. The latter has twoeffects: induces a mutual synchronization between field modes and, at thesame time, produces a decreasing of the influence of thermal fluctuations.For quantum fluctuations (always present even in the critical region, butneglected as overwhelmed by thermal fluctuations) it is now an easy gameto take again the control of the situation and restore a quantum regime.While the details of this recoherence story are strongly dependent bothon the nature of the system taken into consideration and on the kind ofenvironment, we stress that a number of studies evidenced the capital roleof mean-field coupling in driving towards a globally ordered attractor state.This occurs for different kinds of systems and even in presence of noise andchaos, which in some cases enhance the efficiency of such a mechanism(among the many contributions to this topic we quote Shibata and Kaneko

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1997; Shibata et al. 1999; Pikovsky et al. 2001; Topaj et al. 2001; Ito andKaneko 2002; Mikhailov and Calenbuhr 2002, Chapp. 6 and 7; Ito andKaneko 2003; Rosenblum and Pikovsky 2003; Willeboordse and Kaneko2005; Morita and Kaneko 2006).

We can show some features of recoherence story by resorting to our(1+1)-dimensional toy model of phase transition, described above and ruledby Ginzburg–Landau–Langevin equation (70). Without embarking on com-plex computations, we will resort to poor man arguments, based on rathercrude approximations and on the hypothesis of dealing with very small val-ues of ϕ and α. To begin, we will try to show that, after crossing the criticalpoint of phase transition, the amplitude of mean field undergoes a very fastgrowth. To this end, we will perform a spatial discretization of (70), so thatthe stochastic differential equation ruling the time evolution of field valuein the i-th lattice point, here denoted by ϕi, will assume the form:

(1 + 3 ϕ2i )(∂ϕi/∂ t) + (ϕi+1 + ϕi−1 − 2 ϕi)− (α/8) ϕi + (1/2) ϕ3

i = θ (73)

We will now suppose that (we already crossed the critical value) in boththe two neighbouring sites, whose indices are i + 1 and i − 1, the fieldassume a value corresponding to one of the two ground states, for instance(1/2)

√α. We will then check whether this situation can contribute to an

increase of mean field amplitude (obviously related to the chosen groundstate) following a Kuramoto-like mechanism (see Kuramoto 1984; Pikovskyet al. 2002). To simplify the following computations we will assume that theterm 3 ϕ2

i can be neglected with respect to unity, and that the value of ϕi

be so small as to let us neglect even the term (1/2)ϕ3i . Then the Langevin

equation for ϕi will assume the simple form (hereafter we will eliminate theindex i):

dϕ/dt = f(ϕ) + θ. (74)

Because we supposed that α be very small, the function f(ϕ) will be writtenin an approximate way as:

f(ϕ) =√

α− 2 ϕ. (75)

As θ is supposed to denote a Gaussian white noise with zero mean andvariance given by σ2, the probability distribution function P (ϕ, t) will fulfil(we remind that we are still far from a quantum regime) the followingFokker–Planck equation:

∂tP = − ∂ϕ[f(ϕ)P ] + (σ2/2) ∂ϕϕP. (76)


Trivial computations show that the maximum of stationary probability dis-tribution associated to (76) is given, as expected, by:

ϕ = (1/2)√

α. (77)

The mean field amplitude at a given time t can be measured through thedistance between the probability distribution occurring at t and the sta-tionary probability distribution corresponding to (77). As we are interestedin the time evolution of this distance, it is convenient to adopt, for thesolution of (76), the ansatz:

P (ϕ, t) = Ps(ϕ) + Q(ϕ, 0) exp[h(ϕ) t]. (78)

Here Ps(ϕ) denotes the stationary probability distribution, while Q(ϕ, 0)is a measure of the distribution function occurring at t = 0. The (negative)function h(ϕ) is related to the rate of relaxation towards the stationaryprobability distribution. Its absolute value is proportional to the rate ofgrowth of mean field amplitude.

To simplify the things, we will suppose that Q(ϕ, 0) be given by aGaussian function of the form (here and in the following many constantswill be set equal to 1 to shorten the computations):

Q(ϕ, 0) = exp(−ϕ2/2 s2). (79)

This distribution function describes a Gaussian peaked, at time t = 0,around the value ϕ = 0, as it is reasonable to suppose in correspondence tothe critical point, when the values of ϕ are still in the neighbourhood of theground state existing before the phase transition. The value of the variances2 is supposed to be very small. However, we will assume to deal only withvalues of ϕ even smaller of s2 so as to neglect terms of the order of ϕ/s2 orof higher powers of this quantity. If we substitute the ansatz (78), togetherwith (79), into (76), the previous hypotheses allow us to obtain that thefunction h(ϕ) fulfils the following differential equation:

h = K − f(ϕ)(dh/dϕ) + (σ2/2)(d2h/dϕ2) + (σ2/2)(dh/dϕ)2 (80)

where

K = 2 + (σ2/2 s2). (81)

Rather than searching for a full solution of the nonlinear equation (80),we will limit ourselves to deal with a simple approximation, obtained bysupposing that the first derivative dh/dϕ (as well as its higher powers) be

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so small as to allow us to neglect it. Then we will obtain from (80) thesimple linear differential equation:

h = K + (σ2/2)(d2h/dϕ2). (82)

If we adopt the initial conditions:

h(0) = − a2, dh/dϕ (0) = 0 (83)

where a is a suitable constant, elementary computations show that thesolution of (82) is given by:

h = K − (a2 + K) cosh[(2/σ2)1/2ϕ]. (84)

Taking into account that, up to second order terms and for small values ofx, hyperbolic cosine is approximated by the formula cosh(x) ≈ 1 + (x2/2),we can write (84), under the previous hypotheses, as:

h = − a2 − (2/σ2)(a2 + 2) ϕ2. (85)

This formula is our main result. It shows that mean field amplitude growthrate scales as ϕ2. As ϕ values undergo very fast fluctuations close to criticalpoint this in turn entails a very fast growth of mean field amplitude itselfwhich, in a very short time, will dominate the system dynamical evolution.

In presence of a large mean field amplitude the dynamics of order pa-rameter Φ (in our case the difference between field expectation values, inpresence of fluctuations, after and before the critical point) will be de-scribed, in an effective field approximation (here we will neglect, to savespace, finite volume effects) by a Ginzburg–Landau equation having theform:

∂tΦ = ∂xxΦ + [(α/8)− (3/2)⟨ϕ2

⟩] Φ. (86)

Here⟨ϕ2

⟩is the average value of ϕ2 (related just to mean field amplitude).

From (86) we see how the latter affects the dynamics. We remark thata similar equation describes, in a first approximation, even the dynamicsof disturbances ξ around the chosen ground state. Now, if we take intoconsideration the ground state value ϕ = (1/2)

√α and we assume, by

making a crude approximation, that⟨ϕ2

⟩= α/4, we will obtain that ξ will

be ruled by:

∂tξ = ∂xxξ − (α/4) ξ. (87)

A linear stability analysis around the equilibrium state ξ = 0, based on anansatz like the traditional one ξ = exp(λ t) sin(k x), leads to the followingrelationship between λ and k:

λ(k) = −k2 − (α/4). (88)


It shows that, when α < 0 (that is before the critical point α = 0), theequilibrium state is unstable with respect to long wavelength perturbations,while, when α > 0, it becomes asymptotically stable. This entails that, incorrespondence to critical point, we have λ(0) = 0. This corresponds tothe occurrence of long range (here infinite range) modes, which are nothingbut the classical counterpart of Goldstone bosons (Pessa 1988). The latter,thus, have a already a precursor in the classical regime. We can thereforeconclude that recoherence is easy, as all features of quantum regime havealready been set up in the classical stage.

Let us now shortly dwell on the last important problem in phase transi-tion dynamics: the one of interactions between defects and on the ensuingbehaviour of defects themselves, each one viewed, this time, as a wholisticentity constituting the basic unit of a new level of description. This topicis of prime importance when dealing with biological matter, as we need toknow whether a phase transition theory based on QFT is able to accountfor the hierarchical organization of levels observed in living beings and es-sential for their operation. As regards this subject, we recall that most re-search activity has been devoted to topological defects, such as vortices andsolitons. On the latter there is an extensive literature, including both clas-sical (we will limit ourselves to quote Rajaraman 1987; Lakshmanan 1988;Kivshar and Pelinovsky 2000; Nekorkin and Velarde 2002; Scott 2003) andquantum solitons (Ni et al. 1992; Friberg et al. 1996; Werner 1996; Wernerand Friberg 1997; Konotop and Takeno 2001; Manton et al. 2004). How-ever in very few cases there has been some theoretical proposal about theexplicit form of the interaction potential between two defects (a notableexception is Zhang et al. 2006; some related ideas were used to describeinteractions between biological cells; see Schwarz and Safran 2002; Bischofset al. 2004; in a very different context, the one of interactions betweendefects in quantum liquids, explicit expressions for interaction potentialshave been given, for instance, in Recati et al. 2005). In other cases wehave mostly numerical simulations (see, for instance, Hoyuelos et al. 2003;Podolny et al. 2005). However, almost all authors neglect the fundamen-tal question: are the new entities (that is defects) classical or quantum? Itseems that the classical nature of defects be taken for granted and that,when quantum corrections are needed (as in the case of quantum solitons),they depend on the influence of some medium in which defects themselvesare embedded. In this regard we remark that the things could be more com-plex than suspected. Namely the existence of quantum effects is cruciallydependent on the ratio between the amplitudes of quantum fluctuations and

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thermal fluctuations. The former can be expressed through ~ω, where ω issome characteristic frequency of the system under study. But, what is ~? Ifwe deal with atoms, molecules, or elementary particles, the answer is wellknown. But in other cases we could deal with some kind of “effective Planckconstant”, whose numerical value could be very different from the usual one.Such a circumstance could, in principle, fully change the nature of behaviourof defects. Science fiction? In this regard we recall that from a number ofyears the number of researchers introducing an “effective Planck constant”is ever increasing (we can quote, for instance, Artuso and Rusconi 2001;Averbukh et al. 2002; perhaps one of clearer explanations of how to com-pute the effective Planck constant is contained in Angelopoulou et al. 1994).To make a further example, Fogedby (Fogedby 1998; Fogedby and Bran-denburg 2002), when studying the noisy Burgers equation:

∂u/∂t = ν ∇2u + λ u∇u +∇η, (89)

where ν is a damping constant, or a viscosity, λ a coupling coefficient, andη a Gaussian white noise satisfying:

〈η(x, t) η(x′, t′)〉 = ∆ δ(x− x′) δ(t− t′), (90)

found that the theory of this equation could be recast exactly in the formof an equivalent QFT, provided the usual Planck constant ~ be identifiedwith the quantity ∆/ν. However, except a small number of researchers,most scientists ignore the relevance of this possibility.

6. Are we Really Describing Phase Transitions inBiological Matter?

The previous sections evidenced the amazing modeling ability of QFT andstatistical mechanics within the context of phase transition theory. How-ever, it seems restricted to specific (and very limited) physical contexts. Onthe contrary, when looking at the real world of phenomenological modelsof biological phenomena, we encounter things such as neural networks, cellsystems, nonequilibrium phase transitions, species evolution, business cy-cles, without nothing resembling to Hamiltonians, QFT models, partitionfunctions, Goldstone bosons, and like. In other words, all efforts made toremedy the shortcomings of TPT, described in the points 1)–5) listed at theend of third Section, seem to have few contacts with models of biologicalbehaviours. The sensation is that we effectively describe all aspects of somekinds of phase transitions, but they are useless when dealing with phasetransitions in biological matter.


In this regard, there are, in principle, three possible answers:r.1) phenomenological models of biological change are already QFT mod-

els; however, at present they are formulated in such a way that their sim-ilarity to physical models of phase transitions is hidden; anyway, a futuresuitable generalization of QFT will allow us to recognize that both phe-nomenological models and QFT models belong to the same general class;

r.2) phenomenological models of biological change are incomplete; how-ever, by introducing suitable quantization methods for these models, it willbe possible to build more complete models, accounting in a deeper way forthe observed behaviours of biological systems;

r.3) phenomenological models don’t have and will never have any kindof relationships with QFT models; namely biological world is irreducibleto physical world; moreover, the understanding of biological world cannotbe based in an important way on modelling activity which, in the lattercontext, has a limited usefulness.

It seems that, within the scientific community, the answer r.3) had neverbeen given. Namely it would contradict the requirement that science be ableto do predictions and design methods to control phenomena. However, itshould be remarked that, in practice, many scientists work as if they wereholding that r.3) is the right answer. This attitude frees them from theneed of knowing alternative (and often more powerful) modelling methodsand gives them the opportunity of leading a quiet life, avoiding the dangerof interference of other kinds of scientists in their (personal and private)domain of study.

The adoption of answer (r.1) is easier in some kinds of models, typicallyformulated by using a mathematical language not too different from theone used in field theories. In this regard, among the many possible classesof biological emergence models, we will take into consideration a first ex-ample given by reaction-diffusion systems. The latter have been used tomodel, for instance, chemical reactions, species evolution, swarm intelli-gence, morphogenesis. In general models of this kind are constituted by aset of basic components (which, to conform to a common usage, we willconventionally denote as particles) undergoing two types of processes: arandom diffusion (which, for instance, can be modeled as a random walkon a suitable continuous or discrete space) and reactions between particles(the spontaneous decay of a particle, giving rise to new particles, beingconsidered as a special case of a reaction). At any given time instant thesemodels allow a suitable definition of system’s microstate, characterized bythe values of microscopic variables associated to the single particles. As the

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system evolution is stochastic, to each microstate α we can associate itsprobability of occurrence at time t , denoted by p(α, t). The latter mustfulfill an evolution equation of the form:

dp(α, t)/dt =∑

βRβαp(β, t)−

∑γ

Rαγp(α, t). (91)

Here the symbol Rβα denotes the transition rate from the state β into α.The equation above is usually referred to as master equation. It is easy tounderstand that, in general, the particle number is not conserved, owing tothe fact that reactions can, in principle, destroy or create particles. Thiscircumstance lets introduce the so-called formalism of second quantization,widely used in Quantum Field Theory. This can be done (the pioneeringwork on this subject was done by Doi 1976; Peliti 1985) very easily, forinstance, in the case of particles moving on a discrete spatial lattice char-acterized by suitable nodes, labeled by a coordinate i (we will use a singlelabel to save on symbols). In this context we can introduce the numbers ofparticles lying in each node (the so-called occupation numbers) and two op-erators, that is a creation operator a+

i and a destruction operator ai, actingon system microstate and, respectively, raising the number of particles lyingin the i-th node of one unity and lowering the same number of one unity.It is possible to show that these operators fulfill the same commutationrelationships holding in Quantum Field Theory. They can be used to definea state with given occupation numbers, starting from the ground state, inwhich no particle exists, and applying to it, in a suitable way, creation anddestruction operators. If we denote by |α, t〉 a state characterized, at timet , by given occupation numbers (summarized through the single label α)system’s state vector can then be defined as:

|ψ(t)〉 =∑

αp(α, t) |α, t〉. (92)

By substituting this definition into the master equation (91) it is possibleto see that system’s state vector fulfills a Schrodinger equation of the form:

d |ψ(t)〉 /dt = −H |ψ(t)〉 (93)

where H denotes a suitable Hamiltonian, whose explicit form depends onthe specific laws adopted for transition rates. In the case of a lattice of nodes,a suitable continuum limit then gives rise to a field Hamiltonian which canbe dealt with exactly with the same methods used in standard QuantumField Theory (on this topic see Cardy, 1996). The above procedure has beenapplied by a number of researchers to investigate the statistical featuresof models of interacting biological agents (see, among the others, Cardy


and Tauber 1998; Pastor–Satorras and Sole 2001). We, however, underlinethat this procedure has only a formal character and, as it is possible tounderstand from (92), doesn’t lead to a genuine quantum theory. On theother hand, it is not excluded that suitable generalizations of Doi–Pelitimethod can lead to a description in terms of truly quantum processes (see,for instance, Smith 2006).

Other holders of answer (r.1) adopted a very different framework. Forinstance people interested in socio-economic models tried to formulate themin terms of some network dynamics (see, for a review, Albert and Barabasi2002). Then to each network node, characterized by a specific fitness, itwas associated an energy, proportional to the logarithm of fitness, whileeach connection between two nodes was made to correspond to two non-interacting particles, whose energies were given by the energies associatedto the two nodes. Each addition of a new node was thus viewed as equiv-alent to a particle creation. Suitable hypotheses on fitness distribution aswell as on node connection increase rate can give rise to well known dis-tributions of energy levels associated to the nodes, such as, for instance,Bose–Einstein distribution. Then it is possible to map essentially quantumphenomena, such as Bose–Einstein condensation, on phenomena such asnetwork dynamics (cfr. Bianconi and Barabasi 2001).

As regards the holders of answer (r.2) we are forced to limit our con-siderations to the field of quantum neural networks, so far the only onein which this approach was object of an intensive investigation. In thiscontext, we must distinguish between two approaches: the most popularone, in which quantum neurons are introduced from the start as physicalrealizations of quantum systems, such as multiple slits or quantum dots(for reviews see Behrman et al. 2000; Narayanan and Menneer 2000) , andthe less studied one, in which quantum neurons are quantum dynamicalsystems whose laws are obtained by applying a quantization procedure todynamical laws of classical neurons (see Pessa 2004). In any case it is to beremarked that, by choosing (r.1) or (r.2), in both cases we need some formof generalization of QFT, which makes the two approaches complementary,rather than anthitetic.

In order to illustrate the difficulties arising when trying to connect phe-nomenological models with QFT we will use a particular version of a verysimple model of population evolution, introduced by Michod (see, amongthe others, Michod 2006; Michod et al. 2006), and applied to describe theevolutionary biology of volvocine algae. Within this model the i-th indi-vidual of a population is characterized by the values of two variables: its

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generative ability bi and its survival ability vi. In general, as shown by bio-logical data, the two variables b and v are not reciprocally independent, butare connected through a general law expressed by a function v(b) which isdecreasing in b. In this regard, we introduced a specific form of v(b) givenby:

v(b) = [γ (1 + γ)/(b + γ)]− γ (94)

where γ is a suitable parameter. For reasons of convenience we constrainedthe values of v and b within the interval [0, 1] and (94) tells us that v(0) = 1,v(1) = 0. However, as the total number of individuals is not constant (owingto the existence of a reproductive ability), but a function of time N = N(t),we supposed (Michod did not make explicitly this hypothesis), that thevalue of parameter γ appearing in (94) (that is the convexity of the curve)were dependent on the momentarily value of N through a law of the form:

γ = (β0/N) + α0 (95)

where β0 and α0 are other parameters.What should we expect from the behaviour of this simple model? Mi-

chod introduced two different kinds of fitness measure: the average individ-ual fitness of population members, denoted by w, and the total (normalized)population fitness, denoted by W . Their definitions are:

w = (1/N)∑

ibi vi, W = (1/N2) B V, B =

∑i

bi, V =∑

ivi.

(96)From both biological observations and results of computer simulations Mi-chod recognized that, while obviously both fitness measures vary with time,however for most time they have different values. Thus he was lead to intro-duce a quantity, called covariance and here denoted by Cov, which measuressuch a difference through the simple relationship:

Cov = w −W. (97)

When the value of Cov is negative, the total population fitness is greaterthan the average individual fitness. In other words, we are in a situationin which it seems that there is some sort of cooperation (or coordination)among individuals producing an increase of total fitness. Clearly this isnot quantum coherence, but recalls some aspects of the latter. How is thisincreased total fitness reached? The simulations show that it is sometimesdue to a sort of increase in specialization of population members. In thisregard, we remark that every pair of values (v, b) characterizes a singleindividual and that the form of the distribution of these pairs (or better,


only of b values, as (94) lets us find the value of v, once known the valueof b) gives a measure of the degree of specialization present within thepopulation at a given time instant. Namely a flattened distribution meansa strong difference between the individuals, and therefore a high degree ofspecialization, while a strongly peaked distributions means small differencesbetween the individuals and low degree of specialization.

In order to check whether these effects are present even under the hy-potheses we introduced before in (94)–(95), we performed suitable numer-ical simulations of the evolution of populations. They were based on aprevious subdivision of the interval [0, 1] of values of continuous variable b

in a suitable number of equal sub-intervals, in correspondence to each oneof which the value of b was identified with the middle point of the sub-interval. The mechanism of reproduction was random, based on a uniformdistribution, and such that each reproductive value was interpreted as theprobability, at each generation, of producing a number of offspring whichwere a fraction of a maximum possible number fixed in advance by theexperimenter. The produced descendants were assigned at random to thedifferent generative ability sub-intervals. Besides, even the value of v foreach individual was interpreted as the probability of its survival in the nextgeneration.

In the Figure 5 we can see a plot of both kinds of fitness vs generationnumber in a “life history” characterized by 100 generations, an initial totalnumber of individuals given by 50, a number of 100 different sub-intervalsof generative ability, and a maximum allowable number of descendants foreach population member and for each generation given by 5. Moreover themaximum allowable number of individuals for each generative ability sub-interval was fixed to 100, and the initial value of γ was 5. The values ofremaining parameters were α0 = 3, β0 = 100.

As it is immediate to see, the covariance is always negative and thegroup fitness prevails over the average individual fitness. What has been inthis case the effect of population evolution on the distribution of values of b

among the individuals. We remark that at the beginning of this simulationwe chose to put all individuals within the same generative ability class,corresponding to the 50-th sub-interval. This distribution is depicted inthe Figure 6 a). For a comparison we show in the Figure 6 b) the finaldistribution obtained after 100 generations.

As it is possible to see, not only the final distribution deeply differsfrom the initial one, but the former evidence a very high degree of spe-cialization of single individuals. It is easy to understand that this simple

218 Eliano Pessa

Fig. 5. Group fitness (upper curve) and average individual fitness (lower curve) vs time.

Fig. 6. (a) Initial distribution of generative abilities, (b) Final distribution of generativeabilities.

model accounts not only for the evolution of populations of volvocine algae,but, more in general, of the fact that most biological organisms survive ina complex environment just owing to the fact that their components (cellsor organs) are highly specialized and reciprocally cooperating.

Now let us deal with the main question: can this model be dealt withthrough the methods of QFT? If yes, through which algorithms? If not, forwhich reasons? To begin, we could argue that the model was not cast underthe form of a system of differential equations and, therefore, it could notbe dealt with through Hamiltonian-based methods. This argument is, how-ever, very weak as, by using suitable tricks and approximations, it wouldbe possible, in one or in another form, to achieve a mathematical formatcloser to the ones popular in theoretical physics, eventually resorting to the


introduction of suitable noise (this has already been done within the re-search on protocells and on cellular systems; see, among the others, Furu-sawa and Kaneko 2002; Rasmussen et al. 2003; 2004; Takagi and Kaneko2005; Fellermann and Sole 2006; Lawson et al. 2006; Macia and Sole 2006).In any case, this would be a problem only of technical (mathematical) na-ture, and not a serious conceptual obstacle.

A stronger argument takes into consideration the nature of the environ-ment which, as implicitly described in the previous model, has a typicallyreactive character. On the contrary, QFT models are placed within sim-pler environments, such as thermal baths, and the concept itself of fitnessis absent. Besides, in biological world often the action of the environmentoccurs on large scales and influences the behaviour of organisms in a ratherindirect way. This implies, in turn, that control of emergence cannot beimplemented in the simple ways described in the previous section. We cantherefore claim that QFT will never give rise to a reliable description ofphase transitions in biological matter if we will not generalize it so as to in-clude more realistic descriptions of biological environments. Of course, thisgeneralization should also take into account the fact that, owing to previousreasons, phase transitions in biological matter are often of nonequilibriumtype. The doubling mechanism quoted in Section 4 constitutes a first steptowards this direction, but probably further steps are needed.

A third argument deals with the nature of system components. WhileQFT models can be interpreted as describing assemblies of particles, allhaving the same nature, the members of population previously introducedare different individuals. In other words, once translated in the languageof particle creation and annihilation, the model operating according therules (94)–(95) describes creation and annihilation, not only of particles,but even of kinds of particles. In terms of a field language it is equivalent toa theory describing the birth and the disappearing of fields. Within QFTthis would require the introduction of third quantization. This is still asomewhat exotic topic, dealt with almost exclusively within the domainof quantum theories of gravitation, but so far with only very few contactswith the world of QFT models of condensed matter (for a first attemptsee Maslov and Shvedov 1999). However the latter domain could be thebetter context for applying this kind of extension of QFT, owing to thepresence of “effective force fields” which appear and disappear as a functionof environmental constraints. On the contrary, it would be useless within

220 Eliano Pessa

the world of elementary particle physics at high energies, where from thestarting people are searching for evidence of universal force fields whosenature lasts unchanged.

As a consequence of the previous arguments we can claim that, notwith-standing the theoretical efforts quoted in the previous sections, actuallyQFT doesn’t describe phase transitions in biological matter, except in alimited number of cases, related to low-level phenomena. However, pro-vided the generalizations mentioned before were included within a largertheoretical framework, such a description could probably become feasibleand realistic.

7. Conclusions

The long path followed in this paper touched some key problems of actualtheory of phase transitions, having in mind the usefulness of its applicationto describe changes or transitions occurring in biological matter. Owing tothe complexity of this topic we neglected a number of important points,including the roles of noise and disorder within this context. However, suit-able generalizations of QFT and Statistical Mechanics, already availablefrom a number of years, allow to include these aspects within the previousframework without essential conceptual changes, if not of purely technicalnature. At the end of our trip we are full of admiration for the ingenuityof a number of theoreticians who introduced so powerful tools for dealingwith various aspects of phase transitions. Notwithstanding these efforts,it seems that the main problems still remain unsolved. Namely it appearsthat, except a limited number of cases, we still lack a realistic theory ofphase transitions in biological matter. Not only, we are still unable to de-cide whether observed biological changes can or not be described, at leastin principle, within the context of actual phase transition theory, eventuallygeneralized in a suitable way. However, despite this situation, we have rea-sons for being confident in the future developments of the theory. Namelythe progress so far made, part of which shortly described even in this pa-per, gives us solid grounds for believing that in a near future the goal ofachieving a general theory of phase transitions, including both physical andbiological matter, will be reached. In this regard, we expect that this newtheory, though including TPT as a limiting case, will include even other fea-tures, stemming from a sort of back-reaction of knowledge, deriving frombiology, psychology, sociology, economics, on the physics itself. We feel thatthis backreaction will open new horizons even to investigation of physicalworld.


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229

Microcosm to Macrocosm via the Notion of a Sheaf(Observers in Terms of t-topos)

Goro Kato

Mathematics Department, California Polytechnic State UniversitySan Luis Obispo, CA 93407, U.S.A.

[email protected]

The fundamental approach toward matter, space and time is that particles(either objects of macrocosm or microcosm), space and time are all presheafi-fied. Namely, the concept of a presheaf is most fundamental for matter, spaceand time. An observation of a particle is represented by a morphism fromthe observed particle (its associated presheaf) to the observer (its associatedpresheaf) over a specified object (called a generalized time period) of a t-site(i.e. a category with a Grothendieck topology). This formulation provides ascale independent and background space-time free theory (since, for the t-topostheoretic formulation, space and time are discretely defined by the associatedparticle, whose particle-dependency is a consequence of quantum entangle-ment.). It is our basic scheme that the method of t-topos may provide a devicefor understanding and concrete formulation of macro-object and micro-objectinterconnection as morphisms in the sense of t-topos.

Keywords: t-topos; Sheaf, Space-time Free Theory; Quantum GravityPACS(2006): 02.40.k; 04.20.Gz; 04.60.m

1. Introduction

The concept of a sheaf appeared in 1940’s by Leray in algebraic topologyand by K. Oka in the theory of analytic functions in several complex vari-ables. In 1960’s, theory of sheaves (cohomology of sheaves) made tremen-dous development in algebraic geometry, especially by Grothendieck andin algebraic analysis by M. Sato and M. Kashiwara. The notion of a sheafhas been used to obtain global information from local data. The definitionof a sheaf is a contravariant functor from a category with a Grothendiecktopology, called a site, to a category satisfying the sheaf axiom as indicatedin [1], [2], [3]. The notion of a topos, i.e. a category of (pre-) sheaves overa category with a Grothendieck topology has been used to study quantumgravity by Isham as in [4], Mallios–Rapitis as in [5], Guts–Grinkevisch as in

230 Goro Kato

[6], Kato as in [7] and [8] and others. One of the challenges for such a theoryis to explain the physical reality by building a micro-macro theory, whichis scale independent, (e.g. assigning real numbers or complex numbers formeasuring quantities), and background spacetime free. In quantum gravity,such a theory needs to be a background spacetimeless theory. In our theoryof t-topos, we have obtained certain levels of success as shown in [7], [8],[9], [10], [11]. For a general survey on the foundations of physics based onhidden variable theories, see [12], where the reports on various theories andideas with direct experimental applications are thoroughly given.

2. The Very Small to not so Small and the Large

We often divide the physical world into the microcosm where quantumphysics is applied and the macrocosm where classical (or/and relativistic)mechanics is applied. One of the fundamental approaches to obtain a unifiedtheory is to build the global (macro) object information from studying local(micro) objects. The inadequacy of the traditional continuum-based models(e.g. classical differential geometry based on either real numbers or complexnumbers) as final physical theories has been pointed out by various authors.Our approach is to apply the concept of a (pre)sheaf as a candidate for sucha unifying theory. Hence, the fundamental reality is described by a discreteconcept. Via the notions of categories and sheaves, we need to describe theconcepts of wave-particle duality, measurement and quantum entanglementin microcosm and the concepts of a light-cone and the gravitational effecton spacetime in macrocosm. (See the above references.) Here is our initialstep toward such a unification theory of the very small, not so small andthe large. Some of the results have been obtained toward this goal in [7],[8], [9], [10].

By definition, a category consists of objects and morphisms satisfyingcertain axioms as in the above references. Let us recall that a presheafis a contravariant functor from a site S (a category with a Grothendiecktopology. See [1], [2], or [3].) to a category. The basic notion is the triple ofpresheaves (m, κ, τ) associated with a particle, space and time, respectively.Note that presheaves (κ ,τ) are depending upon the particle m. Namely,the space and time depend upon locally m. Recall that this dependency isa consequence of quantum entanglement. (See [7] and [8] for the assertion.)Let m be a presheaf over a site S, i.e. m ∈Ob(S), i.e. m is an object ofthe category Sof presheaves over S. (For definitions of presheaves, sites andtoposes and the notations, see [1], [2], or [3].) Note also that in [7] and [8],the term a “t-site” is used when it is referred to in the theory of t-topos.

Microcosm to Macrocosm via the Notion of a Sheaf 231

An object of t-site S is said to be a generalized time (period). There aretwo types of ur-wave states in the theory of t-topos : first, when the particlepresheaf m is not observed, and secondly, when there are more than onechoice in objects in the t-site S (as in the case of well-known double-slitexperiment), then m is said to be the ur-wave state. Next, we will considerthe case when m is in the ur-particle state. (For the double-slit applicationof t-topos and the duality, see the earlier mentioned papers [9] and [12].)That is, when m is observed by P ∈Ob(S), by definition, there exists ageneralized time period V in t-site S so that there exists a morphism s(V)from m (V) to P(V), (see [7]). However, note that a particle can be in theur-particle state as long as an object in the t-site is specified, regardlesswhether the particle is observed or not. The notions of microlocalizationsof m and V in t-topos theory are decompositions of m and V as describedin [8]. That is, write m as a direct product of finitely many sub-presheavesmi, namely, m =

∏mi , and for V it is to consider a covering Vj of V,

that is, V← Vj, (see [8]). When m is observed by P over V, one obtainsno information of mi. However, when a microcosm-object mi is observedby P over a generalized time period Vj , there exists a morphism s(ij) frommi(Vj) to P(Vj). Note that m may not be defined over Vj . Namely, obtainingan information locally during the generalized time period does not implythat the macro-object m can be observed by P over Vj . In general, onecannot compose the morphisms s(ij) with the canonical projection π(Vj):m(Vj) → mi(Vj) to obtain the morphism from m(Vj) to P(Vj). This isone of the roots of the local-global phenomena: why studying an individualparticle mi does not provide the global information of m =

∏mi.

When m is observed by P over generalized time periods V first andlater U in S. Then there exists a morphism ρ from V to U in S. (See[7]) Then there exist morphisms from m(V) to P(V) and m(U) to P(U),

respectively For each factorization Vρ′−→ W

ρ′′−→ U of the morphism Vρ−→

U in the t-site S, there corresponds a ur-particle state m(W) of m. For allthe possible factorizations of the morphism of ρ, the presheaf m induces

funtorially the corresponding factorizations m(V )m(ρ′)←− m(W )

m(ρ′′)←− m(U)

of m(V )m(ρ)←− m(U) . Those possible “paths” via m(W) between m(V) and

m(U) represent the Feynman paths in the classical sense (See Section 3.1 in[9]). Notice also that the directions of arrows are reversed when evaluatedat the presheaf m. This is because the presheaf m is a contravariant functorby definition. One can generalize the above by formulating its relativisticversion by the notion given in [8].

232 Goro Kato

3. Conclusion

The language of category theory, sheaf theory, i.e. a topos theory can givequalitative analysis in physics, especially in quantum gravity as in [4], [5],[8]. In order to obtain quantitative results, the notion of scaling needs tobe introduced. Traditionally, the fields of real numbers and complex num-bers have been used for scaling. Some have attempted to use the ring ofp-adic integers instead of real or complex numbers. Hence, the categoricalapproach alone is not sufficient for applications to physics. However, oneshould have at least one theory formulating both microcosm and macro-cosm, i.e., a theory on quantum gravity, with one mathematical model. Ourchoice is a category of presheaves from a (t-) site to a product category,which is scale independent and background spacetime free.

Acknowledgment

The author would like to thank Ignazio Licata for his invitation and en-couragement with the preparation of this paper.

References

1. Kashiwara, M., and Schapira, J., (2006), Categories and Sheaves, Springer.2. Gelfand, S., and Manin, Yu. I., (1996), Methods of Homological Algebra,

Springer.3. Kato, G., (2006), The Heart of Cohomology, Springer.4. Isham, C., and Butterfield, J., (1999) Spacetime and the philosophical chal-

lenge of quantum gravity, arXiv:gr–qc/9903072 v1.5. Mallios, A., and Raptis, I., Finitary–Algebraic resolution of the inner

Schwarzschild singularity, arXov:gr-qc/0408045 v3 20 Aug.2004.6. Guts, A.K., and Grinkevich, E. B., (1996), Toposes in general theory of

relativity, arXiv:gr–qc/9610073.7. Kato, G., Elemental principles of t-topos, (2004), Europhys. Lett., 68 (4)

pp. 467-472.8. Kato, G., (2005), Elemental t.g. principles of relativistic t-topos, Europhys.

Lett., 71(2), pp. 172–178.9. Kato, G., and Tanaka, T., (2006), Double-slit interference and temporal

topos, Foundations of Physics, Vol. 36, No.11, pp. 1681–1700.10. Kafatos, M., Kato, G., Roy, S., and Tanaka, T., Sheaf theory and geometric

approach to EPR nonlocality, to be submitted to Foundations of Physics.11. Kato, G., Black horizon and t-topos, in preparation.12. Genovese, M., (2005), Research on hidden variable theories: A review of

recent progresses, Physics Reports, 413, pp. 319–396.

233

The Dissipative Quantum Model of Brain and LaboratoryObservations

Walter J. Freemana and Giuseppe Vitiellob

a Department of Molecular and Cell BiologyUniversity of California, Berkeley CA 94720-3206, USA

b Dipartimento di Matematica e Informatica, Universita di Salerno,and Istituto Nazionale di Fisica Nucleare,

Gruppo Collegato di Salerno, I-84100 Salerno, [email protected] - http://sulcus.berkeley.edu

[email protected] - http://www.sa.infn.it/giuseppe.vitiello

We discuss the predictions of the dissipative quantum model of brain in con-nection with the formation of coherent domains of synchronized neuronal oscil-latory activity and macroscopic functions of brain revealed by functional brainimaging.

Keywords: Quantum Models of Brain; Biophysics; Complex SystemsPACS(2006): 03.65.w; 03.67.a; 87.16.Ac; 87.15.v; 89.75.k; 89.75.Fb

1. Introduction

The mesoscopic neural activity of neocortex appears consisting of the dy-namical formation of spatially extended domains in which widespread co-operation supports brief epochs of patterned oscillations. These “packetsof waves” have properties of location, size, duration and carrier frequenciesin the beta-gamma range (12 − 80 Hz). They re-synchronize, through asequence of repeated collective phase transitions in cortical dynamics, inframes at frame rates in the theta-alpha range (3 − 12 Hz). The forma-tion of such patterns of amplitude modulated (AM) synchronized oscilla-tions in neocortex has been demonstrated by imaging of scalp potentials(electroencephalograms, EEGs) and of cortical surface potentials (electro-corticograms, ECoGs) of animal and human from high-density electrodearrays 1−5. The AM patterns appear often to extend over spatial domainscovering much of the hemisphere in rabbits and cats6,7 , and over the lengthof a 64× 1 linear 19 cm array1 in human cortex with near zero phase dis-persion8,9 . Synchronized oscillation of large-scale neuronal assemblies in

234 Walter J. Freeman and Giuseppe Vitiello

beta and gamma ranges have been detected also by magnetoencephalo-grafic (MEG) imaging in the resting state and in motor task-related statesof the human brain10 .

In this paper we compare the predictions of the dissipative quantummodel of brain11,12 with these neurophysiological data5,13 . It turns outthat the model explains two main features of the EEG data: the texturedpatterns of AM in distinct frequency bands correlated with categories ofconditioned stimuli (CS), i.e. coexistence of physically distinct AM pat-terns, and the remarkably rapid onset of AM patterns into (irreversible)sequences that resemble cinematographic frames. Each spatial AM patternis indeed described to be consequent to spontaneous breakdown of symme-try (SBS)14–16 triggered by external stimulus and is associated with one ofthe unitarily inequivalent ground states. Their sequencing is associated tothe non-unitary time evolution in the dissipative model, as discussed below.

Cellular and molecular models of neuronal function have been formu-lated in the so-called K-set theory: a hierarchical set of models based instudies of single neurons embedded in populations of increasing size andcomplexity17,18 . The purpose of the present paper is to show that the dis-sipative quantum model may describe the entire forebrain dynamics as itselects, adapts, and elaborates generic intentional actions5,13 .

In our presentation we closely follow Refs. 5,13. The question concerningthe mass action in the observed cortical activity as originally pointed outby Lashley is considered in Section II. The dissipative many-body model ofbrain is summarized in Section III and it is there discussed in relation to theevidence of a multiplicity of ground states. The formation and recurrenceof AM oscillatory patterns are further discussed in Section IV. Free energyand classicality of trajectories in brain space are discussed in Section V.Final remarks and conclusions are presented in Section VI.

2. Lashley Dilemma

In the first half of the 20th century Lashley was led to the hypothesis of“mass action” in the storage and retrieval of memories in the brain and ob-served: “...Here is the dilemma. Nerve impulses are transmitted ...form cellto cell through definite intercellular connections. Yet, all behavior seemsto be determined by masses of excitation...within general fields of activity,without regard to particular nerve cells... What sort of nervous organizationmight be capable of responding to a pattern of excitation without limitedspecialized path of conduction? The problem is almost universal in the ac-tivity of the nervous system” (pp. 302–306 of Ref. 19). Many laboratory

The Dissipative Quantum Model of Brain and Laboratory Observations 235

observations in the following years confirmed Lashley’s finding. Pribramextended the concept of mass action by introducing the analogy betweenthe fields of distributed neural activity in the brain and the wave patternsin holograms20 . Observational access to real time imaging of “patternsof excitation” and dynamical formation of spatially extended domains ofneuronal fields of activity has been provided by magnetoencephalogram(MEG), functional magnetic resonance imaging (fMRI), positron electrontomography (PET), blood-oxygen level deletion (BOLD) and single photonemission computed tomography (SPECT). As already mentioned in the In-troduction, observations7 show that cortex jumps abruptly from a receivingstate to an active transmitting state. Spatial AM patterns with carrier fre-quencies in the beta and gamma ranges form during the active state anddissolve as the cortex return to its receiving state after transmission. Thesestate transitions in cortex form frames of AM patterns in few ms, holdthem for 80− 120 ms, and repeat them at rates in alpha and theta rangesof EEG1−8. In such an activity neocortex appears to behave according tofour interrelated properties21,22 : the exchangeability of its ports of sen-sory input; its ability to adapt rapidly and flexibly to short- and long-termchanges; its reliance on large-scale organization of pattern of neural activ-ity that mediate its perceptual functions; the incredibly small amounts ofinformation entering each port in brief behavioral time frames that supporteffective and efficient intentional action and perception.

Four material agencies have been then proposed to account for the pro-cesses involving large populations of neurons. None of them, however, ap-pears to be able to explain the observed cortical activity23 :

– Nonsynaptic neurotransmission21 has been proposed as the mechanismfor implementation of volume transmission to answer the question on howthat broad and diffuse chemical gradients might induce phase locking ofneural pulse trains at ms intervals. Although nonsynaptic transmission isessential for neuromodulation, diffusion of chemical fields of metabolitesproviding manifestations of widespread coordinated firing is, however, tooslow to explain the highly textured patterns and their rapid changes.23 Theobserved high rates of field modulation are not compatible with mediation ofchemical diffusion such as those estimated in studies of spike timing amongmultiple pulse trains (see Refs. 24–26, of cerebral blood flow using fMRIor BOLD27,28 , and of spatial patterns of the distributions of radio-labeledneurotransmitters and neuromodulators as measured with PET, SPECTand optical techniques.

– Weak extracellular electric currents have been postulated as the


agency by which masses of neurons link together29 . They have been shownto modulate the firing of neurons in vitro. However, the current densitiesrequired in vivo to modulate cortical firing exceed by nearly two ordersof magnitude those currents that are sustained by extracellular dendriticcurrents across the resistance of brain tissue17,30 . The passive potentialdifferences are measured as the EEG.

– The intracellular current in palisades of dendritic shafts in corticalcolumns generates magnetic fields of such intensity that they can be mea-sured 4−5 cm above the scalp with MEG. The earth’s far stronger magneticfield can be detected by specialized receptors for navigation in birds andbees,31 leading to the search for magnetic receptors among cortical neurons(Refs. 32,33), so far without positive results.

– Radio waves propagating from the combined agency of electric andmagnetic fields has also been postulated.34 However, neuronal radio com-munication is unlikely, owing to the 80 : 1 disparity between electric per-mittivity and magnetic permeability of the brain tissue and to the lowfrequency (< 100 Hz) and kilometer wavelengths of e.m. radiation at EEGfrequencies.

The conclusion is that Lashley dilemma remains still to be explained:neither the chemical diffusion, which is much too slow, nor the electricfield of the extracellular dendritic current nor the magnetic fields inside thedendritic shafts, which are much too weak, are the agency of the collec-tive neuronal activity. An alternative approach is therefore necessary. Thedissipative quantum model of brain has been then proposed5,11–13 to ac-count for the observed dynamical formation of spatially extended domainsof neuronal synchronized oscillations and of their rapid sequencing.

3. The Dissipative Many-Body Model and Evidence of AMultiplicity of Ground States

A. The many-body modelThe dissipative quantum model of brain is the extension to the dissipa-tive dynamics of the many-body model proposed in 1967 by Ricciardi andUmezawa35,36 . They suggested that the extended patterns of excitationsobserved in the neurophysiological research might be described by meansof the SBS formalism in many-body physics. The fact that the brain is anopen system in permanent interaction with the environment was not con-sidered in the many-body model. It has been therefore extended in a way toinclude the intrinsically open dynamics of the brain. This led to the dissipa-tive many-body model11,12 . Umezawa explains the motivation for using the


quantum field theory (QFT) formalism of many-body physics37 : ‘In anymaterial in condensed matter physics any particular information is carriedby certain ordered pattern maintained by certain long range correlationmediated by massless quanta. It looked to me that this is the only way tomemorize some information; memory is a printed pattern of order supportedby long range correlations...” In QFT long range correlations among thesystem constituents are indeed dynamically generated through the mecha-nism of SBS. These correlations manifest themselves as quanta, called theNambu–Goldstone (NG) boson particles or modes, which have zero massand therefore able to span the whole system. At a mesoscopic/macroscopicscale, due to such correlations in this way established, the system appears inan ordered state. One expresses this fact by saying that the NG bosons arecoherently condensed in the system lowest energy state (the system groundstate) according to the Bose–Einstein condensation mechanism. The densityof the condensed NG bosons provides a measure of the degree of orderingor coherence. One arrives thus to the definition of the order parameter, aclassical field specifying the ordered patterns observed in the system groundstate. This is the way the dynamical formation of extended ordered pat-terns is described in many-body physics. Laboratory observations allow todetect the NG quanta or particles by means of their scattering with ob-servational probes; for example, one can use neutrons as probe to observetheir scattering with phonons in a crystal. The phonons are the NG particleresponsible for the crystalline ordering. They are the quanta associated tothe elastic waves. Other examples of NG particles are the magnons in theferromagnets, namely the quanta of the spin waves.

In the case of the brain, the quantum variables are identified11,38,39 withthe electrical dipole vibrational field of the water molecules and of otherbiomolecules present in the brain structures. The associated NG quanta arenamed the dipole wave quanta (DWQ). These do not derive from Coulombinteraction. They are dynamically generated through the breakdown of therotational symmetry of the electrical dipole vibrational field. Water is morethan the 80% of brain mass and in the many-body model it is thereforeexpected to be a major facilitator or constraint on brain dynamics. Thetheoretically and experimentally well established knowledge of condensedmatter physics thus suggested to Umezawa that memory storage might bedescribed in terms of the coherent Bose condensation process in the systemlowest energy state. It has to be remarked that the neuron and the glia cellsand other physiological units are not quantum objects in the many-bodymodel of brain.


In the many-body model the external input or stimulus acts on thebrain as a trigger for the spontaneous breakdown of the symmetry with theconsequent long range correlation established by the coherent condensationof NG bosons. In the dissipative model it becomes clear that the stimulusaction may trigger the symmetry breakdown only provided the cortex isat or near a singularity (see below), predicting indeed the laboratory ob-servations.40 The density value of the condensation of DWQ in the groundstate acts as a label classifying the memory there created. The memory thusstored is not a representation of the stimulus. This reflects a general featureof the spontaneous breakdown of symmetry in QFT: the specific orderedpattern generated through SBS by an external input does not depend onthe stimulus features. It depends on the system internal dynamics. Themodel thus accounts for the laboratory observation of lack of invariance ofthe AM neuronal oscillation patterns with invariant stimuli .

The recall of the recorded information occurs under the input of a stim-ulus able to excite DWQ out of the corresponding ground state. In themany-body model such a stimulus is called “similar” to the one responsiblefor the memory recording36 . In the model similarity between stimuli thusrefers not to their intrinsic features, but to the reaction of the brain tothem; in other words, to the possibility that under their action DWQ arecondensed into or excited from the ground state carrying the same label.

The many-body model, however, fails in explaining the observed coex-istence of AM patterns and also their irreversible time evolution. Indeed,one shortcoming of the model is that any subsequent stimulus would can-cel the previously recorded memory by renewing the SBS process with theconsequent DWQ condensation, thus overprinting the ’new’ memory overthe previous one (’memory capacity problem’). In order to solve these prob-lems, considering that the brain is an open system, the original many-bodymodel has been extended11 to the dissipative dynamics.B. The dissipative many-body modelThe mathematical structure of QFT provides the possibility of having dif-ferent vacua with different symmetry properties. Indeed, infinitely manyrepresentations of the canonical commutation relations (CCR’s) exist inQFT. They are, with respect to each other, unitarily inequivalent (i.e thereis no unitary operator transforming one representation into another oneof them)41 . Thus they are physically inequivalent: they describe differ-ent physical phases of the system (it is not so in Quantum Mechanicswhere all the representations are unitarily (and therefore physically) equiv-alent)15,42,43 . The existence of infinitely many representations of the CCR’s


is fully exploited in the dissipative quantum model of brain, which is there-fore not a quantum mechanical model but a QFT model43 .

In the dissipative model the environmental influence on the brain istaken into account by performing a suitable choice of the brain vacuumstate (the brain ground state) among the infinitely many of them, eachother unitarily inequivalent. The choice of the effective vacuum is done inthe process of SBS triggered by the external stimulus. As explained above,the specificity of such a “choice” is fully controlled by the internal dynamicsof the brain system; “making” the choice signals that the brain-environmentinteraction is “active”. A change in the brain–environment reciprocal influ-ence then corresponds to a change in the choice of the brain vacuum: thebrain evolution through the vacuum states thus reflects the evolution of thecoupling of the brain with the surrounding world. The vacuum condensateof DWQ, consequent to SBS induced by the external stimulus, is assumed tobe the quantum substrate of the AM pattern observed at a phenomenolog-ical level. In agreement with observations, the dissipative dynamics allows(quasi-)non-interfering degenerate vacua with different condensates, i.e. dif-ferent values for the order parameter (AM pattern textures), and (phase)transitions among them (AM patterns sequencing). These features couldnot be described in the frame of the original many-body model. By ex-ploiting the existence of the infinitely many representations of the CCR’sin QFT, the dissipative model allows a huge memory capacity.

In the QFT formalism for dissipative systems44 the environment is de-scribed as the time-reversal image of the system. This is realized by ‘dou-bling’ the system degrees of freedom. In the dissipative model, the braindynamics is indeed described in terms of an infinite collection of dampedharmonic oscillators aκ (a simple prototype of a dissipative system) rep-resenting the boson DWQ modes11 and by the aκ modes which are the”time-reversed mirror image” (the “mirror or doubled modes”) of the aκ

modes. aκ represent the environment modes. The label κ generically denotesdegrees of freedom such as, e.g. spatial momentum, etc.11,44 .

The breakdown of the dipole rotational symmetry is induced by theexternal stimulus and this leads to the dynamical generation of DWQ aκ.Their condensation in the ground state is then constrained by inclusion ofthe mirror modes aκ in order to account for the system dissipation. Thesystem ground state is indeed not invariant under time translation and thisreveals the irreversible time evolution of the system, namely dissipation.

Although the brain holds itself far from equilibrium, the balance ofthe energy fluxes at the system-environment interface, including heat


exchanges, holds in a sequence of states through which the brain evolves.This is manifested in the regulation of mammalian brain temperature. Atsome arbitrary initial time t0 = 0, we denote the zero energy state (the vac-uum) of the system by |0〉N . This prescribes11 that E0 =

∑κ ~Ωκ(Naκ

−Naκ) = 0. Here, Ωκ is the common frequency of aκ and aκ, andNaκ andNaκ

are the (non-negative integer) numbers of aκ and aκ which are condensedin |0〉N .

This implies that the ”memory state” |0〉N is a condensate of equalnumber of modes aκ and mirror modes aκ for any κ: Naκ − Naκ = 0. Ndenotes the set of integers defining the ”initial value” of the condensate,N ≡ Naκ

= Naκ, ∀ κ, at t0 = 0, namely the label, or order parameter

associated to the information recorded at time t0 = 0. Clearly, balancingNaκ

−Naκto be zero for any κ, does not fix the value of eitherNaκ

orNaκfor

any κ. It only fixes, for any κ, their difference. Therefore, at t0 we may haveinfinitely many perceptual states, each one in one-to-one correspondence toa given N set.

The dynamics ensures that the number (Naκ − Naκ) is a constant ofmotion for any κ (see Ref. 11). The system of aκ mirror modes representsthe system’s Double.

It can be shown that aκ and aκ modes satisfy the Bose–Einstein distri-bution. |0〉N is thus recognized to be a finite temperature state and it canbe shown to be a squeezed coherent state11,15,45,46 .

The important point is that the spaces |0〉N and |0〉N ′ are eachother unitarily inequivalent for different labels N 6= N ′ in the infinite vol-ume limit:

N 〈0|0〉N ′ −→V→∞

0 ∀ N , N ′ , N 6= N ′ . (1)

The whole space of states thus includes infinitely many unitarily inequiv-alent representations |0〉N , for all N ’s, of the CCR’s. A huge numberof sequentially recorded memories may coexist without destructive interfer-ence since infinitely many vacua |0〉N ,∀ N , are independently accessible.In contrast with the non-dissipative model, recording the memory N ′ doesnot necessarily produce destruction of previously printed memory N 6= N ′,which is the meaning of the non-overlapping in the infinite volume limitexpressed by Eq. (1). Dissipation allows the possibility of a huge memorycapacity by introducing the N -labeled “replicas” of the ground state. Themodel thus predicts the existence of textures of AM patterns, as indeedobserved in the laboratory.

The stability of the order parameter against quantum fluctuations is


a manifestation of the coherence of the DWQ boson condensation. Thememory N is thus not affected by quantum fluctuations and in this senseit is a macroscopic observable. The “change of scale” (from microscopic tomacroscopic scale) is dynamically achieved through the boson condensationmechanism. The state |0〉N provides an example of “macroscopic quantumstate”.

In “the space of the representations of the CCR’s”, each representation|0〉N denotes a physical phase of the system and may be thought as a“point” identified by a specific N -set. In the infinite volume limit, pointscorresponding to different N sets are distinct points (do not overlap, cf.Eq. (1)). We also refer to the space of the representations of the CCR’sas to the “memory space” or the brain state space. The brain in relationwith the environment may occupy any one of the multiplicity of groundstates, depending on how the E0 = 0 balance or equilibrium is approached.Or else, it may sit in any state that is a collection or superposition ofthese brain-environment equilibrium ground states. The system may shift,under the influence of one or more stimuli acting as a control parameter,from ground state to ground state in this collection (from phase to phase)namely it may undergo an extremely rich sequence of phase transitions,leading to the actualization of a sequence of dissipative structures formedby AM patterns, as indeed experimentally observed.

Denote by |0(t)〉N the state |0〉N at time t specified by the initial valueN , at t0 = 0. Time evolution of the state |0(t)〉N is represented as thetrajectory of ”initial condition” specified by the N -set in the space of therepresentations |0(t)〉N . The state |0(t)〉N provides the ’instantaneouspicture’ of the system at each instant of time t, or the ’photogram’ at t ina cinematographic sequence.

When the DWQ frequency is time–dependent, κ-components of the N -set with higher momentum have been found to possess longer life–time.Since the momentum is proportional to the reciprocal of the distance overwhich the mode can propagate, this means that modes with shorter rangeof propagation (more “localized” modes) survive longer. On the contrary,modes with longer range of propagation decay sooner. Correspondingly,condensation domains of different finite sizes with different degree of sta-bility are allowed in the model51 .

Finally, let us observe that coherent domains of finite size are obtainedby non-homogeneous boson condensation described by the condensationfunction f(x) which acts as a “form factor” specific for the considered do-main15,47,48 . The important point is that such a condensation function


f(x) has to carry some topological singularity in order for the condensationprocess to be physically detectable. A regular function f(x) would producea condensation which could be easily “washed” away by a convenient fieldtransformation (“gauged” away by a gauge transformation). In a similarway, the phase transition from one space to another (inequivalent) one canbe only induced by use of a singular condensation function f(x). One canshow that this is the reason why topologically non trivial extended ob-jects, such as vortices, appear in the processes of phase transitions.15,47,48

Stated in different words, this means that phase transitions driven by bo-son condensation are always associated with some singularities in the fieldphase (indeterminacy of the phase at the phase transition point). The ob-servational counterpart in the brain dynamics is the decrease in power thatin the perceptual phase transition precedes the formation of a new AMpattern which reflects the reduction in the amplitude of the spontaneousbackground activity40 . The event that initiates a phase transition is thusan abrupt decrease in the analytic power of the background activity tonear zero, depicted as a null spike. This reduction induces a brief state ofindeterminacy in which the amplitude of ECoG is near to zero and phaseof ECoG is undefined, as indeed should be as said above according to thedissipative model.

If a stimulus arrives at or just before this state, then the cortex canbe driven by the input across the phase transition process to a new AMpattern. The response amplitude depends not on the input amplitude, asthe dissipative model also predicts, but on the intrinsic state of the cortex,specifically the degree of reduction in the power and order of the back-ground brown noise. The null spike in the band pass filtered brown noiseactivity is conceived as a shutter that blanks the intrinsic background. Atvery low analytic amplitude when the analytic phase is undefined, the sys-tem, under the incoming weak sensory input, may re-set the backgroundactivity in a new AM frame, if any, formed by reorganizing the existingactivity, not by the driving of the cortical activity by input (except for thesmall energy provided by the stimulus that is required to select an attractorand force the phase transition). The decrease (shutter) repeats in the thetaor alpha range, independently of the repetitive sampling of the environmentby limbic input. In conclusion, the reduction in activity constitutes a singu-larity in the dynamics at which the phase is undefined (in agreement withthe dissipative model requiring the singularity of the boson condensationfunction). The aperiodic shutter allows opportunities for phase transitions.The power is not provided by the input, exactly as the dissipative model


predicts, but by the pyramidal cells, which explains the lack of invarianceof AM patterns with invariant stimuli40 .

4. Emergence and Recurrence of Amplitude Patterns

High-density electrode arrays (typically 8× 8 in a 2D square) fixed on thescalp or the epidural surface of cortical areas and fast Fourier transform(FFT) have been used in order to measure AM pattern textures for whichhigh spatial resolution is required1,49 .

The set of n amplitudes squared from an array of n electrodes (typically64) defines a feature vector, A2(t), of the spatial pattern of power at eachtime step. The vector specifies a point on a dynamic trajectory in brainstate space, conceived as the collection of all possible brain states, essen-tially infinite. Measurement of n EEG signals defined a finite n-dimensionalsubspace, so the point specified by A2(t) is unique to a spatial AM pat-tern of the aperiodic carrier wave. Similar AM patterns form a cluster inn-space, and multiple patterns form either multiple clusters or trajectorieswith large Euclidean distances between digitizing steps through n-space. Acluster with a verified behavioral correlate denotes an ordered AM pattern.The vector A2(t) is taken to be the best available numeric estimator of ourorder parameter, because when the trajectory of a sequence of points entersinto a cluster, that location in state space signifies increased order from theperspective of an intentional state of the brain, owing to the correlationwith a conditioned stimulus. We use the reciprocal of the absolute value ofthe step size between successive values of De(t) = |A2(t)−A2(t− 1)| as ascalar index of our order parameter.

Pattern amplitude stability was proved by small steps in Euclidean dis-tances, De(t), between consecutive points (higher spikes in Fig. 1). Patternphase stability was proved by calculating the ratio, Re(t), of the temporalstandard deviation of the mean filtered EEG to the mean temporal stan-dard deviation of the n EEGs3,4 (lower curve in Fig. 1). By these measuresAM/phase-modulated patterns stabilized just after the phase transitionsand before reaching the maximum in the spatial AM pattern amplitude.Re(t) = 1 when the oscillations were entirely synchronized. When n EEGswere totally desynchronized, Re(t) approached one over the square root ofthe number of digitizing steps in the moving window. Experimentally Re(t)rose rapidly within a few ms after a phase discontinuity and several ms be-fore the onset of a marked increase in mean analytic amplitude, A(t). Thesuccession of the high and low values of Re(t) revealed episodic emergenceand dissolution of synchrony; therefore Re(t) was adopted as an index of


cortical efficiency,50 on the premise that cortical transmission of spatialpatterns was most energy-efficient when the dendritic currents were mostsynchronized.

Fig. 1. Spikes: De stability measure. Curve: Re boson density.3

Re-synchronized oscillations in the beta range near zero lag commonlyrecurred at rates in the theta range and covered substantial portions of theleft cerebral hemisphere under observation2 (exceeding the length of therecording array (19 cm) in human brain).

We conclude that a specific value of the phenomenological order param-eter A2(t) may be assumed to correspond to a specific value of N of theorder parameter predicted by the dissipative model: the observation of theAM pattern textures and of their sequencing finds thus a description inthe dissipative model. The time evolution in the brain space described bythe space of the representations of the CCR’s gives the image of quantumorigin of the trajectories described by the time dependent vector A2(t) inthe brain state space as phenomenologically described above.

A further agreement of the dissipative model with observed features isrecognized by considering the common frequency Ωκ(t) for the aκ and aκ


modes (cf. Eq. (8) in Ref. 51). The duration, size and power of AM pat-terns are predicted to be a decreasing functions of the carrier wave numberκ, as indeed confirmed in the observations. Carrier waves in the gammarange (30− 80 Hz): durations seldom exceeding 100 ms, diameters seldomexceeding 15 mm; low power in the 1/fa relation. Carrier frequencies inthe beta range (12− 30 Hz): durations often exceeding 100 ms; estimateddiameters large enough to include multiple primary sensory areas and thelimbic system; greater power by 1/fa.

5. Free Energy and Classicality

In the frame of the dissipative model one can show that, providedchanges in the inverse temperature β are slow, the changes in the energyEa ≡

∑k EkNak

and in the entropy Sa are related by

dEa =∑

k

EkNakdt =

1β

dSa, (2)

i.e. the minimization of the free energy Fa holds at any t:

dFa = dEa − 1β

dSa = 0. (3)

As usual heat is defined as dQ = 1β dSa. Moreover, the time-evolution of

the state |0(t)〉N at finite volume V is controlled by the entropy vari-ations, which reflects the irreversibility of time evolution (breakdown oftime-reversal symmetry) characteristic of dissipative systems, namely thechoice of a privileged direction in time evolution (arrow of time)11,44 .

Eq. (2) shows that the change in time of the condensate, i.e. of the orderparameter, turns into heat dissipation dQ. Therefore the ratio between therate of free energy dissipation to the rate of change in the order parameteris a good measure of the ordering stability. This is in agreement with obser-vations. Indeed, in terms of the laboratory observables the rate of change ofthe order parameter is specified by the Euclidean distance De(t) betweensuccessive points in the n-space. Typically De(t) takes large steps betweenclusters, decreases to a low value when the trajectory enters a cluster, andremains low for tens of ms within a frame. Therefore De(t) serves as ameasure of the spatial AM pattern stability. Empirically6,7 it was foundthat the best predictor of the onset times of ordered AM patterns was, assuggested by the dissipative model, the ratio He(t) of the rate of free energydissipation to the rate of change in the order parameter because De(t) falls


and A2(t) rises with wave packet evolution:

He(t) =A2(t)De(t)

.

This index is named the pragmatic information index after Atmanspacherand Scheingraber52 .

Our measurements showed that typically the rate of change in the in-stantaneous frequency ω(t) was low in frames that coincided with low De(t)indicating stabilization of frequency as well as AM pattern. Between framesω(t) increased often several fold or decreased even below zero in interframebreaks that repeated at rates in the theta or alpha range of the EEG2 (phaseslip53) .

In the dissipative model as well as in the laboratory observations thetime evolution of the brain state is represented as a trajectory in the brainstate space. The possibility of deriving from the microscopic dynamics theclassicality of such trajectories is one of the merits of the dissipative many-body field model11,12,51,54 . These trajectories are found to be deterministicchaotic trajectories54,55 , and thus the observed changes in the order param-eter become susceptible to be described in terms of trajectories on attractorlandscapes. One can show indeed that in the brain space the trajectoriesare classical and that

i) they are bounded and each trajectory does not intersect itself (tra-jectories are not periodic).

ii) there are no intersections between trajectories specified by differentinitial conditions.

iii) trajectories of different initial conditions are diverging trajectories.The property ii) implies that no confusion or interference arises among

different memories, even as time evolves. In realistic situations of finitevolume, states with different N labels may have non–zero overlap (non-vanishing inner products). This means that some association of memoriesbecomes possible. In such a case, at a “crossing” point between two, ormore than two, trajectories, one can “switch” from one of these trajectoriesto another one which there crosses. This reminds us of the “mental switch”occurring, in the perception of ambiguous figures, or in general, while per-forming some perceptual and motor tasks56,57 as well as while resorting tofree associations in memory tasks58 .

From the property iii) one can derive54 that the difference between κ–components of the sets N and N ′ may become zero at a given time tκ.However, the difference between the sets N and N ′ does not necessarily


become zero. The N -sets are made up by a large number (infinite in thecontinuum limit) of Naκ

(θ, t) components, and they are different even if afinite number (of zero measure) of their components are equal (here θ is aconvenient parameter, see Ref. 54,55). On the contrary, for very small δθκ,suppose that ∆t ≡ τmax− τmin, with τmin and τmax the minimum and themaximum, respectively, of tκ, for all κ’s, be “very small”. Then the N -setsare “recognized” to be “almost” equal in such a ∆t. Thus we see how inthe “recognition” (or recall) process it is possible that “slightly different”Naκ

–patterns are “identified” (recognized to be the “same pattern” even ifcorresponding to slightly different inputs). Roughly, ∆t provides a measureof the “recognition time”.

The relevant point, at the present stage of our research, is that thedissipative model predicts that the system trajectories through its physicalphases may be chaotic54 and itinerant through a chain of ’attractor ruins’,59

embedded in a set of attractor landscapes60 accessed serially or merely ap-proached in the coordinated dynamics of a metastable state61–64 . Themanifold on which the attractor landscapes sit covers as a ”classical blan-ket” the quantum dynamics going on in each of the representations of theCCR’s (the AM patterns recurring at rates in the theta range (3− 8 Hz)).The picture which emerges is that a CS selects a basin of attraction in theprimary sensory cortex to which it converges, often with very little infor-mation as in weak scents, faint clicks, and weak flashes. The convergenceconstitutes the process of abstraction. The astonishingly low requirementsfor information in high-level perception have been amply demonstrated byrecent accomplishments in sensory substitution22,65,66 . There is an indefi-nite number of such basins in each sensory cortical area forming a pliableand adaptive attractor landscape. Each attractor can be selected by a stim-ulus that is an instance of the category (generalization) that the attractorimplements by its AM pattern. In this view the waking state consists of acollection of potential states, any one of which but only one at a time canbe realized through a phase transition.

6. Final Remarks and Conclusions

In this paper we have compared the predictions of the dissipative quan-tum model of brain with the dynamical formation of spatially extendeddomains in which widespread cooperation supports brief epochs of pat-terned oscillations. The model seems to explain two main features of theneurophysiological data: the coexistence of physically distinct AM patternscorrelated with categories of conditioned stimuli and the remarkably rapid


onset of AM patterns into irreversible sequences that resemble cinemato-graphic frames. A relevant role is played by the main ingredients of themodel, namely spontaneous breakdown of symmetry and dissipation: eachspatial AM pattern is indeed described to be consequent to spontaneousbreakdown of symmetry triggered by external stimulus and is associatedwith one of the QFT unitarily inequivalent ground states. Their sequenc-ing is associated to the non-unitary time evolution implied by dissipation.Many other features predicted by the model have been found in agreementwith laboratory observations.

We finally observe that another possible way to break the symmetry inQFT is to modify the dynamical equations by adding one or more terms thatare explicitly not consistent with the symmetry transformations (are notsymmetric terms). In QFT this is called explicit breakdown of symmetry.The system is forced by the external action into a specific non-symmetricstate that is determined by the imposed breaking term. Again this fits wellwith our observation in the response of cortex to perturbation by an im-pulse, such as an electric shock, sensory click, flash, or touch: the evokedor event-related potential (ERP). The explicit breakdown in cortical dy-namics is observed by resort to stimulus-locked averaging across multiplepresentations in order to remove or attenuate the background activity, soas to demonstrate that the location, intensity and detailed configurationof the ERP is predominantly determined by the stimulus, so the ERP canbe used as evidence for processing by the cortex of exogenous information.Contrastingly, in SBS the AM pattern configurations are determined frominformation that is endogenous from the memory store.

The variety of these highly textured AM patterns, their exceedinglylarge diameters in comparison to the small sizes of the component neurons,the long ranges of correlation despite the conduction delays among them,and the extraordinarily rapid temporal sequence in the neocortical phasetransitions by which they are selected, provide the principal justificationfor exploring the interpretation of nonlinear brain dynamics in terms ofdissipative many-body theory and multiple ground states to complementbasin-attractor theory.

In conclusion, much work remains to be done in many research direc-tions, such as the analysis of the interaction between the boson conden-sate and the details of the electrochemical neural activity, or the problemsof extending the dissipative many-body model to account for higher cog-nitive functions of the brain. At the present status of our research, thestudy of the dissipative many-body dynamics underlying the richness of the


laboratory observations seems to be promising. John von Neumann notedthat “...the mathematical or logical language truly used by the central ner-vous system is characterized by less logical and arithmetical depth thanwhat we are normally used to. ...We require exquisite numerical precisionover many logical steps to achieve what brains accomplish in very few shortsteps” (pp. 80–81 of Ref. 67). The dissipative quantum model describingthe textured AM patterns and the sequential phase transitions observed inbrain functioning perhaps opens a window on such a view.

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59. I. Tsuda (2001) Behav. Brain Sci. 24, 793.60. C. A. Skarda and W. J. Freeman (1987) Brain Behav. Sci. 10, 161.61. S. L. Bressler (2002) Current Directions in Psychological Sci. 11, 58.62. A.A. Fingelkurts , A.A. Fingelkurts (2004) Int J Neurosci 114, 843.63. A.A. Fingelkurts , A.A. Fingelkurts (2001) Brain and Mind 2, 261.64. S.L. Bressler and J.A.S. Kelso (2001) Trends Cog. Sci. 5, 26.65. L. G. Cohen, Celnik P, Pascal-Leone A, Corwell B, Faiz L, Dambrosia J,

Honda M, Sadato N, Gerloff C, Catala MD, Hallett M (1997) Nature 389,180.

66. L. Von Melchner, Pallas S.L., Sur M. (2000) Nature 404, 871.67. J. von Neumann (1958) The Computer and the Brain. New Haven: Yale

University Press.

253

Supersymmetric Methods in the Traveling Variable:Inside Neurons and at the Brain Scale

H.C. Rosu a,c, O. Cornejo-Perez b, and J.E. Perez-Terrazas a,d

a Division of Advanced Materials, IPICyT,Apartado Postal 3-74, Tangamanga, 78231 San Luis Potosı, S.L.P., Mexico

b Centro de Investigacion en Matematicas (CIMAT),Apartado Postal 402, 36000 Guanajuato, Gto., Mexico

[email protected]@ipicyt.edu.mx

[email protected]

We apply the mathematical technique of factorization of differential operatorsto two different problems. First we review our results related to the super-symmetry of the Montroll kinks moving onto the microtubule walls as well asmentioning the sine-Gordon model for the microtubule nonlinear excitations.Second, we find analytic expressions for a class of one-parameter solutions ofa sort of diffusion equation of Bessel type that is obtained by supersymme-try from the homogeneous form of a simple damped wave equation derived inthe works of P.A. Robinson and collaborators for the corticothalamic system.We also present a possible interpretation of the diffusion equation in the braincontext.

Keywords: Biophysics; Complex Systems; Quantum Information; Soliton exci-tation; Nonlinear Wave Equation; Brain DynamicsPACS(2006): 87.16.Ac; 87.15.Aa; 87.15.v; 89.75.Fb; 03.67.a; 05.45.Yv; 03.65.Ge

1. Nonlinear Biological Excitations

The possibility of soliton excitations in biological structures has been firstpointed out by Englander et al.1 in 1980 who speculated that the so-called‘open states’ units made of approximately ten adjacent open pairs in longpolynucleotide double helices could be thermally induced solitons of thedouble helix due to a coherence of the twist deformation energy. Since thena substantial amount of literature has been accumulating on the biologicalsignificance of DNA nonlinear excitations (for a recent paper, see Ref. 2).On the other hand, the idea of nonlinear excitations has emerged in 1993in the context of the microtubules (MTs),3 the dimeric tubular polymers

254 H.C. Rosu, O. Cornejo-Perez, and J.E. Perez-Terrazas

that contribute the main part of the eukaryotic cytoskeleton. In the caseof neurons, MTs are critical for the growth and maintenance of axons. Itis known that axonal MTs are spatially organized but are not under theinfluence of a MT-organizing center as in other cells. We also remind that in1995 Das and Schwarz have used a two-dimensional smectic liquid crystalmodel to show the possibility of electrical solitary wave propagation in cellmembranes.4 Nevertheless, there is no clear experimental evidence at themoment of any of these biological solitons and kinks.

2. Supersymmetric MT Kinks

Based on well-established results of Collins, Blumen, Currie and Ross5

regarding the dynamics of domain walls in ferrodistortive materials,Tuszynski and collaborators3,6 considered MTs to be ferrodistortive andstudied kinks of the Montroll type7 as excitations responsible for the en-ergy transfer within this highly interesting biological context.

The Euler-Lagrange dimensionless equation of motion of ferrodistortivedomain walls as derived in Ref. 5 from a Ginzburg–Landau free energy withdriven field and dissipation included is of the travelling reaction-diffusiontype

ψ′′

+ ρψ′ − ψ3 + ψ + σ = 0 , (1)

where the primes are derivatives with respect to a travelling coordinateξ = x− vt, ρ is a friction coefficient and σ is related to the driven field.5

There may be ferrodistortive domain walls that can be identified withthe Montroll kink solution of Eq. (1)

M(ξ) = α1 +√

2β1 + exp(βξ)

, (2)

where β = (α2 − α1)/√

2 and the parameters α1 and α2 are two nonequalsolutions of the cubic equation

(ψ − α1)(ψ − α2)(ψ − α3) = ψ3 − ψ − σ . (3)

Rosu has noted that Montroll’s kink can be written as a typical tanhkink8

M(ξ) = γ − tanh(βξ

2

), (4)

where γ ≡ α1 + α2 = 1 + α1√

2β . The latter relationship allows one to use a

simple construction method of exactly soluble double-well potentials in the

Supersymmetric Methods in the Traveling Variable 255

Schrodinger equation proposed by Caticha.9 The scheme is a non-standardapplication of Witten’s supersymmetric quantum mechanics10 having as theessential assumption the idea of considering the M kink as the switchingfunction between the two lowest eigenstates of the Schrodinger equationwith a double-well potential. Thus

φ1 = Mφ0 , (5)

where φ0,1 are solutions of φ′′0,1 + [ε0,1 − u(ξ)]φ0,1(ξ) = 0, and u(ξ) is the

double-well potential to be found.

u( )

ξξ

ξ

0 ξ

ξL

M

RM

|0 |1

Fig. 1. Single electron within the traveling double-well potential u(ξ) as a qubit. Theelectron can switch from one well to the other by tunneling and the relation between thewavefunctions in the two wells is given by Eq. (5).

Substituting Eq. (5) into the Schrodinger equation for the subscript 1and substracting the same equation multiplied by the switching functionfor the subscript 0, one obtains

φ′0 +RMφ0 = 0 , (6)

where RM is given by

RM =M

′′+ εM

2M ′ , (7)

and ε = ε1− ε0 is the lowest energy splitting in the double-well Schrodingerequation. In addition, notice that Eq. (6) is the basic equation introducingthe superpotential R in Witten’s supersymmetric quantum mechanics, i.e.


the Riccati solution. For Montroll’s kink the corresponding Riccati solutionreads

RM (ξ) = −β2

tanh(β

2ξ

)+

ε

2β

[sinh(βξ) + 2γ cosh2

(β

2ξ

) ](8)

and the ground-state Schrodinger function is found by means of Eq. (6)

φ0,M (ξ) = φ0(0) cosh(β

2ξ

)exp

(ε

2β2

)exp

(− ε

2β2

[cosh(β ξ)

−γβξ − γ sinh(βξ)])

, (9)

while φ1 is obtained by switching the ground-state wave function by meansof M . This ground-state wave function is of supersymmetric type

φ0,M (ξ) = φ0,M (0) exp

[−

∫ ξ

0

RM (y)dy

], (10)

where φ0,M (0) is a normalization constant.The Montroll double well potential is determined up to the additive

constant ε0 by the ‘bosonic’ Riccati equation

uM (ξ) = R2M −R

′M + ε0 =

β2

4+

(γ2 − 1)ε2

4β2+ε

2+ ε0

+ε

8β2

[ (4γ2ε+ 2(γ2 + 1)εcosh(βξ) − 8β2

)cosh(βξ)

−4γ(ε+ εcosh(βξ) − 2β2

)sinh(βξ)

]. (11)

If, as suggested by Caticha, one chooses the ground state energy to be

ε0 = −β2

4− ε

2+

ε2

4β2

(1 − γ2

), (12)

then uM (ξ) turns into a travelling, asymmetric Morse double-well potentialof depths depending on the Montroll parameters β and γ and the splittingε

UL,R0,m = β2

[1 ± 2εγ

(2β)2

], (13)

where the subscript m stands for Morse and the superscripts L and R forleft and right well, respectively.


The difference in depth, the bias, is ∆m ≡ UL0 − UR

0 = 2εγ, while thelocation of the potential minima on the traveling axis is at

ξL,Rm = ∓ 1

βln

[(2β)2 ± 2εγε(γ ∓ 1)

], (14)

that shows that γ = ±1.

An extension of the previous results is possible if one notices that RM

in Eq. (8) is only the particular solution of Eq. (11). The general solutionis a one-parameter function of the form

RM (ξ;λ) = RM (ξ) +d

dξ

[ln(IM (ξ) + λ)

](15)

and the corresponding one-parameter Montroll potential is given by

uM (ξ;λ) = uM (ξ) − 2d2

dξ2

[ln(IM (ξ) + λ)

]. (16)

In these formulas, IM (ξ) =∫ ξφ2

0,M (ξ)dξ and λ is an integration constantthat is used as a deforming parameter of the potential and is related to theirregular zero mode. The one-parameter Darboux-deformed ground statewave function can be shown to be

φ0,M (ξ;λ) =√λ(λ+ 1)

φ0,M

IM (ξ) + λ, (17)

where√λ(λ + 1) is the normalization factor implying that λ /∈ [0,−1].

Moreover, the one-parameter potentials and wave functions display singu-larities at λs = −IM (ξs). For large values of ±λ the singularity movestowards ∓∞ and the potential and ground state wave function recoverthe shapes of the non-parametric potential and wave function. The one-parameter Morse case corresponds formally to the change of subscriptM → m in Eqs. (15) and (16). For the single well Morse potential theone-parameter procedure has been studied by Filho12 and Bentaiba et al.13

The one-parameter extension leads to singularities in the double-well po-tential and the corresponding wave functions. If the parameter λ is positivethe singularity is to be found on the negative ξ axis, while for negative λ it ison the positive side. Potentials and wave functions with singularities are notso strange as it seems14 and could be quite relevant even in nanotechnologywhere quantum singular interactions of the contact type are appropriatefor describing nanoscale quantum devices. We interpret the singularity asrepresenting the effect of an impurity moving along the MT in one direc-tion or the other depending on the sign of the parameter λ. The impurity


may represent a protein attached to the MT or a structural discontinuity inthe arrangement of the tubulin molecules. This interpretation of impuritieshas been given by Trpisova and Tuszynski in non-supersymmetric modelsof nonlinear MT excitations.15 Another biophysical application of this oneparameter extension will be discussed in Section 5.

3. The Sine-Gordon MT solitons

Almost simultaneously with Sataric, Tuszynski and Zakula, there was an-other group, Chou, Zhang and Maggiora,16 who published a paper on thepossibility of kinklike excitations of sine-Gordon type in MTs but in a bi-ological journal. Even more, they assumed that the kink is excited by theenergy released in the hydrolysis of GTP → GDP in microtubular solutions.As the kink moves forward, the individual tubulin molecules involved in thekink undergo motion that can be likened to the dislocation of atoms withinthe crystal lattice.

They performed an energy estimation showing that a kink in the systempossesses about 0.36–0.44 eV, which is quite close to the 0.49 eV of energyreleased from the hydrolysis of GTP.

They assumed that the interaction energy U(r) between two neighboringtubulin molecules along a protofilament is harmonic:

U(r) ≈ 12k(r − a0)2 , (18)

where k = d2U(a0)dr2 and r = xi − xi−1. In addition to this kind of nearest

neighbor interaction, a tubulin molecule is also subjected to interactionswith the remaining tubulin molecules of the MT, i.e. those in the sameprotofilament but not nearest neighbor to it.

Chou et al. cite pages 425–427 in the book of R.K. Dodd et al. (Solitonsand Nonlinear Wave Equations, Academic Press 1982) for the claimingthat this interaction for the ith tubulin molecule of a protofilament can beapproximated by the following periodic effective potential

Ui = U0

(1 − cos

2πξia0

), (19)

where U0 is the half-height of the potential energy barrier and ξi is the dis-placement of the ith tubulin molecule from the equilibrium position withina particular protofilament.


Introducing the new variable φi = 2πa0ξi the following sine-Gordon equa-

tion is obtained

m∂2φ

∂t2= ka2

0

∂2φ

∂x2−

(2πa0

)2

U0 sinφ (20)

that can be reduced to the standard form of the sine-Gordon equation

∂2φ

∂x2− 1c2∂2φ

∂t2=

1l2

sinφ (21)

if one sets c2 = ka20

m and l−2 = 4π2U0ka4

0. Now, it is well known that the

sine-Gordon equation has the famous inverse tangent kink solution

φ = tan−1(exp[±γ

l(x− vt)]

), (22)

where γ = 1√1− v2

c2

is an acoustic Lorentz factor and w = γl is the kink

width.Most interestingly, the momentum of a tubulin dimer is strongly local-

ized:

p =d(mξ)dt

=ma0

π

γv

lsech

[− γ

l(x− vt)

]. (23)

This momentum function possesses a very high and narrow peak at the cen-ter of the kink width implying that the corresponding tubulin molecule willhave maximum momentum when it is at the top of the periodic potential.According to Chou et al. this remarkable feature occurs only in nonlinearwave mechanics.

Interestingly, for purposes of illustration, these authors have assumedthe width of a kink w ≈ 3a0. Therefore, with the kink moving forward, theaffected region always involves three tubulin molecules. For a general case,however, the width w of a kink can be calculated from

w =a0

2π

√ka2

0

U0, (24)

if the force constant k between two neighboring tubulin molecules along aprotofilament, the distance a0 of their centers, and the energy barrier 2U0

of the periodic, effective potential are known. Then the number of tubulinmolecules involved in a kink is given by

w

a0= (2π)−1

√ka2

0/U0 . (25)

It is further known that the tubulin molecules in a MT are held bynoncovalent bonds, therefore the interaction among them might involve


hydrogen bonds, van der Waals contact, salt bridges, and hydrophobic in-teractions.

It was found by Israelachvili and Pashley17 that the hydrophobic forcelaw over the distance range 0-10 nm at 21oC is well described by

FH

R= Ce−D/D0 N/m (26)

where D is the distance between tubulin molecules, D0 is a decay length,and R = R1R2

R1+R2is a harmonic mean radius for two hydrophobic solute

molecules, all in nm. R is 4 nm in the case of tubulin.

3.1. More on the Hydrolysis and Solitary Waves in MTs

Inside the cell, the MTs exist in an unstable dynamic state characterizedby a continuous addition and dissociation of the molecules of tubulin. Thepolypeptides α and β tubulin each bind one molecule of guanine nucleotidewith high affinity. The nucleotide binding site on α tubulin binds GTPnonexchangeably and is referred to as the N site. The binding site on β

tubulin exchanges rapidly with free nucleotide in the tubulin heterodimerand is referred to as the E site.

Thus, the addition of each tubulin is accompanied by the hydrolysis ofGTP 5’ bound to the β monomer. In this reaction an amount of energyof 6.25 × 10−21 J is freed that can travel along MTs as a kinklike solitarywave.

The exchangeable GTP hydrolyses very soon after the tubulin binds tothe MT. At pH = 7 this reaction takes place according to the formula:

GTP 4− +H2O → GDP 3− +HPO2−4 +H+ + ∆HE . (27)

The last mathematical formulation of the manner in which the energy∆HE is turned into a kink excitation claims that the hydrolysis causes adynamical transition in the structure of tubulin18 .

4. Quantum Information in the MT Walls

Biological information processing, storage, and transduction occurring bycomputer-like transfer and resonance among the dimer units of MTs havebeen first suggested by Hamerrof and Watt19 and enjoys much speculativeactivity.20

For the case of sine-Gordon solitons, the information transport has beeninvestigated by Abdalla et al.21 .


Recently Shi and collaborators22 worked out a processing scheme ofquantum information along the MT walls by using previous hints of Lloydfor two-level pseudospin systems.23 The MT wall is treated as a chain ofthree types of two pseudospin-state dimers. A set of appropriate resonantfrequencies has been given. They conclude that specific frequencies of laserpulse excitations can be applied in order to generate quantum informationprocessing.

Lloyd’s scheme uses the driving of a quantum computer by means of asequence of laser pulses. He assumes a 1-dimensional arrangement of atomsof two types (A and B) that could be each of them in one of two states andare affected only by nearest neighbors. Then, information processing couldbe performed by laser pulses of specific frequencies ωKα,β

, that change thestate of the atom of the K kind (A or B type) if in a pair of atoms AB, Ais in α state and B is in state β.

5. Supersymmetry at the Brain Scale

Neuronal activity is the result of the propagation of impulses generated atthe neuron cell body and transmitted along axons to other neurons. Re-cently, Robinson and collaborators25 obtained simple damped wave equa-tions for the axonal pulse fields propagating at speed va between two popu-lations, a and b, of neurons in the thalamocortical region of the brain. Theexplicit form of their equation is

ORφa(t) = S[Va(t)] , (28)

where

OR =(

1ν2

a

d2

dt2+

2νa

d

dt+ 1 − r2a∇2

)(29)

where νa = va/ra, ra is the mean range of axons a, and Va =∑

b Vab is aso-called cell body potential which results from the filtered dendritic treeinputs. Robinson has used the experimental parameters in this equation forthe processing of the experimental data. In the following we concentrate ona particular mathematical aspect of this equation and refer the reader tothe works of Robinson’s group for more details concerning this equation.

5.1. The Homogeneous Equation

We treat first the homogeneous case, i.e. S = 0 and we discard thesubindexes as being related to the phenomenology not to the mathematics.


Let us employ the change of variable z = ax+ by− ct (see, Ref. 26), whichis a traveling coordinate in 2+1 dimensions. This is justified because it wasnoticed by Wilson and Cowan24 that distinct anatomical regions of cerebralcortex and of thalamic nuclei are functionally two-dimensional although ex-tending to three spatial coordinates is trivial. We have the following rescal-ings of functions: φt = −cφz, φtt = v2φzz , φxx = a2φzz, φyy = b2φzz. Then,we get the ordinary differential equation corresponding to the damped waveequation in the following form

OR,zφ ≡(d2

dz2− 2µ

d

dz+ µ2

)φ = α2φ , (30)

where

µ =νc

c2 − ν2r2(a2 + b2), α2 =

ν4r2(a2 + b2)[c2 − ν2r2(a2 + b2)]2

. (31)

The simple damped oscillator equation (30) can be factorized

L2µφ ≡

(d

dz− µ

) (d

dz− µ

)φ = α2φ , (32)

The case c2 < ν2r2(a2 + b2) implies µ < 0 and the general solution of (30)can be written

φ(z) = eµz(Aeαz +Be−αz) . (33)

The opposite case c2 > ν2r2(a2 + b2) will lead to only a change of sign infront of µ in all formulas henceforth, whereas the case c2 = ν2r2(a2 + b2)will be considered as nonphysical. The non-uniqueness of the factorization ofsecond-order differential operators has been exploited in a previous paper27

on the example of the Newton classical damped oscillator, i.e.

Ny ≡(d2

dt2+ 2β

d

dt+ ω2

0

)y = 0 , (34)

which is similar to the equation (30), unless the coefficient 2β is the frictionconstant per unit mass, ω0 is the natural frequency of the oscillator, andthe independent variable is just time not the traveling variable. Proceedingalong the lines of Ref. 27, one can search for the most general isospectralfactorization

(Dz + f(z))(Dz + g(z))φ = α2φ. (35)

After simple algebraic manipulations one finds the conditions f + g = −2µand dg/dz + fg = µ2 having as general solution fλ = λ

λz+1 − µ, whereas


f0 = −µ is only a particular solution. Using the general solution fλ we get

A+λA−λφ ≡(Dz +

λ

λz + 1− µ

) (Dz − λ

λz + 1− µ

)φ = α2φ. (36)

This equation does not provide anything new since it is just equation (31).However, a different operator, which is a supersymmetric partner of (36)is obtained by applying the factorizing λ-dependent operators in reversedorder

A−λA+λφ ≡(Dz − λ

λz + 1− µ

) (Dz +

λ

λz + 1− µ

)φ = α2φ . (37)

The latter equation can be written as follows

ˆOλφ ≡(d2

dz2− 2µ

d

dz+ µ2 − α2 − λ2

(λz + 1)2

)φ = 0 , (38)

or (d2

dz2− 2µ

d

dz+ ω2(z)

)φ = 0 , (39)

where

ω2(z) = µ2 − α2 − λ2

(λz + 1)2(40)

is a sort of parametric angular frequency with respect to the traveling co-ordinate.

This new second-order linear damping equation contains the additionallast term with respect to its initial partner, which may be thought of asthe Darboux transform part of the frequency.28 Zλ = 1/λ occurs as anew traveling scale in the damped wave problem and acts as a modulationscale. If this traveling scale is infinite, the ordinary damped wave problemis recovered. The φ modes can be obtained from the φ modes by operatorialmeans27 .

Eliminating the first derivative term in the parametric damped oscillatorequation (39) one can get the following Bessel equation

d2u

dx2−

(n2 − 1

4

x2+ β2

)u = 0 , (41)

where x = z + 1/λ, n2 = 5/4, and β = iα. Using the latter equation, thegeneral solution of equation (39) can be written in terms of the modifiedBessel functions

φ = (z + 1/λ)1/2[C1I√5/2(α(z + 1/λ)) + C2I−√5/2(α(z + 1/λ))]eµz . (42)


What could be a right interpretation of the supersymmetric partner equa-tion (37) ? Since the solutions are modified Bessel functions, we considerthis equation as a diffusion equation with a diffusion coefficient depend-ing on the traveling coordinate. Noticing that the velocity in the travelingvariable of this diffusion is the same as the velocity of the neuronal pulseswe identify it with the diffusion of various molecules, mostly hormones, inthe extracellular space (ECS) of the brain, which is known to be neces-sary for chemical signaling and for neurons and glia to access nutrients andtherapeutics occupying as much as 20 % of total brain volume in vivo29 .

5.2. The Nonhomogeneous Equation

The source term S in Robinson’s equation (28) is a sigmoidal firing function,which despite corresponding to a realistic case led him to work out exten-sive numerical analyses. Analytic results have been obtained recently byTroy and Shusterman30 by using a source term comprising a combinationof discontinuous exponential coupling rate functions and Heaviside firingrate functions. In addition, Brackley and Turner31 incorporated fluctuatingfiring thresholds about a mean value as a source of noisy behavior31 .

The procedure of Troy and Shusterman can be applied for the paramet-ric damped oscillator equation as well as for the Bessel diffusion equationobtained herein in the realm of Robinson’s brain wave equation with thedifference that the method of variation of parameters should be employed.The detailed mathematical analysis is left for a future work.

Acknowledgment

This work was partially supported by the CONACyT Project 46980.

References

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2. J.D. Bashford (2006) J. Biol. Phys. 32, 27 .3. M.V. Sataric, J. A. Tuszynski and R.B. Zakula (1993) Phys. Rev. E 48,

589.4. P. Das and W.H. Schwarz (1995) Phys. Rev. E 51, 3588 .5. M.A. Collins, A. Blumen, J.F. Currie, and J. Ross (1979) Phys. Rev. B 19,

3630.6. J.A. Tuszynski, S. Hameroff, M.V. Sataric, B. Trpisova, and M.L.A. Nip

(1995) J. Theor. Biol. 174, 371 .7. E.W. Montroll, in Statistical Mechanics, ed. by S.A. Rice, K.F. Freed, and

J.C. Light(1972) (Univ. of Chicago, Chicago).


8. H.C. Rosu (1997) Phys. Rev. E 55, 2038.9. A. Caticha (1995) Phys. Rev. A 51, 4264.

10. E. Witten (1981) Nucl. Phys. B 185, 513.11. B. Mielnik (1984)J. Math. Phys. 25, 3387; D.J. Fernandez (1984) Lett. Math.

Phys. 8, 337; M.M. Nieto (1984) Phys. Lett. B 145, 208 . For review seeH.C. Rosu, Symmetries in Quantum Mechanics and Quantum Optics, eds.A. Ballesteros et al. (1999) (Serv. de Publ. Univ. Burgos, Burgos, Spain)pp. 301-315, quant-ph/9809056.

12. E.D. Filho, (1988) J. Phys. A 21, L1025.13. M. Bentaiba, L. Chetouni, T.F. Hammann (1994) Phys. Lett. A 189, 433.14. T. Cheon and T. Shigehara (1998) Phys. Lett. A 243, 111.15. B. Trpisova and J.A. Tuszynski (1992) Phys. Rev. E 55, 3288 .16. K-C Chou, C.-T. Zhang, and G.M. Maggiora (1994) Biopolymers 34, 143.17. J. Israelachvili and R. Pashley (1982) Nature 300, 341.18. M.V. Sataric and J.A. Tuszynski (2005) J. Biol. Phys. 31, 487 .19. S. Hameroff and R.C. Watt, (1982), J. Theor. Biol. 98, 549.20. See e.g., J. Faber, R. Portugal, L. Pinguelli Rosa, Information process-

ing in brain MTs, contribution at the Quantum Mind 2003 Conference,q-bio.SC/0404007.

21. E. Abdalla, B. Maroufi, B. Cuadros Melgar, and M. Brahim Sedra (2001)Physica A 301, 169.

22. C. Shi, X. Qiu, T. Wu, R. Li (2006) J. Biol. Phys. 32, 413.23. S. Lloyd (1993) Science 261, 1569.24. H.R. Wilson and J.D. Cowan (1973) Biol. Cybernetics 13, 55.25. P.A. Robinson, C.J. Rennie, D.L. Rowe, S.C. Connor (2004) Human Brain

Mapping 23, 53; H. Wu and P.A. Robinson (2007) J. Theor. Biol. 244, 1;P.A. Robinson, Phys. Rev. E, (2005), 72, 011904 .

26. P.G. Estevez, S. Kuru, J. Negro, L.M. Nieto (2006) J. Phys. A 39, 11441.27. H.C. Rosu and M. Reyes(1998) Phys. Rev. E, 57, 2850.28. G. Darboux (1882) C.R. Acad. Sci. 94, 1456.29. R.G. Thorne and C. Nicholson (2006) Proc. Natl. Acad. Sci. U.S.A. 103,

5567 and references therein.30. W.C. Troy and V. Shusterman (2007) SIAM J. Applied Dynamical Systems

6, 263.31. C.A. Brackley and M.S.Turner (2007) Phys. Rev. E 75, 041913 .

267

Turing Systems: A General Model for ComplexPatterns in Nature

R.A. Barrio

Instituto de Fısica, Universidad Nacional Autonoma de Mexico (UNAM),Apartado Postal 20-364 01000 Mexico, D.F., Mexico

[email protected]

More than half a century ago Alan Turing showed that a system of nonlinearreaction-diffusion equations could produce spatial patterns that are station-ary and robust, a phenomenon known as ”diffusion driven instability”. Thisremarkable fact was largely ignored for twenty years. However, in the lastdecade, Turing systems have been a matter of intense and active research, be-cause they are suitable to model a wide variety of phenomena found in Nature,ranging from Turing’s original idea of describing morphogenesis from an egg,and applications to the colouring of skins of animals, to the physics of chemi-cal reactors and catalyzers, the physiology of the heart, semiconductor devices,and even to geological formations. In this paper I review the main properties ofthe Turing instability using a generic reaction-diffusion model, and I give ex-amples of recent applications of Turing models to different problems of patternformation.

Keywords: Biophysics; Complex Systems; Nonlinear Systems; Turing SystemsPACS(2006): 87.64.Bx; 87.64.t; 87.16.Ac; 89.75.k; 89.75.Fb; 82.39.Rt; 43.60.Wy

1. Introduction

Nature shows many systems composed of radically different parts. Usuallythese parts are self-organized in space in a remarkably precise way. Exam-ples are to be found in systems of any size: atoms and molecules in solids,nanostructured materials, living organisms, geological formations and as-tronomical systems.

This organization varies in complexity, for example in living organismsit is extremely complicated at all scales and the precise spatial locationof all the parts is crucial for the functioning of the whole system, that is,for life itself. In other cases the arrangement of parts is a consequence ofvery simple physical interactions that could be described using fundamentalprinciples of Physics, as in simple crystals.

268 R. A. Barrio

In all cases this organization should ultimately be a consequence ofthe physical interactions of the components, independently of the size andcomplexity of the system. However, it is often extremely difficult, or evenimpossible, to model complicated and complex systems using only micro-scopic physical interactions. Coherent macroscopic behaviour is frequentlynot derivable from many-body interactions, and one needs to resort to phe-nomenological models that could capture the specific characteristics of thesystem at the scale one is interested in. To be specific, one needs a simplemechanism able to provide the spatial information needed to accommodatethe parts of the system. For this we recall a universal principle in Physicsthat relates the dynamical behaviour of the system to the symmetries pos-sessed by the fields acting on it, that is, there must be a symmetry-breakingmechanism that results in a stationary spatial pattern.

It is fair to say that the first one who realized that patterning in bio-logical systems could be understood as self-organization was the naturalistD’ Arcy Thompsom1 . The connections that he made between Biology andother fields in Science in his celebrated book of 1917 inspired a continuousstream of theoretical work in complex systems.

Nonlinear systems of differential equations present universal propertiesthat make them particularly suitable to build models of complex systems.Spontaneous symmetry-breaking is one of them, and it appears when theparameters, defining the interactions with the external world, reach certaincritical values. The low-symmetry states are in general extremely robustunder random perturbations, as thermal noise and variations of the initialconditions.

Alan Turing2 predicted that a system of chemical substances diffus-ing and reacting with each other in a medium, can produce concentrationpatterns that are stationary and robust. He attempted to describe verycomplicated systems and processes, as one can infer from reading the ab-stract of his 1952 paper: “The purpose of this paper is to discuss a possiblemechanism by which the genes of a zygote may determine the anatomicstructure of the resultant organism. The theory does not make any new hy-pothesis, but merely suggests that certain well known physical laws sufficefor explaining many facts”.2 Ambitious as it sounds, this phrase abridgesall the basic concepts we need. He coined the word morphogenesis to givethe idea of the processes that lead a system to acquire a precise form, andin consequence, the responsible substances are called morphogens.

The well known physical laws are the kinetics of chemical reactions andthe normal diffusion of diluted chemicals, both deriving from the statistical

Turing Systems: A General Model for Complex Patterns in Nature 269

behaviour of a system with many components. The pattern results whena stationary, highly symmetrical state in the absence of diffusion becomesunstable when particles are allowed to diffuse. This symmetry-breakingmechanism, known as “diffusion driven instability” or simply Turing insta-bility, is surprising. Usually, when a substance spreads out and diffuses oneexpects its concentration gradients to diminish, smearing out any struc-ture. However, in the Turing instability, diffusion is precisely the reasonfor loosing a stable uniform state and for the appearance of a structuredpattern.

Turing models belong to a wide class of systems of differential equa-tions known as reaction-diffusion equations. These systems could present aTuring instability, provided they fulfill a number of restrictive conditions.Reaction-diffusion systems are extremely rich and varied in behaviour, usu-ally made ex professo for the particular phenomenon of interest. However,they all present universal characteristics that can be easily illustrated witha generic model we proposed some time ago3 .

In this paper I will briefly discuss the properties of nonlinear systemsof differential equations, then I shall focus on a generic Turing model andmention some other models widely used in research at present. At the endI shall illustrate pattern modeling in systems belonging to very differentfields in Science.

2. Nonlinear Systems

I shall start by recalling some basic facts from the theory of nonlinear sys-tems, needed to understand the following sections. In general the dynamicalbehaviour of a physical system could be represented as

∂V∂t

= Fr(V, t), (1)

where V is a vector representing the fields, r are parameters describing theinteraction of the fields with the external world, and F is a vector of non-linear functions determining the dynamics and interactions between fields.Eq.1 represents an infinite set of partial differential equations if the phasespace is infinite and continuous. However, one usually utilizes appropriatetechniques and approximations to reduce the problem to a set of ordinarydifferential equations. Suppose that the system has a singular point at V0,that is, the fields do not change in time there. It is reasonable to think thatif the system is driven away from this point by an infinitesimal amount, itsresponse should be linear, independently of the form of the functions F. If

270 R. A. Barrio

the perturbation is V0 + εU, where ε is very small, one gets

dUdt

= LrU, (2)

where Lr is a linear operator. The solution of Eq.2 is U = exp(snt)Xn,implying the eigenvalue equation

LrXn = snXn, (3)

where n labels the normal modes X. The complex eigenvalues sn = (sn)+i(sn) for each normal mode disclose the linear response of the system.

There are various cases(1) If (sn) > 0, then Xn is unstable.(2) If (sn) ≤ 0, then Xn is stable.(3) If (sn) = 0, and (sn) > 0, there are growing oscillations in time.(4) If (sn) = 0, and (sn) > 0, there might be growing spatial fluctuations.

If the eigenmodes form a complete set, any function can be written asa linear combination of normal modes, in particular V =

∑n AnXn, and

the time evolution of the fields is given by the behaviour of the amplitudesof each mode (An). Therefore, Eq.1 reads

∑n

dAn

dtXn =

∑n

snAnXn + N

(∑n

snAn

). (4)

where N is a function that takes care of the nonlinear terms. Then, if thenormal modes form an orthogonal basis,

dAn

dt= snAn + 〈Xn|N(

∑smAm)〉. (5)

Near a bifurcation point most of the eigenvalues sn have negative realpart and die exponentially. Some modes Xc, called marginal, correspondto eigenvalues with a very small positive real part, and they govern thedynamical behaviour of the system at long times. The Center ManifoldTheorem4 states that if (sm) < 0, then

dAm

dt 0,

and this implies that Am = f(Ac; c = 1, 2, ...) (which is the center manifoldequation). Then, the important modes with sc 0, obey

dAc

dt= scAc + N (Ac, Am) , (6)


These are known as the amplitude equations, which is a set of ordinarydifferential equations that can be solved numerically, or analyzed linearlyto investigate the effects of the nonlinearities in the time evolution of thesystem.

3. The Turing Instability

A set of reaction-diffusion equations contains a diffusion term in the entriesof the functions Fr in Eq.1, that is

∂ui

∂t= Di∇2ui + Fi(uj, (7)

where ui are the concentration fields of morphogens, Di are the respectivediffusion constants, and the nonlinear functions Fi give the kinetics of thereactions. Alan M. Turing2 showed that a uniform stable state in the ab-sence of diffusion could become unstable when diffusion is acting, providedthe diffusion constants are not the same.

Despite the fact that a real Turing instability has been reproduced inthe laboratory in a chemical system5 , the existence of Turing patterns inNature is still a controversial issue (see Ref. 6 for a review). There havebeen mainly two serious criticisms: (1) The identification of morphogens ina complex system is a very difficult task, and only few have been identified,but their function is not corroborated yet. (2) The patterns obtained withsimple Turing systems are very simplistic, and hardly bear any resemblancewith the patterns observed in Nature.

The first objection has nothing to do with the theory, and it is a matterof technical experimental difficulties. I am sure that in the near future theseshall be overcome and an explosion of identified morphogens appears. Infact, in some complex biological processes, as the chemical signaling foraxon growth, morphogens have been found7 .

The second objection is more serious, since it would imply that thetheory is far too simple to be applied to real systems, even Turing himself,who invented the computer, could not see the patterns that his theory wouldproduce. Now, computer simulations make it possible to see patterns fromfairly complicated models. In fact, the main purpose of this paper is to showthat Turing models could render a whole zoo of complicated patterns. Inrecent years there has been an exponential increase of the number of paperspublished by a large number of researchers and a widening of the range ofsystems that could be modeled in this way. A magnificent review on variousmodeling techniques in biological systems has been published recently8 .

272 R. A. Barrio

3.1. Generic Turing Model (BVAM)

It is convenient to be more specific now and discuss the properties of Turingsystems. For this I shall use a generic activator-inhibitor model that weproposed in the past3 . I shall refer to it as the BVAM model from nowon. It consists of a set of two reaction-diffusion equations for two chemicalswith concentrations U , and V . Contrary to the usual procedure, we werenot aiming to model a specific chemical reaction, but we wanted to finda general form for the kinetics by assuming that there is a fix point at(Uc, Vc) and Taylor-expanding around this fix point, keeping terms up tothird order. The specific form I shall consider is:

∂u

∂t= Dδ∇2u + αu(1 − r1v

2) + v(1 − r2u) (8)

∂v

∂t= δ∇2v + βv(1 +

αr1

βuv) + u(γ + r2v)

where δ is a scaling factor, D is the ratio between diffusion coefficients , u =U−Uc and v = V −Vc, so there is a uniform stationary solution of Eq.(8) atthe point (u, v) = (0, 0). The special arrangement of the coefficients followsfrom conservation relations between chemicals. There are two interactionparameters, r1 and r2, corresponding to the cubic and a quadratic terms,respectively. In zero dimensions, or in the absence of diffusion, Eq. (8) showsstationary uniform solutions at

v =−(α + γ)

1 + βu.

For clarity I shall assume that α = −γ, so all the singular points collapseat the origin. The analysis in the general case has been made elsewhere.11

In these circumstances the eigenvalues of the linear operator in Eq. (2) are

s = (α + β)/2 ±√

[(α + β)/2]2 − α(β + 1).

The real part of s could be negative in certain regions of the parameterspace (α, β) and the uniform solution is stable. When diffusion is added, thelinear operator has eigenfunctions of the form Xk = expk · r. Therefore,there is a dispersion relation

s(k)2 − Bs(k) + C = 0 (9)

where B = δk2(1 + D) − (α + β), and C = (α − δDk2)(β − δk2) + α.


There are unstable states with (s(k)) > 0 and (s(k)) = 0 for certainfinite values of the wave number k, if the following conditions are met

α + β < 0

α(β + 1) > 0

Dβ + α > 0

(Dβ + α)2 − 4Dα(β + 1) > 0

, (10)

This is called the Turing region, where perfectly stationary patternsappear. The maximum of the real part of the dispersion relation is situatedat

kc =D(α − β) − (D + 1)

√αD

δD(D − 1).

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Fig. 1. (a-l) Patterns obtained from the BVAM model with varying parameters andboundary conditions (bc): Top row shows zero flux bc, in the second row there areperiodic bc and in third row free bc. Columns are from left to right: First and secondwith parameters such that kc = 0.45. The third and fourth columns with kc = 0.89. Inthe first and third columns r1 = 3.5, r2 = 0 (favours stripes), and in the second andfourth columns r1 = 0.02, r2 = 0.2 (favours spots). In all twelve patterns the randominitial conditions and the scale (δ = 10) were the same.

274 R. A. Barrio

In Fig. 1 there are examples of numerical calculations showing basicpatterns obtained by choosing two different sets of parameters: (1) D =0.516, α = −γ = 0.899, and β = −0.91 giving kc = 0.42, for the two leftcolumns; (2) D = 0.122, α = 0.398, and β = −0.4, giving kc = 0.84, for thetwo right columns. Observe that the patterns on the left half of the figurepresent structures with a larger wavelength (λ = 2π/kc) than those on theright half, where the wavelength is smaller.

With this simple model and zero flux, or periodic boundary conditions ina plane, the patterns are fairly simple. The matter of pattern selection hasto do with the nonlinearities, we have established that the cubic interactionr1 favors stripes and that the quadratic one r2 produces spot patterns3 .

In three dimensions there are more possibilities, one could have orderedspheres, disordered spheres, lamellae, layered patterns or labyrinthine pat-terns of tunnels. In Fig. 2 we show an example taken from Ref. 10.

Fig. 2. Examples of numerical calculations on the BVAM model producing complicatedpatterns in three dimensions.

There are other possible bifurcations, for instance if (s) > 0 and(s) = 0 at k = 0, there is a Hopf bifurcation. Also, there could be tworegions, one at k = 0, and another at k = 0 presenting unstable states, andthis is known as a Turing-Hopf bifurcation, in which the spatial patternsoscillate in time.9 The BVAM model also admits traveling wave solutionsif the condition α = −β is relaxed. In a recent paper we have shown thatthe system presents bistability and solitons appear.11 The parameter spacealso shows regions where limit cycle solutions produce rotating spirals ortargets.

In Fig. 3 there are some examples of patterns that look more compli-cated in two dimensions. The first two are labyrinthine stationary patterns,


(A) (B) (C) (D)

Fig. 3. Patterns obtained by relaxing the α = −γ condition. (A) and (B) are stationary,(C) shows traveling stripes and (D) is an oscillating pattern.

obtained by exploring other regions of the Turing space, the third shows aTuring pattern under anomalous diffusion conditions, and the last presentsrotating spirals similar to the ones seen in the Belousov–Zhabotinskii reac-tion.

It is clear that even one of the simplest Turing systems is rich enough tobe applied to the most varied situations. The same could be said of othermodels.

3.2. Other Turing Models

In this section I discuss a selected group of Turing models that have beenwidely used in the past. Some of these models can be considered as specialcases of the BVAM model presented before.

In Table 1 the kinetics of some famous models is written. By famous Imean that they are known by a specific name and that they have been usedsuccessfully to either reproduce or predict the behaviour of real systems.The notation used in the table corresponds to the convention of writingthem as follows:

∂u

∂t= Du∇2u + F (u, v, w, ...) (11)

∂v

∂t= Dv∇2v + G(u, v, w, ...),

∂w

∂t= Dw∇2w + H(u, v, w, ...),

...

... etc.

As a reference, the first row of Table 1 is the BVAM model already dis-cussed, but conveniently written in adimensional form. Then, the simplestmodels have only one field, and the simplest non-trivial kinetics would be

276 R. A. Barrio

Model Equations Ref.BVAM F (u, v) = η(u + av − Cuv − uv2)

G(u, v) = η(bv + hu + Cuv + uv2) 3

Fisher equation F (u) = µu(1 − u/K) 12

Lotka Volterra F (u, v) = au − buv

G(u, v) = −cu + duv 14

Gierer Meinhardt F (u, v) = ρu − µuu + ρu2/v

G(u, v) = ρv − µvv + ρu2 15

Gray-Scott F (u, v) = f(1 − u) − uv2

G(u, v) = −(f + κ)v + uv2 17

Brusselator F (u, v) = a − (b + 1)u + u2v

G(u, v) = bu − u2v 21

Oregonator F (u, v, w) = 1ε [qv − uv + u(1 − u)]

G(u, v, w) = 1ε′ [−qv − uv + fw]

H(u, v, w) = u − v 22

Schnakenberg F (u, v) = γ(a − u + u2v

)G(u, v) = γ(b − u2v) 24

Lengyel-Epstein or F (u, v) = a − u − 4uv/(1 + u2)Brandeisator G(u, v) = δ

(u − uv/(1 + u2)

)26

FitzHugh-Nagumo F (u, v) = −v + u + u3

G(u, v) = ε (u − αv − β) 27

Table 1: Famous Models that present the Turing instability.

the logistic map. This is the well known Fisher equation, which presentstraveling wave solutions. This equation was originally devised by R.A.Fisher to model the spread of an advantageous gene over a population.There have been recent applications of this model to study bacterial dy-namics and epidemics of viruses.13 This model could be thought of as aspecial case of the BVAM model if u = v, µ = η, C = 1/K and all theother parameters are zero.

One of the most famous models in theoretical biology is the one dueto the collaboration between the chemist Alfred Lotka and the biologistVito Volterra, the Lotka–Volterra model, which was originally devised todescribe the predator-prey interactions between two populations of animals,adding external resources for feeding. The usual result is that under certainconditions both populations oscillate in time, and when diffusion is addedin two dimensions, patchy patterns appear also in space. This model and itsvariants has been used many times to model the most varied interactions


between populations and it is still one of the main tools in Ecology. Fromthe table it is obvious that this model could be also encompassed in theclass of the BVAM variations.

The importance of the Turing mechanism in biology was first realizedin 1972 when Gierer and Meinhardt15 proposed their activator-inhibitormodel to account for patterning in biological systems. Usually the autocat-alytic production u2 is replaced by a term with saturation u2/(1 + κu2).This model has been very successful in reproducing the patterns found inmarine snail shells16 , in which the appearance of oblique lines is due to theexistence of traveling waves.

The Gray Scott model was first built to model glycolysis in cells. Pear-son18 performed extensive numerical calculations that showed a wide varietyof spatial patterns. This model has been used extensively in many prob-lems due to the transparency of the autocatalytic reaction terms and theirsimplicity. Recently we have used this model to propose the formation of aprepattern in the interneuronal medium that provides chemical pathwaysto promote axon growing by chemotaxis19 .

The Brusselator model was proposed by Ilya Prigogine and co-workersto describe autocatalytic reactions, which are ubiquitous in Nature, partic-ularly in living organisms. Despite its simplicity this model presents a richphase diagram of bifurcations that makes it suitable for numerous applica-tions, particulary in multilayered systems.

In the Belousov–Zhabotinskii (BZ) reaction a mixture of potassium bro-mate, cerium(IV) sulfate and citric acid in dilute sulphuric acid produce os-cillations of the concentrations of the cerium(III) and cerium(IV) ions. Thisreaction was studied by Noyes, Field and Koros at the University of Oregon.They proposed a model with eighteen chemical reactions, later simplifiedto five reactions only. A further simplification with only three chemicals23

is known now as the Oregonator model written on the table. This modelhas been extensively used recently to produce complicated patterns.

The simplest possible reaction with two variables that models the BZreaction is the Schnackenberg model, which could be regarded as a simplifi-cation of the Oregonator, used in many occasions to study two dimensionalTuring patterns.25

The BZ reaction inspired the chlorine-iodide-malonic acid (CIMA) re-action,5 which was the first experimental uncontestable realization of aTuring pattern in a chemical reactor. The Lengyel–Epstein model was ex-pressly created to mimic the CIMA reaction and has been universally usedever since as a basis to model many chemical systems. Finally, it is worth

278 R. A. Barrio

mentioning the FiztHugh–Nagumo model27 , which is a model of a model.Its origin is the famous Hodgkin-Huxley model for nerve membranes, whichis mathematically complicated, yet the FiztHugh–Nagumo model capturesthe essential phenomena and it only needs three fields. Further reductionto two variables is not unreasonable for many models of excitable media.In the table we have extracted a model in a Turing form, which is one ofthese reductions28 .

3.3. The Complex Ginzburg–Landau Equation

The Complex Ginzburg–Landau equation (CGLE) is one of the most fa-mous and well studied systems in applied mathematics. It describes theevolution of amplitudes of unstable modes for any process resulting froma Hopf bifurcation, and gives qualitative (and often quantitative) infor-mation on a wide variety of phenomena, including nonlinear waves, secondorder transitions, Rayleigh–Benard convection and superconductivity. Neara Hopf bifurcation, all reaction-diffusion equations could be reduced to theCGLE20 . The equivalent reduction for ordinary differential equations iscalled the Stuart-Landau equation. The CGLE can be written as

∂tA = A + (1 + iα)∇2A − (1 + iβ)|A|2A, (12)

were the amplitudes A are complex. We now show the procedure for theBVAM model. We shall set h = −1 (see the first row of Table 1) so all fixedpoints collapse at the origin. A Hopf bifurcation is found in zero dimensionswhen b ≥ bc = −1 and a > 1. The bifurcation parameter is µ = (b − bc)/bc

and the BVAM model could be written as

∂ρ

∂t= D∇2ρ + ηLρ + CMρρ + Nρρρ, (13)

where ρ is the perturbation around the fixed point, L = L0+µL1 is a linearoperator, and M and N are nonlinear operators defined by the kinetics. Onemay write µ = ε2ξ, where ξ = sgn(µ), and make a change of scale τ = ε2t

and e = εr. The perturbation could be expanded in powers of ε, that isρ = ερ1 + ε2ρ2 + ε3ρ3 + ..., and one then substitutes everything in Eq. 13.


Equating coefficients with the same power of ε yields the following(∂

∂t− L0

)ρ1 = 0; (14)(

∂

∂t− L0

)ρ2 = Mρ1ρ1;(

∂

∂t− L0

)ρ3 = −∂ρ

∂τ+ ξL1ρ1 + D∇2

eρ1

+2Mρ1ρ2 + Nρ1ρ1ρ1.

The eigenvectors of the linear operator L0 are X1 = (1 + s,−1) andX2 = X1

∗, with s = +i√

(a − 1). The solvability condition for Eqs. 14gives an amplitude equation analogous to Eq. 6:

∂A1

∂τ= ξλ1A1 + D∇2

eA1 − Γ|A1|2A1, (15)

where λ1 = ηX1†L1X1, and

Γ = η[2X1

†MX1(L0 − 2sI)−1MX1X1

+4X1†MX1L0

−1MX1X2

−3X1†NX1X1X2

](16)

Further change of variables allows to write Eq. 15 exactly as Eq. 12.This exercise is useful because the solutions of the CGLE have been

analyzed in depth. It could be performed to study pattern selection undercertain circumstances. The same procedure could be followed when relaxingthe condition h = −1 to investigate the behaviour of the system aroundthe other fixed points.

4. Applications

4.1. The Coat Pattern of Marine Fish

In 1995 Kondo and Asai29 published an interesting paper suggesting thatthe coat pattern of certain marine fish of the order Pomacanthus could beexplained as a Turing pattern. They based their claim on the peculiar waythe wave length of the pattern is conserved as the fish grows. A one dimen-sional calculation in a Turing system could predict the observed phenomena.These fish in their juvenile form present a vertical or semicircular stripedpattern, common to all species in the order, thought to prevent selectivepredation by the cannibal adults. When they grow their pattern changesdramatically to the specific colors and shapes of the different species. We

280 R. A. Barrio

made extensive numerical simulations in two dimensions, using the BVAMmodel to obtain the skin pattern of the fish Pomacanthus imperator as itgrows. In Fig.4 we show these results, taken from Ref. 30.

Observe that the two mechanisms that conserve the wave length ofthe pattern are precisely those predicted by the Turing model, namely thesplitting and insertion of modulated regions.

(a)

(b)

Fig. 4. (a) Juvenile form of the fish Pomacanthus imperator. The insets show numericalpatterns obtained in a two-dimensional domain resembling the shape of the fish. (b) Adultform of the same fish. Numerical calculations on the left show patterns in domains ofgrowing size.

These findings were corroborated by observing the changes on the coatof a real fish as it grew. After 4 years without growing (these fish stopgrowing if the environment does not provide enough space), suddenly thechanges were produced very rapidly (when the fish was kept in a big enoughtank). In Fig. 5 we show a series of snapshots of these events, spanningapproximately 6 months.

The patterns obtained with the original BVAM model were modified bysuggesting that there must be sources of morphogen in some places that


Fig. 5. Series of photographs taken every month of our juvenile fish.

promote the alignment of the fringes in certain directions, in such a waythat the combination of boundary conditions with the shape of the domainwould produce the observed patterns. We also proposed that a modulationof the diffusion coefficients in some directions would produce alternate spotsand stripes, as observed in other species of fish. More complicated patternscould be obtained by coupling two Turing systems by an interaction termthat could be linear or not. These ideas have been useful in reproducing thecoats of many other fish.30 In Fig. 6 there are few examples comparing thenumerical calculations from the coupled model with the observed patternsin fish.

4.2. Patterns in Curved Domains

When a two dimensional domain changes its shape in time and space, theTuring equations have to be modified. We showed the general transforma-tion of Turing patterns in growing and curved domains.32 Basically thedifferences are the appearance of an advection term in the time derivativesand the replacement of the Laplacian by the Laplace–Beltrami operator

282 R. A. Barrio

(a) (b)

(c) (d)

(e) (f)

Fig. 6. Various marine fish whose coats were modeled. The numerical patterns areshown in the insets, and were obtained by coupling two BVAM models in different ways.(a) Zebrasoma veliferum, (b) Synchiropus picturatus, (c) Hypostomus plecostomus, (d)Pomacanthus maculatus (taken from Ref. 31), (e) Siganus vermiculatus, (f) Ostracionmeleagris.

that takes into account the effects of local curvature.It is interesting to observe patterns formed in growing domains when

the changes due to growth take place in a time scale of the same order asthe characteristic times for the pattern to form.

In Fig. 7(a) there is a calculation showing a spot pattern in a linearlygrowing square. Notice that the two mechanisms that conserve the wave-length are in action, namely, splitting of spots and insertion of new ones inbetween. In Fig. 7(b) one can see a pattern of spots forming on a growingcone. The pattern is affected by the geometry of the conical domain and incertain regions appears as aligned stripes, just as it is seen in the tails ofmany reptiles, as lizards.

Patterns in domains of constant curvature are of particular interest


(a) (b)

Fig. 7. (a) Pattern obtained with Eq.8 on a growing square domain. (b) Pattern ob-tained in a growing cone.

Fig. 8. (a) and (b) Patterns obtained with the BVAM model on spheres of two sizes withonly cubic nonlinearities. (c) and (d) same as before but adding quadratic nonlinearities.

because they are observed in many small organisms, as radiolarians andviruses. We used the BVAM model to numerically find the symmetriesof Turing patterns produced on a spherical surface33 . It was found thatspots always accommodate themselves in regular positions, depending onthe radius of the sphere. Some icosahedral symmetries were ubiquitous.Stripe patterns are very interesting because of the existence of singulari-ties, not necessarily at the poles. When the radius of the sphere matchesthe wavelength of the pattern some target patterns appear. Examples ofthese calculations are shown in Fig. 8.

284 R. A. Barrio

4.3. Oscillating Turing Patterns

Most of the systems that present a Turing instability also show other kindsof bifurcations. We already showed that the BVAM model could presentHopf and Turing–Hopf instabilities. There has been a renewed interest instudying the latter instability, since it produces patterns in which nonlin-ear standing waves exist. This is due to the interplay between the Turinginstability, which produces spatially stable patterns, and a Hopf instabil-ity, that causes oscillations in time. These oscillating Turing patterns arequite complicated and represent a step forward modeling real processes incomplex systems. I shall give two important examples.

4.3.1. The Faraday Experiment and Sea Urchins

First, there is an example of an application of the BVAM model that isunexpected and its consequences far reaching. It is the famous experimentmade by Faraday in 183134 . This experiment is deceptively simple, it con-sists of agitating vertically a vessel containing a liquid. Then, when the am-plitude (A) is zero, the uniform steady state corresponds to a flat surface,and it becomes unstable when A = 0, and the surface acquires a compli-cated shape. The theoretical treatment of this phenomenon in the linearregime was first made by Lord Rayleigh, who described surface waves thatare known as Rayleigh waves ever since. In a cylindrical container withzero-flux boundary conditions the linear solutions are Bessel functions ofthe first kind.

Much more interesting are the nonlinear solutions, and we repeatedthe experiment enhancing the nonlinear interactions, by using a rare liq-uid (fluorinert FC-75) with a high density and a very low surface tensioncoefficient. We observed that the nonlinear patterns are centrosymmetric,and that their symmetry axis depends on the bifurcation parameter, whichis a combination of the amplitude and the frequency. We devised a modelbased on the Euler equation with friction, adding a nonlinear mechanismresulting from the interaction between the vertical and horizontal velocityfields35 .

We also showed that the model could be written as a reaction-diffusionsystem which always presents a Hopf bifurcation and that could be drivento a Turing–Hopf instability. Details are given in Ref. 35. There are othermodels of the Faraday experiment, for example, a phenomenological one40

based on the Swift–Hogenberg equation36 , or a treatment based on anextension of the same idea37 .


With our model one could follow the bifurcation tree of any symmetryaxis. The actual photographs from the experiment of a sequence of 5-foldsymmetry bifurcations are reproduced in Fig. 9, and they are not quasicrys-talline (in fact, quasicrystalline patterns have been obtained in the Faradayexperiment38). M. Torres pointed out that these patterns are very similarto the shells of sea urchins in subsequent geological times of evolution.

Fig. 9. Photographs taken from our experiment, sequentially following the bifurcations

of symmetry five.

Five-fold symmetry is remarkable, not only because of the existence ofquasicrystals, but also because all known vertebrates in this planet present,while developing their limbs, the following scheme of symmetry-breakingprocesses 1 → 2 → 5 (see for instance Ref. 42). Furthermore, 5-fold sym-metry is not only seen in sea urchins but in all the phylum echinodermata.

Therefore, it is important to investigate the robustness of symmetry 5 infinite domains. We performed a series of numerical calculations, using theBVAM model, in a circular domain in two dimensions varying its radius39

. This can be easily done by manipulating δ in Eq. 8. In Fig. 10 we show ahistogram with the results. Observe that for a wide range of small radii, thepatterns are 5-fold, and then they go to the more frequently found 6-fold.

The results of this model allow to envisage the morphogenesis of asea urchin: the egg can be considered as symmetry 1, then the animal

286 R. A. Barrio

Fig. 10. Results of the symmetry found in the centro-symmetric patterns calculated indisks of different sizes.

undergoes metamorphosis to a larva state that is roughly 2-fold symmetric.This larva starts developing a flat circular belly that grows. At a certainsize five primary podia appear on the disk as bulges, from which the sym-metry of the whole animal developes to the adult state. One could say thenthat the podia appear as a result of a Turing mechanism, then the podiaare calcified and the 5-fold symmetry is conserved in further stages.

In Fig. 11 we show a numerical calculation obtained by producing a5-fold pattern and then growing the disk assuming that the spots are mor-phogen sources that maintain the level of u constant. The shape is comparedwith the actual shell.

This model predicts that if one is able to delay the appearance of theprimary podia in the circular belly of the urchin, until it is larger, theanimal should present 6-fold symmetry.

One may ask at this point: What is the relation between the evolutionof animals and the disturbed surface of a liquid?. The most suitable answercould be taken from Faraday himself, who pointed out that the universalityof a Theory, as the Classical Theory of Vibrations, allows a deep under-standing of the phenomena being modeled, regardless of the nature of theparticular system. In the present case, we could infer that the mechanismsacting on the evolution of species could be better described following the bi-furcations of a nonlinear system. The far reaching consequence is that oneneeds to rethink the meaning of natural selection and adaptability when


Fig. 11. (a) Numerical calculation on a small disk while growing to a larger size, main-taining the central spots as sources of morphogen. (b) Photograph of the shell of anadult sea urchin.

talking about Evolution.The nonlinear regime in the Faraday experiment is extremely rich and

complicated patterns could be obtained in different systems. For instance,Faraday patterns could be produced in quantum systems and one couldharness very robust spatial arrangements of quantum states. In fact, veryrecently a paper in which Faraday patterns are seen in a Bose–Einsteincondensate has been published41 .

4.3.2. Layered Coupled Systems

There has been a series of papers dealing with oscillating patterns. Theyare produced by linearly coupling two sets of Turing equations using theBrusselator model43 , the Oregonator model44 , and the Lengyel–Epsteinmodel45 . These models consist of two chemically active layers, separatedby an intermediate chemically inert layer that allows transport of chemicalsthrough it. The coupling made in such a way that the stable fixed pointin the absence of diffusion is preserved, and the parameters are chosen asto enhance resonating modes at two different wavelengths, resulting in su-perimposed patterns. For instance, spots contained in stripes or vice versa,or patterns with spots of two different sizes (wavelengths), and many othercomplicated oscillating patterns46 . In a recent study47 an experimental setup of a bylayer reactor system has been realized.

However, there are situations in which the intermediate layer could bechemically active (as it is the case of biological membranes), and the inter-layer interaction be more complicated than just a simple selective diffusionof chemicals, probably involving further chemical reactions or catalyzers.

288 R. A. Barrio

Such situations could be modeled by coupling two Turing systems nonlin-early, which as such serves as an interesting extension to the studies of thelinear coupling case. We had proposed that in the original paper where theBVAM model appeared, but now it is interesting to explore the patternsin the region where the Turing–Hopf bifurcation takes place, and using theresonance conditions. For comparison with the linear coupling case we alsoinvestigated cubic and quadratic couplings using the Brusselator model.48

Extensive numerical calculations, guided by linear stability analysis, werecarried out to search for new complex patterns. In Fig. 12 there is an exam-ple of the comparison between linear and cubic coupling. The two patternslook very similar, although this is only true if the coupling strength is verysmall.

(a) (b)

Fig. 12. (a) Oscillating pattern of dots inside stripes, obtained with the linearly coupledmodel of Ref. 43. (b) Pattern obtained with the same model but with cubic coupling.The coupling parameter is q=0.01.

We found that the patterns are very much dependent on the couplingstrength, contrary to linear coupling. If the coupling is very small, reso-nances appear resulting in superposition of patterns. For very strong cou-pling, the system with largest wavelength dominates, and the pattern be-comes very simple. In the intermediate region new patterns with differentsymmetries are found. In this patterns there are new phenomena beyondsimple superposition and resonances, as expected, because nonlinear cou-pling changes the kinetics of the original systems. In Fig. 13 we show twopatterns obtained in the same cubic system with two different values of thecoupling parameter, the figure is taken from Ref. 48.


(a) (b)

Fig. 13. (a) Oscillating pattern of stripes inside dots, obtained with a layered systemcoupled cubically with strength q = 0.09. (b) A pattern of “boats” obtained with q =0.15.

4.4. Bistability and Travelling Waves

In all former applications the BVAM model has been used as a true Turingsystem meeting all the conditions for a Turing instability. Furthermore, ithas been assumed that γ = −α, which ensures that there is only one fixedpoint at the origin (u, v) = (0, 0). When γ = −α, two symmetric fixed pointsappear at v = ±√(αg − 1)/αgr1, and u = −gv, where g = (β +1)/(α+γ).Then, if these points are stable, one could have a situation analogous toa thermodynamic system driven to a spinodal decomposition situation, inwhich two phases coexist and evolve in time. We have shown11 that thesolutions of this bistable system are travelling wave fronts, or solitons.

If one chooses C = 0, D = 0, η = 1, and h = −2.5 in the model on thefirst row of Table 1, one gets the phase diagram shown in Fig. 14

In region 2 the central fixed point at (0,0) is unstable and the other twofixed points are stable. We can predict the existence of traveling wave fronts,an almost universally recognized feature of bistable reaction-diffusion sys-tems42 . We expect heteroclinic trajectories in phase space connecting thetwo stable fixed points with the central hyperbolic point. This was corrob-orated by numerical calculations in one dimension, where one sees kinksmoving in both directions with constant velocity. The velocity depends onthe integral of the fields over the whole space, if this is zero, implying thatthe profile of a kink is symmetric around zero, the kink does not move,but of course, this situation is not stable. Another important feature isthat there is a characteristic length that dictates the average separation ofkinks.

290 R. A. Barrio

Fig. 14. Phase diagram of Eq. 8 when there are two symmetrical fixed points at v0 =±√b/g − h = ±√

f and u = −gv. The character of the eigenvalues changes in the regionslabeled by numbers. In region 2 there are oscillating stable modes and in region 5 thesemodes become unstable. Taken from Ref. 11.

Much more interesting is the situation in two dimensions, because thevelocity of the front depends on its local curvature. This results in havingrotating fronts around the point in which the curvature changes sign. Thisis shown in the numerical calculation depicted in Fig. 15

These patterns are very similar to the ones visualized in the electricalactivity of chicken hearts49 , and we are developing a model to tackle thisproblem. One of the important features of the heart is that when it doesnot function well it goes to fibrillation, producing spiral waves. In the bor-der between region 2 and region 5 of Fig. 14 one has limit cycle solutionsand the possibility of spiral waves, as the ones appearing in the Belousov–Zhabotinskii reaction. It is then very simple to model the malfunctioning ofthe heart by only changing the parameter f . In Fig. 16 there is an exampleof this.

4.5. Anomalous Diffusion in Turing Systems

There are many real situations when diffusion is anomalous, in the sensethat the mean square displacement does dot grow linearly in time. Thisoccurs in many fields of science, as physics50 , biology51 , geology and others.Therefore, it is interesting to investigate the diffusion driven instabilityunder anomalous conditions.

Normal diffusion is a consequence of having completely random walks,which by the virtue of the Central Limit Theorem produce a normal


(a) (b)

(c) (d)

Fig. 15. Four snapshots of a pattern consisting of solitary wave fronts. Observe theappearance of a characteristic length and the peculiar way in which the pattern seemsto rotate around some points. Taken from Ref. 11.

Fig. 16. The pattern on the left (from Fig.15(d)) was used as the initial conditions toproduce the pattern on the right. The parameters were g = 0.165, h = −2.5, and thecritical value fc = 0.5043.

Gaussian distribution of step-lengths in the thermodynamic limit, butthis is not always the case. For example, diffusion could be anomalous in

292 R. A. Barrio

inhomogeneous or fractal media. For instance, Levy flights present a dis-tribution with diverging second moment. One way of describing anomalousdiffusion is by replacing the normal derivatives in Fick’s equation by frac-tional derivatives, either in time or in space. In the latter case52 one replacesthe usual Laplacian operator by ∇2 → (aDνx

x +a Dνyy +a Dνz

z ), where thefractional operators for each coordinate are defined as

aDνxx =

1Γ(2 − νx)

d2x

∫ x

a

u(x′)(x − x′)νx−1

dx′. (17)

One of the important features of these fractional operators is that theFourier components are eigenfunctions of these operators, for instance,

−∞Dνxx eikx = (ik)νxeikx. (18)

This shows that the mean square displacement grows in time to a powergiven by the anomalous diffusion parameter ν. There is a discrete form of theRiemann–Liouville expression (Eq.17), convenient to be used in numericalcalculations, known as the Grunwald-Letnikov formula53 .

We studied anomalous diffusion in the BVAM model in one and twodimensions54 , with parameters similar to the ones producing Fig. 1. Wefound two important effects:

(1) Asymmetric anomalous diffusion (only one sign for k in Eq. 18)produces moving patterns. In two dimensions their velocity depends onthe anomalous exponents νx and νy, with direction given by the relativedeviation of these exponents from 2 (when the patterns are stationary).This in consequence facilitates the healing of local defects in the pattern.The hexagonal dotted patterns are perfect, and the striped patterns aredislocation free.

(2) The range of D, for which a Turing instability is possible, widensup. One can find Turing patterns even when D = 1.

Fig. 17. Anomalous diffusion pattern rotating on a sphere.


To illustrate the effects of anomalous diffusion in 2 dimensions, in Fig.17 we show a rotating pattern obtained on a spherical surface and νφ = 2.Observe that as the velocity of the front is constant, curls are formed atthe poles.

5. Final Remarks

I hope I have shown that Turing models can be used to describe a wide rangeof phenomena in many fields of Science. However, I would like to stress thatnot all striped or spotted arrangements with a conserved wavelength needto be produced by a Turing instability. This is more noticeable in Geologyand in Biology. It turns out that banded patterns of vegetation could beexplained by a Turing model, but the ecological variables at play are notclear. Diffusion, chemical reactions and precipitation are important in ge-ological formations, as the Liesegang Rings, but this does not necessarilymean that a simple Turing model would account for all facts. Particularlycumbersome is the field of mathematical biology, where the relevance ofTuring mechanisms is still debatable. The general credo that organizationin living organisms is dictated solely by the genes is well spread now, al-though not necessarily correct. The conspicuous bands in the abdomen ofthe fruit fly, at first glance, look very much as a Turing pattern, yet it hasbeen demonstrated that the actual mechanism that produces these bands ispurely genetic. The case of Drosophila is a shameful failure of Turing mod-els and a warning to theorists, as Einstein put it very rightly, “a Theoryshould be as simple as possible, but not simpler”. However, very recently aninterplay between a Turing mechanism and genes has been demonstratedin the formation of the pattern of hair follicles in mice55 . I am sure that inthe near future we will see more evidences like this, since “not yet very wellknown physical laws” should facilitate the genes mission to organize theenormous number of components to reach a final anatomic form withoutmajor errors.

We, physicists, are living very exciting times modeling phenomena inmany fields that were not touched by us before. Particularly in Biology,where qualitative models are not helpful anymore to understand the com-plexity of biological processes. Quantitative knowledge is needed, and thisis obtainable not only from elaborate experiments, but also from mathe-matical models.

294 R. A. Barrio

Acknowledgments

I am grateful to Ignazio Licata for the invitation to write this paper, andto the Helsinki University of Technology, Laboratory of ComputationalEngineering (LCE) for an adjunct professorship. Most of the work pre-sented here was financed by the project F-40615 from CONACyT. I alsothank my many collaborators, particularly C. Varea, J.L. Aragon, M. Torres(r.i.p.), K. Kaski, Teemu Leppanen, Mikko Kartunen, Klaus Kytta, DamianHernandez, Faustino Sanchez, and P.K. Maini.

References

1. D’ Arcy W. Thompson (1961) On Growth and Form (abridged edition),Cambridge Univ. Press, UK.

2. A. M. Turing (1952) Phil. Trans. R. Soc. Lond. B237, 37.3. R.A. Barrio, C. Varea, J.L. Aragon, and P.K.(1999) Maini Bull. Math. Biol.,

61, 483 .4. A. Kelly (1967) J. Diff. Eqns, 3, 546.5. V. Castets, E. Dulos, J.. Boissonade, and P.D. Kepper (1990) Phys. Rev.

Lett. 64, 2953.6. P.K. Maini, K.J. Painter, H.N.P. Chau (1997) Faraday Transactions, 93

(20), 3601-3610.7. B.J. Dickson (2002) Science 298, 1959.8. S. Schnell, R. Grima, P.K. Maini (2007) Multiscale modeling in biology,

Amer. Sci., 95, 134.9. R.T. Liu, S.S. Liaw and P.K. Maini (2007) J. Korean Phys. Soc., 50, 234.

10. T. Leppanen (2005) in Current Topics in Physics, R.A. Barrio, and K.K.Kaski, eds., Imperial College Press p. 199.

11. C. Varea, D. Hernandez, and R.A. Barrio (2007) Soliton behaviour in abistable reaction diffusion model, J. Math. Biol (in press).

12. R.A. Fisher, Ann. Eugenics (1937) 7, 353., see also Ref. 14 p. 237.13. V.M. Kenkre (2004) Physica A, 342, 242.14. J.D. Murray (1980) “Mathematical Biology”, Springer-Verlag, Berlin, pp.63–

68.15. A. Gierer, H. Meinhardt, Kibernetic (1972) 12, 30.16. H. Meinhardt, and M. Klingler (1987) J. Theor. Biol., 126, 63.17. P. Gray and S.K. Scott (1985) J.Chem.Phys., 89, 22.18. J.E. Pearson (1993) Science 261, 189.19. O. Avila, D. Hernandez, R.A. Barrio, and Limei Zhang, Modeling Neurite

growth and guidance, (to be published).20. Y. Kuramoto (2003) “Chemical Oscillations, Waves, and Turbulence”,

Dover Publ. Inc., New York.21. I. Prigogine, R. Lefever (1968) J. Chem. Phys. 48, 1695.22. R.J. Field and R.M. Noyes (1974) J.Chem.Phys., 60, 1877.


23. J.J. Tyson (1981) “On scaling the Oregonator equations in nonlinear Phe-nomena in Chemical Dynamics” (C. Vidal and A. Pacault, Eds.), Springer-Verlag, Berlin, pp. 222–227.

24. J. Schnackenberg (1979) J. Theor. Biol. 81, 389.25. V. Dufiet and J. Boissonade, J.Chem.Phys., 96, 664 (1992).26. I. Lengyel and I.R. Epstein (1991) Science, 251, 650.27. R. FitzHugh (1961) Biophysics J., 1, 445 ; J. S. Nagumo, S. Arimoto, and

S. Yoshizawa (1962) Proc. IRE, 50, 2061.28. Kyoung J. Lee (1997) Phy. Rev. Lett., 79, 2907.29. S. Kondo and R. Asai (1995) Nature, 376, 765–768.30. J.L. Aragon, C. Varea, R. A. Barrio and P.K. Maini (1998) FORMA 13 154.31. Stanislav Frank (1973) “Encyclopedie Illustree des Poissons”, Grund, Paris

p. 389.32. R. Plaza, F. Sanchez-Garduno, P. Padilla, R.A. Barrio, and P.K. Maini

(2004) Jour. of Dynamics and Differential Equations 16, 1093 .33. C. Varea, J.L. Aragon and R.A. Barrio (1999) Phys. Rev. E, 60, 4588.34. M. Faraday (1831) Philos. Trans. R. Soc. London 121, 299.35. R.A. Barrio, J.L. Aragon, C. Varea, M. Torres, I. Jimenez, and F. Montero

de Espinosa (1997) Phys. Rev. E, 56, 4222.36. J.B. Swift and P.C: Hogenberg (1977) Phys. Rev. A, 15, 319.37. S.V. Kiyashko, L.N. Korsinov, M.I. Rabinovich, and L.S. Tsimring (1996)

Phys. Rev. E, 54, 5037.38. S. Fauve (1995) in Dynamics of nonlinear and disordered systems, (Edited

by G. Martinez-Mekler and T.H. Seligman, World Scientific, Singapore) p.67.

39. J.L. Aragon, M. Torres, D. Gil, R.A. Barrio, and P.K. Maini (2002) PhysRev. E, 65, 051913.

40. R. Lifshitz and D.M. Petrich (1997) Physi. Rev. Lett., 79, 1271.41. P. Engels, C. Atherton, and M.A. Hoefer(2007) Phys. Rev. Lett., 98, 095391.42. J.D. Murray (2003) “Mathematical Biology II: Spatial Models and Biomed-

ical Applications”, Springer-Verlag, Berlin.43. L. Yang, M. Dolnik, A. Zhabotinsky, and I. Epstein (2002) Phys. Rev. Lett.,

88, 208303 .44. L. Yang and I. Epstein (2003) Phys. Rev. Lett., 90, 178303.45. L. Yang and I. Epstein (2004) Phys. Rev. E, 69, 026211.46. for an animated exhibition of these patterns see for instance:

http://hopf.chem.brandeis.edu/yanglingfa/pattern/research frm.html.47. I. Berenstein, M. Dolnik, L. Yang, A. Zhabotinsky, and I. Epstein (2004)

Phys. Rev. E, 70, 046219.48. K. Kytta, K. Kaski, and R.A. Barrio, Complex Turing patterns in nonlin-

early coupled systems (to be published).49. G. Bub, A. Shrier, and L. Glass (2002) Phys. Rev. Lett. 88, 058101.50. K.W: Gentle, et al. (1995) Phys. Plasmas, 2, 2292.51. A.T: Winfree (1997) Int. J. Bifur. Chaos Appl. Sci. Eng., 7, 487.52. B.I. Henry and S.L. Wearne (2002) SIAM J. Appl. Math. 62, 870.

296 R. A. Barrio

53. I. Podlubny (1999) Fractional Differential Equations, Ac. Press, (SanDiego).

54. C. Varea and R.A. Barrio (2004) J. Phys.: Condens. Matter, 16, S5081.55. S. Sick, S. Reinker, J. Timmer, and T. Schlake (2006) Science, 314, 1447.

297

Primordial Evolution in the Finitary Process Soup

Olof Gornerupa and James P. Crutchfieldb

aComplex Systems Group, Department of Energy and EnvironmentChalmers University of Technology, 412 96 Goteborg, Sweden

bCenter for Computational Science & Engineering and Physics Department,University of California, Davis, One Shields Avenue, Davis CA 95616, USA

[email protected]@cse.ucdavis.edu

A general and basic model of primordial evolution—a soup of reacting fini-tary and discrete processes—is employed to identify and analyze fundamentalmechanisms that generate and maintain complex structures in prebiotic sys-tems. The processes—ε-machines as defined in computational mechanics—andtheir interaction networks both provide well defined notions of structure. Thisenables us to quantitatively demonstrate hierarchical self-organization in thesoup in terms of complexity. We found that replicating processes evolve thestrategy of successively building higher levels of organization by autocataly-sis. Moreover, this is facilitated by local components that have low structuralcomplexity, but high generality. In effect, the finitary process soup sponta-neously evolves a selection pressure that favors such components. In light ofthe finitary process soup’s generality, these results suggest a fundamental lawof hierarchical systems: global complexity requires local simplicity.

Keywords: Structural Complexity; Entropy; Information; Computational Me-chanics; Population Dynamics; Hierarchical Dynamics; Emergence; Evolution;Self-Organization; Autocatalysis; AutopoiesisPACS(2006): 89.75.k; 89.75.Fb; 82.39.Rt; 65.40.Gr; 05.70.a; 02.70.c

1. Introduction

The very earliest stages of evolution—or rather, pre-evolution—remain amystery. How did structure emerge in a system of simple interacting objects,such as molecules? How was this structure commandeered as substrate forsubsequent evolution—evolution that continued to transform the objectsthemselves? One wonders if this recursive interplay between structure anddynamics facilitated the emergence of complex and functional organiza-tions.

298 Olof Gornerup and James P. Crutchfield

Since these questions concern the most fundamental properties of evo-lutionary systems, we explore them using principled and rigorous methods.

To build a suitable model a few basic ingredients are required. First,one needs some type of elementary objects that constitute the state ofthe system at its finest resolution. Second, one needs rules for how theobjects interact. Third, one needs an environment in which the objectsinteract. Fourth, one needs quantitative and calculable notions of structureand organization. These requirements led us to the finitary process soupmodel of primordial evolution1 . Simply stated, the soup’s ingredients are,in order, ε-machines, their functional composition, a flow reactor, and thestructural complexity Cµ of ε-machines.

After explaining each of these ingredients, we will relate the model toclassical replicator dynamics by reducing the soup to a special case. We thenmove on to contrast the limited case with the full-fledged finitary processsoup as a constructive, unrestricted dynamical system.

2. Objects: ε-machines

Here we employ a finite-memory process called an ε-machine2–4 , as ourpreferred representation of an evolving information-processing individual.Using a population of ε-machines is particularly appropriate in studyingself-organization and evolution from an information-theoretic perspectiveas they allow quantitative measurements of storage capacity and random-ness. Rather than using the abstraction of a formal language—an arbitraryfinite set of finite length words—we consider a discrete-valued, discrete-timestationary stochastic process described by a bi-infinite sequence of randomvariables St over an alphabet A:

↔S= ...S−1S0S1.... (1)

A process stores information in a set of causal states that are equiva-lence classes of semi-infinite histories that condition the same probabilitydistribution for future states. More formally, the causal states S of a pro-

cess are the members of the range of the map ε :←S 7→ 2

←S from histories

to sets of histories:

ε(←s ) = ←s ′|P(

→S | ←S=

←s ) = P(

→S | ←S=

←s′) , (2)

where 2←S denotes the power set of

←S. Further, let S ∈ S be the current

casual state, S ′ its successor, and→S

1

the next symbol in the sequence (1).The transition from one causal state Si to another Sj that emits the symbol


s ∈ A is given by a set of labeled transition matrices: T = T (s)ij : s ∈ A,

where

T(s)ij ≡ P(S ′ = Sj ,

→S

1

= s|S = Si). (3)

The ε-machine of a process is the ordered pair S,T . One can show thatit is the minimal, maximally predictive causal representation of the pro-cess4 . Unlike a general probabilistic ε-machine, for simplicity, here we takecausal-state transitions to have equal probabilities. The finitary ε-machinescan be thought of as finite-state machines with a certain properties4 : (1)All states are start states and accepting states; (2) All recurrent states forma single strongly connected component; (3) All transitions are determinis-tic: A causal state together with the next value observed from the processdetermines a unique next causal state; And (4) the set of causal states isminimal. Here we use an alphabet of input and output pairs over a binaryalphabet: A = 0|0, 0|1, 1|0, 1|1. This implies that the ε-machines work asmappings between sets of strings. In other words, they are transducers5 .

A B

1|0

0|1

A B

0|0

1|1

A 1|10|0

TB TCTA

Fig. 1. Three examples of ε-machines. TA represents the identity function and has thecausal state A. TB has two causal states (A and B), accepts the input string 1010 . . . or0101 . . ., and operates by flipping 0s to 1s and vice versa. TC has the same domain andrange as TB , but maps input strings onto themselves.

In contrast to prior models of pre-biotic evolution, ε-machines are simplyfinitely-specified mappings. More to the point, they do not have two sepa-rate modes of representation (information storage) or functioning (transfor-mation). The advantage is that there is no assumed distinction between geneand protein6,7 or between data and program8–11 . Instead, one recovers thedichotomy by projecting onto (i) the sets that an ε-machine recognizes andgenerates and (ii) the mapping between these sets. Examples of ε-machinesare shown in Figure 1.

3. Interaction: Functional Composition

The basic pairwise interaction we use in the finitary process soup is func-tional composition. Two machines interact and produce a third machine—


their composition. Composition is not a symmetric operation. Machine TA

composed with another TB does not necessarily result in the same machineas TB composed with TA: TB TA 6= TA TB .

The upper bound on the number of states of the composition is theproduct of the number of states of the parents: |TB TA| ≤ |TB | × |TA|.Hence, there is the possibility of exponential growth of states and machinecomplexity if machines are iteratively composed. A composition, though,may also result in a machine with lower complexity than those of its parents.

3.1. Interaction Networks

We represent the interactions among a set of ε-machines with an interactionnetwork G which is a a graph whose nodes correspond to ε-machines andwhose transitions correspond to interactions. If Tk = Tj Ti occurs in thesoup, then the edge from Ti to Tk is labeled Tj . One may represent G withthe binary matrices:

G(k)ij =

1 if Tk = Tj Ti

0 otherwise.(4)

Consider the ε-machines in Fig. 1, for example. They are related via com-position according to the interaction graph shown in Fig. 2, which is givenby the matrices

G(A) =

1 0 00 0 00 0 0

, (5)

G(B) =

0 1 01 0 10 1 0

, (6)

and

G(C) =

0 0 10 1 01 0 1

. (7)


TA

TB TC

TB TC

TB

TB

TA

TA, TC TA, TC

Fig. 2. Interaction network of the ε-machines in Fig. 1. There is a transition, for ex-ample, that is labeled TC from the node TA to the node TC , since TA composed withTC results in TC (in fact, each transition from TA has the same label as the label of itsrespective sink node since TA is the identity function).

3.2. Meta-machines

For a machine to survive in its environment somehow it needs to producecopies of itself. This can be done directly by self-reproduction, e.g. TATA =TA, or it can be done indirectly in cooperation with other machines: e.g.,TA facilitates the production of TB , which facilitates the production of TC ,which, in turn, closes the loop by facilitating the production of TA. Inother words, there can be sets of machines that interact with each otherin such a way that they collectively self-reinforce the overall productionof the set. This leads to the notion of an autonomous and self-replicatingentity, which we call a meta-machine. Inspired by Maturana and Varela’sautopoietic set12 , Eigen and Schuster’s hypercycle13 , and Fontana andBuss’ organization14 , we define a meta-machine Ω to be a connected set ofε-machines whose interaction matrix consists of all and only the members ofthe set. That is, a set Ω is a meta-machine if and only if (1) the compositionof two ε-machines from the set is always itself a member of the set:

∀Ti, Tj ∈ Ω ; Tj Ti ∈ Ω, (8)

(2) all ε-machines in the set can arise from the composition of two machinesin the set:

∀Tk ∈ Ω ; ∃Ti, Tj ∈ Ω, Tk = Tj Ti, (9)

and (3) there is a nondirected path between every pair of nodes in Ω’sinteraction network GΩ. The third property ensures that there is no subsetof Ω that is isolated from the rest of Ω under composition. Consider, for


example, the union of two self-replicators, TA and TB , for which TB TA =TA TB = T∅. According to property (3), they are not a meta-machine.

4. Complexity Measures: Cµ

In previous computational pre-biotic models, the objects have been rep-resented by, for example, assembly language codes8–11 , tags15,16 , λ-expressions17 and cellular automata18 . We employ ε-machines insteadmainly for one reason: there is a well developed theory (computationalmechanics) of their structural properties. Assembly language programs andλ-expressions, for instance, are computational universal representations andso one knows that it is not possible to calculate their complexity5 .

For finitary ε-machines, in contrast, complexity can be readily definedand analytically calculated in closed form. Define the stochastic connectionmatrix of an ε-machine M = S, T as T ≡ ∑

s∈A T (s). The probabilitydistribution pS over the states in S—how often they are visited—is givenby the normalized left eigenvector of T associated with eigenvalue 1.

The structural complexity Cµ of M is the Shannon entropy of the dis-tribution given by pS ,

Cµ(M) ≡ −∑

v∈Sp(v)S log2 p

(v)S . (10)

The structural complexity of an ε-machine is the amount of informationstored in the distribution over S, which is the minimum average amount ofmemory needed to optimally predict future configurations4 .

To measure the diversity of interactions in the soup we define the inter-action network complexity Cµ(G) to be the Shannon entropy of the distri-bution of effective transition probabilities in the graph G. We consider, inparticular, only the transitions that have occurred between machine typesthat are present. That is,

Cµ(G) ≡ −∑

pi,j,k 6=0

υkij log2 υk

ij , (11)

where

υkij =

pipj/

∑υk

ij if Tk = Tj Ti has occurred

0 otherwise.(12)

and pi is the fraction of machines of type i in the population. To monitorthe emergence of actual and functional reproduction paths, we consideronly those interactions that occurred in the population.


5. Framework: The Soup

TA TB

TC

TDTR

Fig. 3. A schematic illustration of the finitary process soup. Two ε-machines, TA andTB , are composed and produce a third machine TC , or a random machine TR is intro-duced to the soup. In either case, another randomly selected machine TD is removed tomaintain a fixed population size. Note that this is a well stirred setting, and so there isno spatial relationship in the population.

The ε-machines interact in a well stirred reactor with the following iterateddynamics:

(1) Production and influx :

(a) With probability Φin generate a random ε-machine TR.(b) With probability 1− Φin (reaction):

i. Select TA and TB randomly.ii. Form the composition TC = TB TA.

(2) Outflux :

(a) Select an ε-machine TD randomly from the population.(b) Replace TD with the ε-machine produced in the previous step—

either TC or TR.

TR is uniformly sampled from the set of all two-state ε-machines in oursimulations (see below). This sampling is also used when initializing thepopulation. The insertion of TR corresponds to an influx while the removalof TD corresponds to an outflux. The latter keeps the population size con-stant. See Fig. 3 for a schematic illustration. There is no spatial dependence


in this version of the soup as ε-machines are sampled uniformly from thepopulation for each replication and removal.

6. Closed Population Dynamics

To familiarize ourselves with the model we first examine a simple base case:a soup with no influx that is initialized with machines taken from a finite setwhich is closed under composition. This case is also intended to work as abridge between classical population dynamics and the general, constructivedynamics of the finitary process soup. The closure with respect to compo-sition enables us to describe the system’s temporal dynamics of ε-machineconcentrations by a coupled system of ordinary differential equations. Inthe limit of an infinite soup size, the rate equation of concentration pk ofmachine type Tk is given by

pk = ψk − Φoutpk, k = 1, ..., n, (13)

where the conditional production rate ψk is the probability that Tk is pro-duced given that two randomly sampled machines are paired:

ψk =n∑

i,j=1

αkijpipj , (14)

and αkij is a second-order reaction rate constant:

αkij =

1 if Tk = Tj Ti

0 otherwise.(15)

The outflux Φout equals the total production rate of the soup—i.e., theprobability that a reaction occurs given that two ε-machines are paired. Itkeeps the size of the soup constant:

Φout(t) =n∑

i=1

ψi. (16)

Given a soup with no influx, Φin = 0, that hosts machines which are mem-bers of a set that is closed under composition, the distribution dynamicscan alternatively be predicted from by its interaction network:

p(k)t = pt−1 · G(k)

ij · pTt−1Z

−1, (17)

where p(k)t is the frequency of ε-machine type k at time t and Z−1 is a

normalization factor. This approximates the soup’s elements as updatingsynchronously.


We illustrate the closed case by initiating the soup with machinesthat consist of only a single state—mono-machines. There are 15 mono-machines, the null (transition-less) transducer is excluded, and they forma closed set under composition. See Fig. 4 for the temporal dynamics oftheir respective frequencies. Nine machine types remain in the populationat equilibrium. They form a meta-machine M with Cµ(M) = 5.75 bits.In this case, since Cµ(Ti) = 0 for all mono-machines Ti, the population’sstructural complexity derives only from its interaction network.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

p

t/N

Fig. 4. A simple base case: Machine type frequencies of mono-machines as functions oftime. N (= 100, 000) denotes the population size. Dashed lines: simulation; solid lines:Eq. (17). (Reprinted with permission from [1])

7. Open Population Dynamics

We now move on to the general case of a soup with positive influx rateconsisting of ε-machines of arbitrary size. The soup then constitutes a con-structive dynamical system where there is a mutual dependence betweenits equations of motion and the individuals. Due to the openness, Eqs.(13)-(17) do not necessarily apply. We therefore turn to simulations.

In order to study dynamics that is ruled solely by compositional trans-formations we first set the influx rate to zero. A fine-grained description ofthe soup’s history on the ε-machine level is given by a genealogy—a recordof descent of machine types. By studying the example in Fig. 5, a simula-tion with N = 100 individuals, one important observation is that nearlyall the ε-machine types that are present in the soup’s initial populationare replaced over time. Thus, genuine novelty emerges, in contrast to the


closed soup just described. Initially, there is a rapid innovation phase inwhich novel machines are introduced that displace the bulk of the initialmachines. The degree of innovation flattens out, along with the diversity ofthe soup, and eventually vanishes as the population becomes increasinglyclosed under composition.

20

40

60

80

100

120

140

160

180

0 200 400 600 800 1000

Mac

hine

Typ

e

t

Fig. 5. Genealogy of ε-machine types in a soup with 100 machines. A solid line denotesthat a machine type is present in the soup. Dashed lines (drawn from the parents tothe child) denote composition. Note that almost the whole set of initial ε-machine types(with one exception) is replaced by the dynamics.

To monitor the soup’s organization over time, we superimpose Cµ(G)time series from several runs in Fig. 6. One sees that plateaus are formed.These can be explained in terms of meta-machines. In addition to captur-ing the notion of self-replicating entities, meta-machines also describe aninvariant set of the population dynamics. That is, formally,

Ω = G Ω, (18)

where Ω is the set of ε-machines present in the population and G is theirinteraction network. These invariant sets can be stable or unstable underthe population dynamics.

Consider, for example, the meta-machine in Fig. 2. It is unstable, sinceTAs are only produced by TAs, and will decay over time to the meta-machineof Fig. 7. This also illustrates, by the way, how trivial self-replication isspontaneously attenuated in the soup.

The plateaus at Cµ(G) = 4 bits, Cµ(G) = 2 bits, and Cµ(G) = 0 bits


Fig. 6. Decomposition of meta-machines in a soup with no influx. Superimposed plotsof Cµ(G) from 15 separate runs with N = 500. Cµ(G) is bounded by 4 bits while a4-element meta-machine (shown), denoted Ω4, is the largest one in the soup. Ω4 decaysto Ω2, a 2-element meta-machine (shown) due to fluctuations, that in turn decays to Ω1,a single self-reproducing ε-machine.

TB TC

TB

TB

TC TC

Fig. 7. The resulting meta-machine when the meta-machine in Fig. 2 decays under thepopulation dynamics of Eq. (17).

correspond to the largest meta-machine that is present at that time. Sincea meta-machine by definition is closed under composition, it itself does notproduce novel machines; thus, one has the upper bound of Cµ(G). As ameta-machine is reduced due to an internal instability or sampling fluctu-ations by the outflux, the upper bound of Cµ(G) is lowered. This results ina stepwise and irreversible succession of meta-machine decompositions.Fig. 6 shows only three plateaus. In principle, however, there is one plateaufor every meta-machine that at some point is the largest one in the popula-tion. The diagram in Fig. 8 summarizes our results from a more systematicsurvey of spontaneously generated meta-machine hierarchies in simulationsof soups with 500 ε-machines.

We now examine the effects of influx by studying the population-averaged ε-machine complexity 〈Cµ(T )〉 and the run-averaged interactionnetwork complexity 〈Cµ(G)〉 as a function of t and Φin, see Fig. 9.

The average ε-machine complexity 〈Cµ(T )〉 increases rapidly initially


1

2

3

4

5

6

8

7

Decompositio

n

9

10Compositio

n|M| ⇒

Fig. 8. Composition and decomposition hierarchy of meta-machines. Dots denote self-

replicating ε-machines, solid lines denote TATB−→ TC transitions and dashed lines denote

equivalent TBTA−→ TC transitions. The label of the source node and the transition label

are interchanged in the latter transition type. This results in a redundant representationof the interaction network, which is used to show how the meta-machines are related.The interaction networks are shown in a simplified way according to Ω4; cf. Fig. 6.

before declining to a steady state. The average interaction network com-plexity 〈Cµ(G)〉 is relatively high where the average structural complexityof the ε-machines is low, and is maximized at Φin ≈ 0.1. Higher influx rateshave a destructive effect on the populations’ interaction network due to thenew individuals’ low reproduction rate. 〈Cµ(T )〉 is, in contrast, maximized


0 20

40 60

80 t/N 10-3

10-2 10-1

100

Φin

0

1

2

3

4

<Cµ(T)>

(a)

0 20

40 60

80 t/N 10-3

10-2 10-1

100

Φin

0

1

2

3

4

<Cµ(G)>

(b)

Fig. 9. (a) Population- and run-averaged ε-machine complexity 〈Cµ(T )〉 and (b) run-averaged interaction network complexity 〈Cµ(G)〉 as a function of time t and influx rateΦin for a population of N = 100 objects. (Reprinted with permission from1).

at a relatively high influx rate (Φin ≈ 0.75) at which 〈Cµ(G)〉 is relativelysmall. The maximum network complexity Cµ(G) of the population growslinearly at a positive rate of approximately 7.6 · 10−4 bits per replication.

8. Discussion

We presented a conceptual model of pre-biotic evolution: a soup consistingof objects that make new objects1 . The objects are ε-machines and theygenerate new ε-machines by functional composition. The soup constitutesa constructive dynamical system since the population dynamics is not fixedand may itself evolve along with the state space it operates on. Specifically,the dimension of the state space changes over time, which is reminiscent of


the constructive population dynamics associated with punctuated equilib-ria19 .

In principle, this allows for open-ended evolution. The quantitative es-timate quoted above for the linear growth of the interaction network com-plexity supports this intriguing possibility occurring in the open finitaryprocess soup. In the case of no influx, though, the system reaches a steadystate where the soup consists of only one self-replicator. Growth and main-tenance of organizational complexity requires that the system is dissipative;i.e., that there is a small, but steady inflow of random ε-machines. Notably,in this case, the soup spontaneously evolves hierarchical organizations inthe population—meta-machines that in turn are organized hierarchically.

These hierarchies are assembled from noncomplex, general individual ε-machines. In this way, the soup’s emergent complexity derives largely froma network of interactions, rather than from the unbounded increase in thestructural complexity of individuals. It appears, therefore, that higher-ordercomplex organization not only allows for simple local components but, infact, requires them.

Acknowledgments

This work was supported at the Santa Fe Institute under the NetworksDynamics Program funded by the Intel Corporation and under the Com-putation, Dynamics and Inference Program via SFI’s core grants from theNational Science and MacArthur Foundations. Direct support was providedby NSF grants DMR-9820816 and PHY-9910217 and DARPA AgreementF30602-00-2-0583. O.G. was partially funded by PACE (Programmable Ar-tificial Cell Evolution), a European Integrated Project in the EU FP6-IST-FET Complex Systems Initiative, and by EMBIO (Emergent Organisationin Complex Biomolecular Systems), a European Project in the EU FP6-NEST Initiative.

References

1. Crutchfield, J. P. and Gornerup, O. (2006) J. Roy. Soc. Interface 3, 345–349.2. Crutchfield, J. P. and Young, K. (1989) Phys. Rev. Let. 63, 105–108.3. Crutchfield, J. P. (1994) Physica D 75, 11–54.4. Crutchfield, J. P. and Shalizi, C. R. (1999) Physical Review E 59(1), 275–283.5. Brookshear, J. G. (1989) Theory of computation: formal languages, au-

tomata, and complexity, Benjamin/Cummings, Redwood City, California.6. Schrodinger, E. (1967) What is Life? Mind and Matter, Cambridge Univ.

Press, Cambridge, United Kingdom.


7. vonNeumann, J. (1966) Theory of Self-Reproducing Automata, University ofIllinois Press, Urbana.

8. Rasmussen, S., Knudsen, C., Feldberg, P., and Hindsholm, M. (1990) InEmergent Computation : North-Holland Publishing Co. pp. 111–134.

9. Rasmussen, S., Knudsen, C., and Feldberg, R. (1992) In Artificial Life II: Pro-ceedings of an Interdisciplinary Workshop on the Synthesis and Simulationof Living Systems (Santa Fe Institute Studies in the Sciences of Complexity,Vol. 10) : Addison-Wesley.

10. Ray, T. S. (1991) volume XI, of Santa Fe Institute Stuides in the Sciences ofComplexity Redwood City, California: Addison-Wesley. pp. 371–408.

11. Adami, C. and Brown, C. T. (1994) In Artificial Life 4 : MIT Press pp.377–381.

12. Varela, F. J., Maturana, H. R., and Uribe, R. (1974) BioSystems 5(4), 187–196.

13. Schuster, P. (1977) Naturwissenschaften 64, 541–565.14. Fontana, W. and Buss, L. W. (1996) In S. Casti and A. Karlqvist, (ed.),

Boundaries and Barriers, : Addison-Wesley pp. 56–116.15. Farmer, J. D., Packard, N. H., and Perelson, A. S. (1986) Phys. D 2(1-3),

187–204.16. Bagley, R. J., Farmer, J. D., Kauffman, S. A., Packard, N. H., Perelson, A. S.,

and Stadnyk, I. M. (1989) Biosystems 23, 113–138.17. Fontana, W. (1991) volume XI, of Santa Fe Institute Stuides in the Sciences

of Complexity Redwood City, California: Addison-Wesley. pp. 159–209.18. Crutchfield, J. P. and Mitchell, M. (1995) Proc. Natl. Acad. Sci. 92, 10742–

10746.19. Crutchfield, J. P. (2001) When evolution is revolution—origins of innovation

In Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality,Accident, and Function Santa Fe Institute Series in the Sciences of Complex-ity pp. 101–133 Oxford University Press.

313

Emergence of Universe from a Quantum Network

Paola A. Zizzi

Dipartimento di Matematica Pura ed Applicata Via Trieste, 63 35121 PadovaItaly

[email protected]

We find that the very early universe can be described as a growing quantumnetwork, where the nodes (quantum logic gates) are the quantum fluctuationsof the metric, the connecting links are virtual black holes and the free linksare qubits. The quantum fluctuations of the metric and the virtual black holes,which make up the quantum foam, are the connected part of the network,while the qubits are the disconnected part. At each time step (here time isquantized in Planck time units), the net grows because of the inverse relationbetween the quantum information and the cosmological constant. In fact, thevirtual black holes carry virtual information, and a certain number of virtualstates are added to the memory register. These virtual states are transformedinto qubits (available quantum information) at the gates. The growth of thequantum network is responsible of the cosmic inflation. The connected part ofthe graph plays the role of the ”environment”, while the disconnected part playsthe role of the quantum state. The N-qubits state undergoes environmentaldecoherence at the end of inflation but, as the quantum foam is itself partof the network, we are in fact in presence of a kind of self-decoherence. Thequantum growing network of the early universe is strictly related to the problemof the cosmological constant Also, this model gives some hints to the solutionof the ”information loss” puzzle. Any computational system must obey thequantum limits to computation, given by the Margolous/Levitin theorem. Weshow that the quantum growing network saturates these bounds, thus it can beconsidered the ultimate network, just as black holes can be considered ultimatecomputers (as claimed by Seth Lloyd). Hence, in a sense, quantum space-timeitself turns out to be the most powerful tool for quantum computation andquantum communication.

Keywords: Quantum Gravity; Early Universe; Quantum Growing Network;Quantum Information; Complex SystemsPACS(2006): 04.60.m; 04.60.Kz; 04.60.Pp; 04.62.+v; 04.90.+e; 98.80.Cq;03.67.a; 89.75.k

314 Paola A. Zizzi

1. Introduction

Quantum space-time is space-time at the Planck scale. It is widely believedthat at that scale space-time is discrete, and has a foamy structure. How-ever, the proper structure of quantum space-time is not yet completelyunderstood. Once we will know the structure of quantum space-time, wewill also get the theory of quantum gravity, the theory which should recon-cile General Relativity and Quantum Mechanics, as the quantum featuresof gravity should appear just at the Planck scale. The two main theoriescandidate to become the final theory of quantum gravity are string theoryand loop quantum gravity. However, quite recently, it seems that a newstrand to quantum gravity is available, that of quantum computation.

As the whole universe at its very beginning (more precisely, the earlyinflationary universe) had a Planck size, it is strongly believed that the un-derstanding of the early universe could give fundamental hints to formulatethe theory of quantum gravity.

Our approach to quantum gravity in fact consists in applying quantuminformation theory to the very early universe.

In [1] we showed that the holographic principle [2] is strictly related toquantum information encoded in quantum space-time. However, any com-putational system must obey the quantum limits to computation, given bythe Margolous/Levitin theorem [3]. Thus, quantum computing space-time,should obey these bounds as well. Indeed, quantum space-time saturatesthese bounds, as it will be illustrated in the following.

Recently, Seth Lloyd [4] suggested that black holes can be regarded asultimate computers, as they saturate the quantum limits to computation.Moreover, Ng showed [5] that space-time foam [6] provides the quantumlimits to computation.

Finally, in a recent paper [7], we described the early inflationary universeas a quantum growing network (QGN), which saturates the quantum limitsto computation at each time step.

The QGN graph is made of two parts: the connected part which isthe ensemble of the connecting links and the nodes, and the disconnectedpart, which is made of free links out-going from the nodes. The connectinglinks are virtual black holes, quantum fluctuations of the vacuum, whichcarry virtual information (because of the inverse relation between quan-tum information and the quantized cosmological constant). The nodes arequantum fluctuations of the metric, which operate like quantum logic gates,and transform the virtual information into available information. The freelinks are the qubits. The ensemble of virtual black holes and quantum


fluctuations of the metric is what is called here ‘quantum foam’. Then, theultimate network appears to be an artefact of space-time foam.

If it is true that the QGN saturates the quantum limits to computation,it does that at each time step, because of its growth. This is a different casefrom that of the ultimate computer, where in principle the input power(which limits the speed of computation) is known once for all. In our case, itis not possible to know, for example, whether there is a maximum amountof quantum information processed by the QGN. The knowledge of thatwould be necessary to state a priori when the QGN decohered, givingrise to a classical universe (we recall moreover, that decoherence of theQGN would coincide with the end of inflation [7]). Nevertheless, there area few hints (considerations on the actual entropy of our universe) whichallow us to identify the speed of computation and memory space at themoment of decoherence. At this point, one could raise objections concerningdecoherence of an isolated system. However, as we said, the QGN can beconsidered as made of two subsystems: the connected part, which is thequantum foam, and the disconnected part which is the quantum state ofN qubits. We believe that decoherence is due to the quantum foam whichplays the role of the environment. But, as the quantum foam is part of theQGN, the QGN as a whole undergoes self-decoherence.

It should be noticed that, if the quantum state of N qubits did notdecohere at some earlier time, the present amount of entropy would behuge.

2. The Inflationary Universe as A Quantum GrowingNetwork

In a recent paper [7], the early inflationary universe was described as aquantum growing network (QGN). The speed up of growth of the network(inflation) is due to the presence of virtual qubits in the vacuum state ofthe quantum memory register. Virtual quantum information is created byquantum vacuum fluctuations, because of the inverse relation [7] betweenthe quantized cosmological constant Λn[8], and quantum information I:Λn = 1

I l2P, ( n=0,1,2,3. . . ) , where the quantum information I is [1]: I ≡

N = (n + 1)2. In this model, time is quantized in Planck time units:tn = (n+1)tP , (where tP ∼= 10−43 sec is the Planck time). At each time steptn, then, the increase of quantum information is given by: ∆I = −∆Λ

l2P Λ2 =2n + 3.

That is, at each time step tn, there are 2n+3 extra bits (virtual states)in the vacuum state of the quantum memory register.

316 Paola A. Zizzi

The 2n+3 virtual states occurring at time tn, are operated on by a quantum

logic gate Un [7] at time tn′ = tn+1, where: Un =v=2n+3∏

j=1

Had(j), and Had

(j) is the Hadamard gate Had = 1√2

(1 11 −1

)operating on bit j. Then, the

virtual states, are transformed into 2n’+1 qubits at time tn′ .This results into a quantum growing network, where the nodes are the

quantum logic gates, the connecting links are the virtual states, and thefree links are the qubits.At each time step, the total number of qubits (free links) is N = (n + 1)2.The rules of the growing quantum network that we consider, are resumedbelow.

At the starting time (the unphysical time t−1 = 0), there is one node,call it -1. At each time step tn, a new node is added, which links to theyoungest and the oldest nodes, and also carries 2n+1 free links. Thus, atthe Planck time t0 = tP , the new node 0 is added, which links to node -1and carries one free link. At time t1 = 2tP , the new node 1 is added, whichlinks to nodes -1 and 0, and carries three free links, and so on. In general,at time tn, there are:1) n + 2 nodes but only n+1 of them are active, in the sense that theyhave outgoing free links (node -1 has no outgoing free links).2) N = (n + 1)2 free links coming out from n+1 active nodes3) 2n+1 links connecting pairs of nodes4) n loops.

The N free links are qubits (available quantum information), the 2n+1connecting links are virtual states, carrying information along loops, then+1 active nodes are quantum logic gates operating on virtual states andtransforming them into qubits. In fact, notice that the number of free out-going links at node n is 2n+1, which is also the number of virtual states(connecting links) in the loops from node -1 to node n.

As it was shown in [7], the QGN saturates the quantum limits to com-putation [3] [4] [5], thus it can be viewed as the ultimate network. Theultimate computer [4] being a black hole, it would not be surprising thatthe ultimate network is quantum space-time itself.

3. Quantum Gravity Registers

A quantum memory register (the memory register of a quantum computer)is a system built of qubits. In [8] we showed that the early universe can beconsidered as a quantum register.


In our case, moreover, the quantum register grows with time. In fact, ateach time step tn = (n+1)tP ( with n=0,1,2. . . ), where tP ∼= 5.3×10−44 secis the Planck time, a Planckian black hole, which is in fact the 1-qubit state,acts as a creation operator and supplies the quantum register with extraqubits.

At time t0 = tP the quantum gravity register will consist of 1 qubit,at time t1 = 2tP the quantum gravity register will consist of 4 qubits, attime t2 = 3tP , the quantum gravity register will consist of 9 qubits, andso on. In general, at time tn, the quantum gravity register will consist ofN = (n + 1)2 qubits.

We will call |N〉 the quantum state of N=(n + 1)2 qubits.Now, let us consider a de Sitter horizon|Ψ(tn)〉 at time tn, with a dis-

crete area: An = (n + 1)2L2P of N pixels (where LP

∼= 1.6× 10−33cm is thePlanck length).

By the quantum holographic principle [1] we associate N qubits to thenth de Sitter horizon: |N〉 ≡ |Ψ(tn)〉.

The 1-qubit state is given by : |1〉 = Had |0〉 where Had is the Hadamardgate, which is a very important gate for quantum algorithms. Then, thestate |N〉 can be expressed as: |N〉 = (Had |0〉)N .

As time is discrete, there will be no continuous time evolution, there-fore there will not be a physical Hamiltonian which generates the timeevolution according to Schrodinger’s equation. In [1] we considered dis-crete unitary evolution operators Enm between two Hilbert spaces Hn andHm associated with two causally related ‘events’ |Ψn〉 and |Ψm〉. These”events” are de Sitter horizon states at times tn and tm respectively, withthe causal relation: |Ψn〉 ≤ |Ψm〉, for t≤n tm. The discrete evolution opera-tors: Enm : Hn → Hm are the logic quantum gates for the quantum gravityregister: Enm = (Had |0〉)M−N (with M = (m + 1)2, N = (n + 1)2.The discrete time evolution is: E0n |0〉 = |N〉 = |Ψfin〉.

As the time evolution is discrete, the quantum gravity register resemblesmore a quantum cellular automata than a quantum computer. Moreover,the quantum gravity register has the peculiarity to grow at each time step(it is self-producing).

4. Quantum Gravity Computation and Cybernetics

The quantum gravity registers are not proper quantum computers. Basi-cally, they better resemble quantum cellular automata, as time is discrete.

Quantum gravity registers do perform quantum computation, but ina rather particular way, that we shall call quantum gravity computation

318 Paola A. Zizzi

(QGC). The peculiarity of a quantum system which performs QGC, isthat it shares some features of self-organizing systems. We recall that self-organization is a process of evolution taking place basically inside the sys-tem, with minimal or even null effect of the environment.

In fact, the dynamical behaviour of quantum gravity registers followssome cybernetic principles:i) Autocatalytic growthAt each computational time step, the presence of a Planckian black hole(which acts as a creation operator), makes the quantum gravity registergrow autocatalytically.As N qubits represent here a de Sitter horizon with an area of N pixels, theautocatalytic growth, in this case, is exponential expansion, i.e., inflation.ii) AutopoiesisThe quantum gravity register produces itself. The components of the quan-tum gravity register generate recursively the same network of processes(applications of the Hadamard gate to the vacuum state) which producedthem.In this case recursion is defining the program in such a way that it may callitself:

|N〉 = (Hadamard |0〉)N .

iii) Self-similarityThis model of the early inflationary universe is based on the holographicprinciple [2] more in particular, on the quantum holographic principle [1].But each part of a hologram carries information about the whole hologram.So, there is a physical correspondence between the parts and the whole.iv) Self-reproduction

Can a quantum gravity register, as a unit, produce another unit witha similar organization? This possibility, which could be taken into accountbecause the quantum gravity register is an autopoietic system, (and onlyautopoietic systems can self-reproduce), is in fact forbidden by the no-cloning theorem, i.e. quantum information cannot be copied [9].

However, there is a way out. When the selected quantum gravity registercollapses to classical bits, it is not just an ordinary quantum register whichcollapses, but an autopoietic one. The outcomes (classical bits) carry alongthe autopoiesis. The resulting classical automaton is then autopoietic and,in principle, can self-reproduce.


5. Entanglement with the Environment

This superposed state will collapse to classical bits by getting entangledwith the emergent environment (radiation).

This entanglement process with the environment can be interpreted [1]as the action of a XOR (or controlled NOT) gate, which gives the output ofthe quantum computation in terms of classical bits: the source of classicalinformation in the post-inflationary universe. This phase can be illustratedin terms of Cellular Automata.

Cellular automata (CA) were originally conceived by von Neumann [10]to provide a mathematical framework for the study of complex systems.A cellular automata is a regular spatial lattice where cells can have anyof a finite number of states.The state of a cell at time tn depends only onits own state and on the states of its nerby neighbors at time tn−1 (withn ∈ Z). All the cells are identically programmed. The program is the setof rules defining how the state of a cell changes with respect of its currentstate, and that of its neighbours.

It holds that the classical picture of holography (given in terms of clas-sical bits) can be described by a classical CA.States: 0 or 1 (‘off’ or ‘on’).Neighbours: n at each time step tn = (n + 1)tP (this CA is autopoieticand grows With time).Rules: As there are two possible states for each cell, there are 2n+1 rulesrequired.

t0 = tP•

2 rules:1 → 10 → 0

t1 = 2tP• 4 rules:

11 → 010 → 101 → 100 → 0

t2 = 3tP• 8 rules:

000 → 0001 → 1100 → 1101 → 1

,

010 → 1011 → 0110 → 0111 → 0

t3 = 4tP• 16 rules,. . . and so on.

The rules force patterns to emerge (self-organization).By taking into account the ‘classical’ holographic principle, we are lead

to believe That at the end of inflation, the universe starts to behave asa classic CA, which self -organizes and evolves complexity and structure.

320 Paola A. Zizzi

We call it ‘Classical Holographic Cellular Automata’ (CHCA).It should be noted that the CHCA is made out of the bits which are

the outcomes of the collapse of the qubits of the quantum gravity registerwhich is an autopoietic quantum system. Then, the CHCA is an autopoieticclassical system.

There are two important consequences.i) The CHCA, being autopoietic, undergoes autocatalytic growth, and theclassical universe is still expanding. However, as classical computation isslower than quantum computation, the expansion is not anymore exponen-tial .ii) The CHCA, being a classical autopoietic system, can self-reproduce.The produced units will be able to perform both quantum and classicalcomputation.

We conclude by saying that in our model, the post-inflationary universefollows the laws of classical complex adaptive systems (systems at the edgeof chaos).

6. Quantum Computational Aspects of Space-time Foam

It was recently shown by Jack Ng [5], that space-time foam provides limitsto quantum computation. Here we investigate the quantum computationalaspects of spacetime foam in the context of the QGN model.

We identify the node ‘n’ as the quantum fluctuation of the metric onthe nnt slice [8] that is: ∆gn = 1

n+1 . The energy of node ”n” is then theenergy of the nnt quantum fluctuation of the metric: En = EP

n+1 , whereEP ≈ 1019GeV is the Planck energy.

Thus, the energy of node ‘n’ is the Planck energy divided by the num-ber n+1 of virtual Planckian black holes (connecting links of the kind:L−1,0, L0,1, L1,2, ....Ln−1,n). The energy of the nodes decreases as thenetwork grows. This fact is due to the increase of the number of virtualblack holes.

In particular, node ‘0’ has the maximum energy E0 = EP .The energy of the nodes decreases as the network grows.

This fact is due to the increase of the number of virtual black holes.We define ‘quantum foam’ the ensemble of the quantum fluctuations ofthe metric (the nodes) and the virtual black holes (connecting links:L−1,0, L0,1, L1,2, ....Ln−1,n...). What we call here ‘quantum foam’ is basedon the definition of ‘space-time foam’ due to Wheeler’ [6], where quantumfluctuations of the metric induce fluctuations of the topology. Moreover,we enclose the quantum fluctuations of the vacuum (virtual black holes) in


the definition of the ‘quantum foam’ as the two kinds of fluctuations arestrictly related to each other in this context. Hawking [10] also consideredvirtual black holes as constituents of space-time foam. He showed that thetopology of virtual black holes is: S2 × S2 (the 4-dimensional space-timebeing then a simply connected manifold). It should be noticed that these”bubbles” are not solutions of Einstein’s field equations. However, in ourcontext at least, this is not the whole story. In fact, at node ”0” one Eu-clidean Schwarzschild black hole of Planck size comes into existence, and itis not virtual, as it will give rise [8] to a quantum de Sitter space-time. Itstopology is: R1

(space) × S1(time) × S2.

This micro-black hole at node ‘0’ has an horizon area of one pixel (oneunit of Planck area) which, by the quantum holographic principle [1], en-codes one unit of quantum information (one qubit). Our claim is that,virtual black holes carry virtual information which is transformed into realinformation by the quantum fluctuations of the metric (the logic gates).In the transformation from virtual to real information, the topology itselfmust change.

Space-time foam invokes the existence of a minimum length scale [11],which is the Planck length. The Planck length is the length scale of quantumgravity. Given the Planck length as the minimum length scale, the properdistance between two events will never decrease beyond Planck length, andthe uncertainty relation holds: ∆x ≥ lP .

Moreover, in our case, an analogous uncertainty relation holds for timeintervals: ∆t ≥ tP , as we are dealing with time quantized in Planck timeunits.Then we get the simultaneous bound: ∆x∆t ≥ lP tP . For quantized space-time in Planck units, it is: ∆x = xm−xn = (m−n)lP ; ∆t = tm−tn = (m−n)tP , and the simultaneous bound becomes: ∆x∆t = (m−n)2lP tP ≥ lP tP .This bound is saturated for m=n+1.

Notice that we can also write: xn = ∆g−1n lP =

√I lP ; tn = ∆g−1

n tP =√I tP , from which it follows: xntn = ~∆g−2 = ~I, where I is the quantum

information. Thus, the quantum information encoded in quantum space-time, is a consequence of the foamy structure of quantum space-time itself.Notice that for n=0, we get: x0t0 = ~, with I=1 (at the very Planck scale,quantum information encoded in the foam is one qubit).

7. The QGN as the Ultimate Network

The speed of computation ν of a system of average energy E , is boundedas [3]: ν ≤ 2E/π~. Moreover, entropy limits the memory space I [4] as

322 Paola A. Zizzi

the maximum entropy is given by: SMax = kBI ln 2, where kB = 1.38 ×10−23J/K is the Boltzmann’ s constant. As Seth Lloyd showed [4], thesebounds are saturated by black holes, which can then be regarded as theultimate computers. In what follows, however, we will use the notations in[5], which are more suitable to our purpose. As Ng showed [5], the bounds onspeed of computation ν and information I can be reformulated respectivelyas: ν2 ≤ P

~ ; I ≤ ~Pt2P

, where P is the mean input power.These two bounds lead to a simultaneous bound [5] on the information I

and the speed of computation ν: Iv2 ≤ 1t2P

. It is easily shown that the speedof computation νn and the quantum information In of the QGN, saturatethe three bounds above at each time step tn:

v2n =

Pn

~=

En

~tn=

1t2n

; In =~

Pnt2P=

~En

tnt2P

= (n + 1)2; Inv2n =

1t2P

.

As we have seen, at each time step, the QGN saturates the quantumlimits to computation. Then, it can be regarded as a ultimate Internet, inthe same sense that black holes can be regarded as ultimate computers.Moreover, following Ng [5], we define the number of operations per timeunit: v = Iv.

In the context of the QGN, we have then: vn = Invn.Moreover, it holds. vn/ν0 = n + 1 =

√I, where v0 is the number of

operations per qubit at n=0: v0 = I0v0; v0 = t−1P = 1043 sec−1; I0 = 1,

from which it follows: v0 = 1043 sec−1.If the average amount of energy is the Planck energy: EP ≈ 1019GeV ≈109J , then, a Planckian black hole saturates the bound: vP = 2EP

π~ ≈109J

10−34J sec ≈ 1043 sec−1.This is the maximum speed of computation available in nature.Also, from the simultaneous bound, it follows that the memory space of

a Planckian black hole is one qubit: IP = t−2P

ν2P

= 1. Then, the qubit, the unitof quantum information, is the minimum amount of information availablein nature.

This result is in agreement with the quantum holographic principle [1],which states that a Planckian black hole, having a horizon area of one pixel,can encode only one qubit.

8. The QGN and the Problem of Λ

We calculate the present value of the quantized cosmological constant with:nnow = 9× 1060, and we get: Λnow = 1.25× 10−52m−2.


Also, we obtain: ΩΛ = Λc2

3H20≈ 1

3Λc2t2now where tnow ≈ H−10 ≈ 3 ×

1017h−1 sec, H0 being the Hubble constant and h a dimensionless parameterin the range: 0.58 ≤ h ≤ 0.72. By choosing h=0.65, we get: ΩΛ ≈ 0.7. Fromthe relation: ΩΛ = ρΛ

ρc, where ρΛ is the vacuum energy density and ρc is the

critical density, it follows: ρΛ ≈ 0.7ρc in agreement with the Type Ia SNobservation data [12].

From this result, the connecting links (virtual states) of the QGN looklike to be still active. But the value of the total entropy is too big, fornnow = 1060: it would be Snow = 10120 ln 2, indeed a huge amount of en-tropy. We believe that at some earlier time, the QGN decohered. In thisscenario, the free links were not activated anymore by the nodes, since de-coherence time. We can visualize the QGN after decoherence as a regularlattice, the connected part of the QGN itself.The energy of nodes at present time would be: ENODES

now = EP

nnow=

10−41GeV ≈ 10−51J .One would expect that at present, such a weak energy of the nodes would

prevent quantum computation. But this is not true: for n → ∞, quantuminformation I will grow to infinity as n2, while the energy of nodes willdecrease to zero as 1/n.In fact, a low energy just reflects in a low speed of computation ν but not ina low amount of information. The speed of computation at present wouldbe: νnow

∼= 10−17 sec−1 . It should be noticed that in our case, ν−1now is the

age of the universe: ν−1now ≈ 1017 sec .

Then, from the simultaneous bound to computation we get: Inow ≤10120, where in fact the bound is saturated because of the previous ar-guments. We would get the huge total entropy Snow = 10120 ln 2 if themachinery did not stop at some earlier time.

In [13] Lloyd computed the maximum possible number of bits whichthe universe registered since the Big Bang until now, which he found to be10120, equal to the number of elementary operations, which he calculated bythe use of the relation: number of bits = (t/tP )2, where in our case t is thequantized time tn, and the relation above gives explicitly: (tn/tP )2 = (n +1)2 ≡ N . The main difference is that Lloyd does not consider decoherenceat the end of inflation, (while we do, as it will be shown in the next section).

9. Self-decoherence

The QGN can be theoretically divided into two sub-systems: the connectedpart, made of connecting links and nodes, (quantum fluctuations of the

324 Paola A. Zizzi

vacuum and quantum fluctuations of the metric respectively) and the dis-connected part: the free links (qubits). The connected part can be thoughtas the ‘environment’, while the disconnected part can be thought as thequantum state.

The virtual black holes in between space-time slices, together with thequantum fluctuations of the metric on the slices, constitute space-timefoam.

Decoherence, here, can only be caused by space-time foam.The fact thatvirtual black holes can induce decoherence was suggested by Hawking [14].

However, as the quantum foam (virtual black holes and quantum fluc-tuations of the metric) resides in the connected sub-graph, which is partof the QGN, the QGN as a whole, undergoes self-decoherence. Althoughthe details of the mechanism of self-decoherence are not yet properly un-derstood, we can get a clue by considering entropy arguments, as it wasalready pointed out in [1] and [7].

The quantum entropy of N=I qubits is: S=Iln2.Thus, if the QGN did never decohere, now the maximum entropy would

be S = 10120 ln 2 ( with nnow = 1060). To get the actual entropy, one shouldcompute it as: Snow = 10120 ln 2/Sdecoher = 10120/IMax.

If one agrees with Penrose who claims [15] that the entropy now shouldbe of order 10101, this corresponds to the maximum amount of quantuminformation at the moment of decoherence: IMax = 1019 (ncr ≈ 109).where ncr stands for the critical number of nodes which are needed to pro-cess the maximum quantum information, IMax. It follows that the earlyquantum computational universe decohered at tdecoh ≈ 10−34 sec .

Moreover, we find that at the moment of decoherence (n = 109) the meanenergy is:Edecoh ≈ 1010GeV ≈ 1J (corresponding to a rest mass mdecoh ≈ 10−13g)and that the number of operations per qubit is:vdecoh = Imaxt

−1decoh ≈

1053 sec−1.Finally, it should be noticed that, as the QGN describes the early in-

flationary universe, decoherence time corresponds to the time of the end ofinflation, and the decoherence energy corresponds to the reheating energy.

References

1. P. A. Zizzi (2000) Holography, Quantum Geometry and Quantum Informa-tion Theory, gr-qc/9907063, Entropy 2 , 39.


2. G. ’ t Hooft, Dimensional reduction in Quantum Gravity, gr-qc/9310026;G. ’ t Hooft, ”The Holographic Principle”, hep-th/0003004;L. Susskind, ”The World as a Hologram”, hep-th/9409089.

3. N. Margolous, L. B. Levitin (1988) Physica D 120, 188.4. S. Lloyd (2000) Ultimate physical limits to computation, quant-ph/9908043;

Nature 406, 1047.5. Y. J. Ng, Clocks, computers, black holes, spacetime foam and holographic

principle, hep-th/0010234;Y. J. Ng, From computation to black holes and space-time foam, gr-qc/0006105.

6. J. A. Wheeler (1962) Geometrodynamics, Academic Press, New York.7. P. Zizzi, The Early Universe as a Quantum Growing Network, gr-

qc/0103002.8. P. A. Zizzi (1999) Quantum Foam and de Sitter-like Universe, hep-

th/9808180; Int. J. Theor. Phys., 38, 2333.9. W.K. Wooters, W.H. Zurek (1982) Nature, 299, pp 802–819.

10. S. W. Hawking, D. N. Page and C. N. Pope (1980) Quantum gravitationalbubbles, Nucl. Phys. B 170 , 283; S. W. Hawking (1996) Virtual black holes,Phys. Rev.D 53, 3099.

11. Luis J. Garay, Quantum gravity and minimum length, gr-qc/9403008.12. S. Perlmutter et al. (1998) Nature 391, 51; B. P. Schmidt et al. (1998)

Astrophys. J. 507, 46.13. Seth Lloyd, Computational Capacity of the Universe, quant-ph/0110141.14. S. W. Hawking (1993) Black Holes and Baby Universes and other Essays,

Bantam, New York.15. R. Penrose (1989) The Emperor ’s New Mind, Oxford University Press.

327

Occam’s Razor Revisited: Simplicity vs.Complexity in Biology

Joseph P. Zbilut

Department of Molecular Biophysics and PhysiologyRush University Medical Center

1653 West Congress, Chicago, IL 60612 [email protected]

It would seem that the topics of complexity and emergence have become con-nected insofar as emergence would appear to be a unique feature of complexity.Yet it would also seem that the topic of simplicity should also be discussed asa natural counterpoint. In fact, all of these concepts are often presented as cer-tainties without sufficient explication of their bases. It is suggested that therehas been inadequate attention paid to these concepts vis-a-vis modern cogni-tive science and the theory of mathematics. By approaching these topics fromthis perspective it may be that an alternate understanding can be obtainedwhich devolves from a less mechanistic view of dynamics.

Keywords: Biophysics; Complex Systems; Nonlinear DynamicsPACS(2006): 87.15.v; 89.75.k; 82.39.Rt; 89.75.Fb; 05.45.a

1. Introduction

Although the idea of complexity and emergence has found some favor inbiological disciplines to describe the myriad processes found in organisms,many questions remain. The explosion of data concerning molecular biologyhas forced many fatigued scientists to ask if there are any over-all guidingprinciples, or “laws” akin to those sought by physicists and chemists. Im-plicit in this query is the additional question of how detailed must a modelbe for it to accurately describe a biological phenomenon. Clearly, someguiding principles could allow for reduced models which could be managedmore easily. This is especially the case where mathematical models are thecase. Currently, many mathematical models are high dimensional requiringconsiderable computational time. Yet the results of such models are oftenquestioned for the obvious reason that even very small errors can quickly bemagnified. There is also the very real “noise” problem: how does one model

328 Joseph P. Zbilut

the naturally occurring noise found in biological systems? Different kinds ofnoise at different length scales makes the problem extremely important[1].

One can readily appreciate why many have invoked the “science of com-plexity” in this arena. If complexity exhibits a universal paradigm, and itis appropriate for the biological arena, then perhaps there may be a wayof identifying guiding principles to reduce the dimensionalities of these bi-ological problems. To this end, many have looked for signatures of thiscomplexity, such as scale-free, network dynamics. Scale-free networks havebeen claimed for metabolic processes, organ systems, and regulatory struc-tures. Very often these networks are determined in the context of “powerlaws,” whereby some feature of the observed phenomenon falls off as anexponential power law.

The difficulty with such observations is that they are simply an observa-tion of a kind of distribution, in the same way that it can be said that somevariable exhibits a normal distribution, or a Poisson distribution. It saysnothing in particular about the underlying mechanisms of the distribution.Indeed, such power laws are found in many phenomena and with a myr-iad of different mechanisms responsible for them—they are not necessarilyunique.

2. Mathematics and Perceptions

Implicit in such observations is the belief that the mathematics reveals someunderlying universal properties not clearly seen in the empirical observa-tions. This is not surprising given that the foundation of classical mathe-matics is Platonist. Universals and laws are quite at home in Plato’s cave.There is, however, a new challenge to this Platonistic foundation to mathe-matics, and it comes from one of the objects biologists wish to understand;namely, the human brain [2].

For a long time, it has been assumed that logic (or a logic) is independentof a thinker: “correct” reason is not dependent upon the peculiarities ofa human brain–it transcends it and can be verified by any other correctlythinking human brain. However, in recent decades, cognitive neuroscientistshave challenged this indirectly. Many studies have shown that the humanbrain often sees patterns where there are none. And as David Ruelle haspointed out, our mathematics comes out of physical reality, filtered by thebrain. In other words, we do the kind of mathematics we do because we“see” the mathematics in our reality—whatever this might be (see, e.g. [3]).

This is an important consideration relative to the quest for biologi-cal universals, because it parallels the experience of physicists. Perhaps


Heisenberg became the first widely known person to express it in his Un-certainty Principle, but the point is that an “observer” changes the thingobserved by virtue of the observation. This is historically a remarkablestatement which has often been glossed over by many scientists, althoughthere have been a few who have attempted to comment upon it–mainly inthe context of quantum physics. This observation puts into question theentire notion of “objectivity.” And certainly, it has been the progenitor ofmany so-called “weird” effects of quantum science.

Although the biological experience is not quite the same as the physi-cist’s, it points out, to a certain extent, that the scientist as observer, “fil-ters” perceptions through the activity of the brain. It is a brain which likesto see patterns (meaning ?) even where there is none. Can this be the rea-son why so many scientists still pursue mechanistic mathematical modelseven when quantum physics has placed serious questions on such an en-deavor? As is well known, strictly speaking, Newton’s Laws are incorrect(as compared to quantum physics), but the errors are minimal and canthus be used. But does this gloss allow for a continued myopic view of vari-ous phenomena—especially in the life sciences. Consider that the Ptolemaicview of the Universe held sway because in terms of its predictions, the errorswere few. The fact that biological systems must be continually adaptable,and in a sense “contingent” would tend to mitigate the universalist view.And what about the multiple causes of “noise” in the biological world. Of-ten, noise is lumped into one big “black box” of unknowns and sequesteredin differential stochastic equations.

Yet the biologist is confronted with the incontrovertible fact that higherbiological systems coordinate many millions of processes simultaneously ina sea of noise to effect what are obviously deterministic processes. Thusneuronal synapses have a degree of randomness associated with them, asdo ion channels, etc., yet they are capable of developing and coordinat-ing “messages” that can ultimately result in such things as coordinatedmovement and thought.

3. Simplicity

While complexity has been put forward as an important paradigm for bi-ological systems, there is still the old dictum of the medieval Franciscanphilosopher, William of Occam, that states, “pluritas non est ponenda sinenecessitate,” [4] otherwise known as “Occam’s razor.” And indeed, scien-tists of different fields are wont to quote this dictum of parsimony whendeciding on multiple potential models. It was not however until the 1940s


and 50s that there was significant attention paid to the idea of complexity[5]. To be sure, if simplicity has been such an important guiding principle,how has its complement suddenly come to the fore? Clearly, simplicity hasnot been abandoned, inasmuch as such luminaries as Einstein and Hawkinghave invoked it. But where does complexity fit in?

Perhaps an analogy with medieval tapestries may clarify the issue:simple things are not necessarily more intelligible than complex ones. Atapestry is composed of numerous threads of different colors, which maybe continuous throughout, but which appear at various points to trace outa figure or letter, etc. Thus apparent intricacy is really specious, and fromsingle threads emerges a unique story. And certainly emergent propertiesare often cited as being a hallmark of complex systems: traits not seen inthe individual parts are demonstrated only in the complete system. Yetwhat could be simpler than the periodic table of elements: proton numberaccounts for a periodicity and electron shell completeness to account for avery simple ordering of elements. From this basic arrangement emerge theelements’ visibly different characteristics. It is still not clear how all thisallows for such astonishingly different elemental attributes and perceptions.These attributes would appear to be “emergent” properties.

If the research on cognitive neurophysiology is to be believed, it mayserve as a cautionary tale for a whole-hearted acceptance of bio-complexity.At the very least, the issue is far from decided in that a universally accepteddefinition of complexity is not at hand. What is known, is that it is a bigproblem based on massive amounts of data. Still, it should be recalled thatjust because something is big, does not necessarily mean that it is complex;nor does it mean that just because something is small does not mean it isnot complex.

The noted artificial intelligence pioneer and Nobel laureate, Herbert Si-mon, in his dissertation work discovered that when confronted with massiveamounts of data, the human mind was inclined to pull back to a smallersubset of data which was well appreciated to make decisions [6]. A simi-lar tendency may be operative with the suddenness of massive amounts ofmolecular data. There may be a psychological comfort in looking for “basicprinciples.”

Still there may yet be another approach. Given that there is at leastsome determinism, and some “noise,” an approach which favors “piece-wisedeterminism may provide a way which recognizes the essential “buildingblock” nature of many processes, while at the same time recognizing thatbuilding blocks are not always easy to assemble.


It has been argued that if the Lipschitz conditions for differential equa-tions are relaxed, this allows for a simpler way to construct processes. At thesame time it can more easily model the massive asynchronous parallelismfound in biological processes [7,8].

4. Piece-wise Determinism

The governing equations of classical dynamics based upon Newton’s laws:ddt

∂L∂qi

=∂L∂qi

− ∂R∂q i

, i = 1, 2, ..., n (1)

where L is the Lagrangian, and q,qi are the generalized coordinates andvelocities, include a dissipation function R(qiqj) which is associated withthe friction forces:

Fi(q1, q2, · · · qn) = −∂R∂qi

. (2)

But the functions (2) do not necessarily follow from Newton’s laws, and,strictly speaking, some additional assumptions need to be made in orderto define it. The “natural” assumption (which has been never challenged)is that these functions can be expanded in Taylor series with respect to anequilibrium state

qi = 0. (3)

Obviously this requires the existence of derivatives:

| ∂Fi

∂qj|< ∞ at qj → 0 (4)

i.e. Fi must satisfy the Lipschitz condition. This condition allows one todescribe the Newtonian dynamics within the mathematical framework ofclassical theory of differential equations. However, there is a certain price tobe paid for such a mathematical “convenience”: the Newtonian dynamicswith dissipative forces remain fully reversible in the sense that the time-backward motion can be obtained from the governing equations by timeinversion, t → −t. In this view, future and past play the same role: nothingcan appear in future which could not already exist in past since the trajec-tories followed by particles can never cross (unless t → ±∞). This meansthat classical dynamics cannot explain the emergence of new dynamicalpatterns in nature.

To simplify the exposition, consider a one-dimensional motion of a par-ticle decelerated by a friction force:

mv = F (v) (5)


in which m is mass, and v is velocity. Invoking the assumption (4 ) one canlinearize the force F with respect to the equilibrium v = 0:

F → −αv at v → 0, α = −(∂F∂v

)v=0 > 0 (6)

and the solution to (5) for v → 0 is:

v = v0e− α

m t → 0 at t →∞, v0 = v(0). (7)

As follows from (7), the equilibrium v = 0 cannot be approached infinite time. The usual explanation of such an effect is that, to accuracyof our limited scale of observation, the particle “actually” approaches theequilibrium in finite time. In other words, eventually the trajectories (7)and v = 0 become so close that we cannot distinguish them. The sametype of explanation is used for the emergence of chaos: if two trajectoriesoriginally are “very close,” and then they diverge exponentially, the sameinitial conditions can be applied to either of them, and therefore, the motioncannot be traced.

Hence, there are variety of phenomena whose explanations cannot bebased directly upon classical dynamics: in addition, they require some addi-tional qualifications or distinctions, e.g. about a scale of observation, “veryclose” trajectories, etc.

A different structure of dissipation forces can be described which elim-inates the paradox discussed above and makes the Newtonian dynamicsirreversible. The main properties of the new structure are based upon aviolation of the Lipschitz condition (4). Turning to the example (5), let usassume that

F = −αv − α1vk, α1 ¿ α, k =

pp + 2

< 1, p À 1 (8)

in which p is an odd number.By selecting large p, one can make k close to 1 so that (6) and (8)

will be almost identical everywhere excluding a small neighborhood of theequilibrium point v = 0, while, as follows from (8), at this point:

| ∂F

∂v|= (α + kα1v

k−1) →∞ at v → 0, i.e. F → −α1vk at v → 0. (9)

Consequently, the condition (4) is violated, the friction force growssharply at the equilibrium point, and then it gradually approaches thestraight line (6). This effect can be interpreted as a jump from static tokinetic friction.

It appears that this “small” difference between the friction forces (6),(8) leads to fundamental changes in Newtonian dynamics.


Firstly, the time for approaching the equilibrium v = 0 becomes finite.Indeed, as follows from (5) and (9):

t0 = −∫ 0

v0

− mdvα1vk

=mv1−k

0

α1(1− k)< ∞. (10)

Obviously this integral diverges in the classical case.Secondly, the motion described by (5), (8) has a singular solution v ≡ 0

and a regular solution

v = [v1−k0 − α1

m(1− k)t]

11−k . (11)

In a finite time the motion can reach the equilibrium and switch to thesingular solution, and this switch is irreversible. It is interesting to notethat the time-backward motion

v− = [v1−k0 − α

m(1− k)(−t)]p+21/2 (12)

is imaginary. [One can verify that the classical version of this motion (7) isfully reversible if t < ∞].

The equilibrium point v = 0 of (8) represents a “terminal” attractorwhich is “infinitely” stable and is intersected by all the attracted transients.Therefore, the uniqueness of the solution at v = 0 is violated, and themotion for t < t0 is totally “forgotten.” This is a mathematical implicationof irreversibility of the dynamics (8).

So far we were concerned with stabilizing effects of dissipative forces.However, as well-known from dynamics of non-conservative systems, theseforces can destabilize the motion when they feed the external energy intothe system (e.g. the transmission of energy from laminar to turbulent flowin fluid dynamics, or from rotations to oscillations in dynamics of flexiblesystems). In order to capture the fundamental properties of these effects inthe case of a “terminal” dissipative force (8) by using the simplest math-ematical model, consider(5) and assume that now the friction force feedsenergy into the system:

mv = α1vk, k =

pp + 2

< 1, v → 0. (13)

One can verify that for (13) the equilibrium point v = 0 becomes aterminal repeller, and since

dv

dv=

kα1

mvk−1 →∞ at v → 0 (14)


it is “infinitely” unstable. If the initial condition is infinitely close to thisrepeller, the transient solution will escape it during a finite time period:

t0 =∫ vo

ε→0

mdvα1vk

=mv1−k

o

α1(1− k)< ∞ (15)

while for a regular repeller, the time would be infinite.As in the case of a terminal attractor, here the motion is also irreversible:

the solution

v = ±[α1

m(1− k)t]

11−k (16)

and the solution (11) are always separated by the singular solution v ≡ 0,and each of them cannot be obtained from another by time inversion: thetrajectory of attraction and repulsion never coincide.

But in addition to this, terminal repellers possess even more surprisingcharacteristics: the solution (16) becomes totally unpredictable. Indeed, twodifferent motions described by the solution (16) are possible for “almost thesame” (v0 = +ε → 0, or v0 = −ε → 0 at t → 0) initial conditions. Themost essential property of this result is that the divergence of these twosolutions is characterized by an unbounded rate:

σ = limt→t0

(1t

lnt1/1−k

| v0 | ) →∞ at | v0 |→ 0. (17)

In contrast to the classical case where t0 → ∞, here σ can be definedin an arbitrarily small time interval t0, since during this interval the initialinfinitesimal distance between the solutions becomes finite. Thus, a terminalrepeller represents a vanishingly short, but infinitely powerful “pulse ofunpredictability” which is pumped into the system via terminal dissipativeforces. Obviously failure of the uniqueness of the solution here results fromthe violation of the Lipschitz condition (4) at v = 0.

5. Neural Example

Traditional “neural net” models of cognition have some difficulties, includ-ing difficulties dealing with noise, parallelism, etc. A “piece-wise determin-istic,” or “terminal” model of neurodynamics can exhibit a stochastic at-tractor as a universal tool for generalization.

Indeed, in contradistinction to chaotic attractors of deterministic dy-namics, stochastic attractors can provide any arbitrarily prescribed proba-bility distributions by an appropriate choice of (fully deterministic!) synap-tic weights Tij .


The information stored in a stochastic attractor can be measured bythe entropy H via the probabilistic structure of this attractor:

H(Xi, X2, ..., Xn) = −∑

X1

...∑

Xn

f(x1, ..., xn) log f(x1, ..., xn) (18)

where the joint density f(x1, ..., xn) is uniquely defined by the synapticweights Tij . Random neurodynamical systems can have several, or even, an

Fig. 1. Piece-wise dynamics with multiplicative probabilities (top) result in a stochasticattractor (bottom).

infinite number of stochastic attractors (Fig. 1). For instance, a dynamicalsystem

x1 = γ1 sink[√

ω sin(x1 + x2)] sin ωt (19)

x2 = γ2 sink[√

ω sin(x1 + x2)] sin ωt (20)

has stochastic attractors with the following densities:

f(x1, x2) = 0.5 | cos(x1 + x2) cos(x1 − x2) | (21)


πm1

2< x1 + x2 <

π(m1 + 2)2

,πm2

2< x1 − x2 <

π(m2 + 2)2

(22)

m1 = · · · −7,−3, 1, 5, 9, etc.; m2 = · · · −5,−1, 3, 5, 7, · · · etc. (23)

The solution (21) represents a stationary stochastic process, which at-tracts all the solutions with initial conditions within the area (22). Eachpair m1 and m2 from the sequences (23) defines a corresponding stochasticattractor with the joint density (21).

Hence, the dynamical system (19–20) is capable of discrimination be-tween different stochastic patterns, and therefore, it performs pattern recog-nition on the level of classes.

There is some experimental evidence that this may be the case for realneural systems. Since this would be a multiplicative process, one wouldexpect power law behaviour of the observed variables as well as a lognormaldistribution. In fact, this is what has been routinely observed in neuralinvestigations [9].

Table 1.

Non-Lipschitz DynamicsDependent upon singularityNo attractor in the usual senseUncertainty at singular pointsDynamics is well behaved away from singular pointsRandom spread of points in region of phase space (stochastic attractor)Information infinite at singularity (knowledge of past is zero)Countable combinatorial setPredictability based on probability evolutionEasily controlled at singularity (not on trajectory)

6. Metabolic Nets

Considerable interest has been paid to metabolic scale free metabolic net-works. The architecture of these metabolic nets is in many ways similarto neural nets. Indeed, several studies based on real data point to lognor-mal distributions and power laws [10,11]. Thus there is a good reason toconsider the possibility of underlying multiplicative processes. Furthermoreit should be stressed that the benefit of piece-wise modeling is that suchdynamics are much more forgiving of “errors” (Table 1).


7. Conclusion

What is notable about piece-wise determinism is that it acknowledges thata given corpus can constitute a very large amount of inter-related data,while at the same time posing the linkages in a way which does not tie thedeterministic pieces to the past. A further benefit is that “control” of thedynamics is exquisite at the singularities, while being relatively imperviousto the effects of noise once a trajectory is chosen. The stochastics of onelevel evolve into a progressively “deterministic” process at a higher levelas can be seen through lognormal distributions. Thus a very large problemcan be reduced to one of combinatorics and probabilities. This last pointneeds to be emphasized since there is a persistent tendency to view manyof the implied dynamics through the legacy of Laplace. This is to say thatthere is a tendency to model these dynamics with traditional differentialequations. As has been suggested, however, this need not be so. Piece-wise deterministic dynamics can produce similar results, but are relativelysimpler, and perhaps heuristically more satisfying, given the underlyingstochastics [12].

It should be also noted that although the singularities have been mod-eled with Lagrangians, different types of mathematics may be more appro-priate with other systems. We note that topology admits numerous singular-ities without metrics. The experimental indications are that in intransitivesituations (dynamical systems with fixed points of rotation and translation)3 spatial dimensions could best represent physical situations if the metricsignature is not Euclidean geometry, but instead of the Minkowski typeNon-Euclidean geometry [13].

Acknowledgments

Discussions with Sandro Giuliani, Robert Kiehn, David Ruelle, Chuck Web-ber, and Michail Zak have been useful and greatly appreciated.

References

1. E.F. Keller (2007) Nature, 445, 603.2. D. Ruelle (2000) Nieuw Archief voor Wiskunde, 5/1: 30-33.3. P. Sterzer, P. A. Kleinschmidt (2007) PNAS, 104, 323-328.4. W.M. Thorburn (1918) Mind, 217, 345–353.5. J.P. Zbilut, A. Giuliani (2005) Encyclopedia of Nonlinear Science, A. Scott

(Ed.), New York and London, Routledge, pp 7–96. H. Simon (1997) Administrative Behavior: A Study of Decision-Making Pro-

cesses in Administrative Organizations. 4th Ed. New York, The Free Press.


7. M. Zak, J.P. Zbilut, R.E. Meyers (1997) From Instability to Intelligence:Complexity and Predictability in Nonlinear Dynamics, (Lecture Notes inPhysics: New Series m 49). Springer Verlag, Berlin Heidelberg New York.

8. J.P. Zbilut (2004) Unstable Singularities and Randomness. Elsevier, Ams-terdam.

9. P. Cao, Y. Gu, S-R. Wang (2004) J Neurosci 24, pp 7690–7698.10. K. Kaneko (2003) Phys Rev E 68, 031909.11. C. Furusawa, K. Kaneko (2006) Phys Rev E 73, 011912.12. D. Mumford (1999) Dawing of the Age of Stochasticity [Lecture delivered to

the Accademia Nazionale dei Lincei, May], in Mathematics: Frontiers andPerspectives, (V. Arnold, M. Atiyah, P. Lax, B. Mazur, Eds.)(2000) AMS.

13. R. Kiehn, Non-Equilibirum Thermodynamic Processess, Vol. 1: Non-Equilibrium Systems and Irreversible Processes. Copyright R. Kiehn, 2003–2006. Lulu Enterprises Inc., Morrisville, NC; (http//:www.lulu.com/kiehn).

339

Order in the Nothing: Autopoiesis and the OrganizationalCharacterization of the Living

Leonardo Bicha and Luisa Damianob

aCE.R..CO.-Center for Research on the Anthropology and Epistemology ofComplexity, University of Bergamo,

Piazzale S. Agostino 2, 24129 Bergamo, Italyb Biology Department, University of RomaTre,

V.le G. Marconi 446, 00146 Rome, [email protected]

An approach which has the purpose to catch what characterizes the specificityof a living system, pointing out what makes it different with respect to physi-cal and artificial systems, needs to find a new point of view — new descriptivemodalities. In particular it needs to be able to describe not only the single pro-cesses which can be observed in an organism, but what integrates them in aunitary system. In order to do so, it is necessary to consider a higher level of de-scription which takes into consideration the relations between these processes,that is the organization rather than the structure of the system. Once on thislevel of analysis we can focus on an abstract relational order that does not be-long to the individual components and does not show itself as a pattern, but isrealized and maintained in the continuous flux of processes of transformation ofthe constituents. Using Tibor Ganti’s words we call it “Order in the Nothing”.In order to explain this approach we analyse the historical path that generatedthe distinction between organization and structure and produced its most ma-ture theoretical expression in the autopoietic biology of Humberto Maturanaand Francisco Varela. We then briefly analyse Robert Rosen’s (M,R)-Systems,a formal model conceptually built with the aim to catch the organization of liv-ing beings, and which can be considered coherent with the autopoietic theory.In conclusion we will propose some remarks on these relational descriptions,pointing out their limits and their possible developments with respect to thestructural thermodynamical description.

Keywords: Autonomy; Autopoiesis; (M,R)-Systems; Organization; RelationalBiologyPACS(2006): 89.75.k; 89.75.Fb; 82.39.Rt; 87.10.+e; 87.16.Ac

340 Leonardo Bich and Luisa Damiano

1. Introduction: Epistemological Remarks

“More is different” (P.W. Anderson)

In this paper we analyse, conceptually and historically, a descriptive modal-ity which could allow to identify, even if not yet to explain at its presentstate of development, what is specific of a living being. It is a relationalqualitative description based on the systemic thesis that what is peculiar ofan organism is not the single physical processes which take place in it, butthe way these are related in order to produce and maintain the integratedbiological unity they belong to. The main characteristic of the descriptiveapproach we derive from this hypothesis is to focus on the organization ofthe system. With this term we mean the topology of relations which allow usas scientific observers to identify a system as a unity belonging to a certainclass, that is the class of living systems.

Such a theoretical definition expresses one of the main epistemologicaloutcomes of contemporary science: the impossibility to consider the scien-tific knowledge as independent from the activity of observation and catego-rization performed by the observer. With this acknowledgment comes thecollapse of the classical idea which makes of the scientist an “absolute wit-ness” of nature, whose cognitive point of view, neutral and external fromthe natural world, guarantees the power of intelligibility that able to ac-cess and represent a reality in itself. In the epistemology of modern sciencethis amounts to capturing and representing the laws of nature, which areconsidered as pre-existent to the observer and discovered through an act ofapprehension of a reality external to him (Prigogine and Stengers, 1979).

A first breaking of this classical epistemology was operated by quantumphysics. By showing the inseparability of the activity of observing from theobject observed, it caused the subsiding of the objectivistic view of mod-ern science. This crisis of the scientific objectivity was amplified by somebranches of the contemporary science, which have been led to re-orientatethe traditional epistemological axis from the classic representationism intoa radical constructivism. Among them we can find a twentieth century tra-dition of research dedicated to the study of the biological organization, andknown for having given birth to the concept of self-organization (Stengers,1985). This branch of science has the peculiarity of having developed a sci-entific epistemology alternative to the modern tradition (Damiano, 2007),according to which the scientific observer does not have a direct access toreality, for he has an active role in the determination of the object he inves-tigates. He interacts with the natural world through theoretical categorieswhich instead of representing pre-existing and pre-defined objects build

Autopoiesis and the Organizational Characterization of the Living 341

reality as a set of defined objects of research. From this perspective nomodel or theoretical apparatus can express the reality as independent fromthe activity of the scientific observer (von Foerster and Zopf, 1962; Pri-gogine and Stengers, 1979; Maturana, 1988; 2000). The object treated bythe science is co-construed: the observer gives it an objectual form throughthe categories he resorts to, while the reality, limiting the range of theirapplicability, defines the area in which nature can be handled as made ofobjects categorised by the observer.

This epistemological thesis is specifically connected to the problem ofthe correlations between scientific disciplines, in particular those specificallyconcerned by this tradition: physics and biology. It is a non-reductionisticapproach oriented to the establishing of communicative circuits betweenthese disciplines: bidirectional transfers of models, questions, theoreticalstructures (Morin and Piattelli-Palmarini, 1974; Domouchel and Dupuy,1983).

The thesis of this paper, about the necessity of the construction of newkinds of models specific for biology, goes in fact in this direction, rejectingthe reductionistic approaches which move from physics to biology such asthe one promoted by molecular biology, which tried to find what is perti-nent in order to explain the living on a different level from the one wherelife manifest itself, missing the crucial problem of the systemic unity, as thename itself shows. Biology instead can undergo a development that followsits own modalities, derived from the kind of problems it faces, that is thenecessity of producing models able to explain the experience of the observerinteracting with a living system. And also, the attention given to the prob-lems which emerge from biological research can lead to new approaches tophysics and consequently to open new lines of research.

The first to understand this opportunity was Erwin Schrodinger, aphysicist that explicitly faced the main question of biology, what is life?, inhis seminal essay on the living (Schrodinger, 1944). He tried to characterizethe order proper of living systems, that he called “order from order” em-bedded in a particular rigid structure, in opposition with the statistic orderof physical systems. The new descriptive modality was not supposed to bedue to a new sort of force, but to the attention focused on the way livingsystems are built, expressed by a different concept of order. Even if he forhe didn’t face the problem of the unity that constitutes the living system,he had the merit to introduce the hypothesis that looking for the laws ofbiology could have opened the way to the development of a new physics.

This was in fact what Ilya Prigogine did (Prigogine and Stengers, 1979).


Starting from the concept of open systems proposed by von Bertalanffy’sorganicistic biology in the twenties (von Bertalanffy, 1949) as one of theproperties characterizing the living, he opened the way to the developmentof a new branch of thermodynamics, that of dissipative structures.

Coherently with this theoretical line, and referring explicitly toSchrodinger’s insight (Rosen, 2000), Robert Rosen faced the problem ofthe integration of the biological unity (Rosen, 1958a; 1958b, 1959, 1972),putting into evidence the epistemological consequences of his approach forscientific knowledge (Rosen, 1991). One main example is the inclusion intoscience of circular causal loops, considered as the characteristic peculiar tobiological systems, with the purpose to widen the range of scientific expla-nation from the specificity of mechanistic systems to the higher generalityof complex ones (Mikulecky, 2001a). Following this line of research he builtnew kinds of models able to deal with phenomena not explainable by themeans of the physical description (Stewart 2002) and he developed newapproaches to the study of complex systems also in disciplines other thanbiology.

2. Order in the Nothing

The distinction of a unity from its background is the primitive epistemicaction which an observer performs in order to study a natural object. Itis the procedure which separates, according to some criteria, the objectunder study from the medium with which it interacts and so specifies itsdomain of existence. As George Spencer Brown wrote: “a universe comesinto being when a space is severed or taken apart [. . . ] The act is [. . . ] ourfirst attempt to distinguish different things in a world where, in the firstplace, the boundaries can be drawn anywhere we please. At this stage theuniverse cannot be distinguished from how we act upon it”(Spencer Brown,1969).The kind of unity that is distinguished depends on the observer’soperation of distinction, which relies on his purpose and his point of view.Consequently it is an epistemological action, which puts some boundariesthat can be topological or functional. An example is the study of thermo-dynamical physical systems, where the boundary can be the container usedfor the experiment.

What do we distinguish while interacting with a living system? Thestudy of living systems has a particular aspect. When we interact withthem we can perform distinctions according to a manifold criteria, like theproduction of a certain enzyme or of waste products or the identificationof a certain reaction pathway inside of it. But in order to identify them as


living unities, which is the main problem faced in this work, we performa special kind of distinction. In fact the identification of these systems op-erated by the observer depends on an operation of distinction that definestheir identity in the same domain where they specify it through their in-ternal operations. The organism produces itself and by so doing specifiesits topology in such a way that its topological and functional boundariescoincide. In the interaction with a living system we can observe that itis characterized by the maintenance of the unity as such in a continuousprocess of transformation of components. A special kind of homeostasis isinvolved here which does not concern single processes but the whole systemwhich is maintained stable. What makes the set of processes which occurin a living organism a unity, or better a system, needs to be found at alevel different from that of the physical and chemical description. Materialcomponents indeed are continuously produced, transformed and degradedin such a way that what characterizes the identity of the system, is thatwhat is maintained in this change, does not belong to their epistemologicaldomain.

Starting from these remarks and following Schrodinger’s insight aboutthe necessity of characterizing the order peculiar of living systems, we posethe following question: which kind of order characterizes the continuous andintertwined flux of processes of production and transformation of compo-nents which a scientific observer sees as realizing a living being?

The difficulty in catching what makes a living system a unity of somekind consists exactly in this: we are dealing with a system which is char-acterised by a continuous change and nevertheless maintains itself. Unlikea machine, where the material parts are in an order visually observable,in the organism the chemical components are mixed in a continuous fluxof processes that occurs at the same time in a fluid network of reactions.The order is not positional, it is not localized, as everything is dissolved ina field-like way. Also, there is a difference from some physical systems likefor example a twister where there is an order in spite of the movements ofcomponents. In organisms indeed components are not pre-given but theyare produced and transformed by the system itself.

The first step towards a characterization of the order proper from theliving is to differentiate it from purely statistical order or order from

disorder (Schrodinger, 1944). This way of producing regularities in naturalsystem is not absent from biology, but it is not the factor which catcheswhat happens in a living system and that differentiates it from other naturaland artificial systems. Jean Jacques Kupiec and Pierre Sonigo (Kupiec and


Sonigo, 2000), in their critics of the concept of gene, refer to statistic order asthe crucial element to describe the living. They propose a sort of “molecularmechanics” which interiorizes the Darwinian selection at the molecular leveltogether with chance. Implicitly, they develop the line of research of thetradition of Francis Galton, Wilhelm Roux, and August Weissmann whoused the term “intrabiontic selection” (Pichot, 1999). Kupiec and Sonigoconsider organization as an epiphenomenon, the result of the dynamics ofthe ontogenesis, in a way similar to the processes of pattern formation,without any sort of causal role in conserving and realizing the system.Their theoretical model is more similar to the twister referred above, anddepends directly on the materiality of the parts contained in the systemobserved. They have the great merit to put into evidence some problems inthe point of view on metabolism assumed by molecular biology, especiallyabout instructive functional interactions operated by the genes or by theenvironment, which are typical of a machine-like approach. But a livingsystem is not only a visible shape, an ordered pattern. On the contraryit shows autonomy and produces its own components. The difficulty ofexplaining its difference from a phenomenon like the one giving rise toa self-organizing (or more correctly a self-ordering) system as a twisteror Bernard’s cells, comes out because the starting point is not the basicquestion “What is life”, but just a description of some chemical processeshappening in the organism. The problem of the integrated unity is not takeninto consideration.

Maturana and Varela instead, belong to a systemic tradition focusedon the problem of the relational unity of the living, which has its originsin Claude Bernard’s concept of milieu interieur (Bernard, 1865). In one oftheir early papers on the problem of the description proper to the living(Varela and Maturana, 1972), they propose the thesis according to whichmachines and organisms are different from physical systems in that they de-pend on how their elements are connected: “what makes physics peculiar isthe fact that the materiality per se is implied; thus, the structures describedembody concepts which are derived from materiality itself, and don’t makesense without it. [...] the definitory element in the living organization [andin machines] is a certain structure independent of the materiality that em-bodies it; not the nature of the components but their interrelations” (Varelaand Maturana, 1972).

The problem of how things are connected in a living system is also atthe core of Schrodinger’s conceptual model. But another step is needed,that is the distinction between organisms and artificial systems. Both are


organized systems whose qualitative properties are generated by the waytheir components are interconnected: that is by their internal organization.But machine are characterized by a positional order. Hard automata in factare organized by geometrical spatial constrains which unlike some physi-cal objects like crystals, are not characterized by symmetries, but havea dynamical functioning. Von Neumann’s self-reproducing automata (vonNeumann, 1966) constitute an example. They don’t really produce them-selves, but just recreate a certain spatial disposition. They are neverthelessat the basis of the metaphor of the organism as an “information processingmachine” that conceptually influenced molecular biology.

The order characterizing the machine metaphor is of the same kind ofthe one proposed by Schrodinger as fundamental in the living. His order

from order is in fact realized by an aperiodic solid characterized by apositional structure. The living processes are controlled by an ordered groupof atoms that maintains itself and transmits his structural order to othermolecular structures. Schrodinger’s conceptual model is that of a clock-like system with mechanisms interconnected in a way that the positionsare what is relevant (Schrodinger, 1944). He solidify the biochemical fluxin a sort of chemical mechanism in which the microscopic order of thegenetic code is transmitted positionally to the macroscopic order of theliving. Both in organisms and in mechanisms the shape, the structure, iswhat is conserved and transmitted in ontogenesis and phylogenies. Thisidea is conceptually very close to von Neumann’s one, as the aperiodicsolid performs the role of the program in a computationalist view.

But living systems are not solid, they are a flux of processes and trans-formations in a solution that is almost homogeneous from the point of viewof matter and energy. That is the reason why considering processes andespecially interactions between processes is crucial. The macroscopic or-der is not due to the shape of the microscopic structure of componentsbut depends on the shape of the interconnections between the processesof transformations. Their proper order differentiates itself from the othertwo kinds and can be found on a different level of abstractions over theflux of change. It is an epistemologically higher level of analysis where wecan find a kind of order which with Tibor Ganti’s words we call “orderin the nothing” (Ganti, 2003). What is produced and maintained in thecontinuous change of components that we observe in living systems, andwhat produces and maintain this same change is the shape of the relationsbetween the processes. Consequently, in order to understand the living as asystemic unity we need to put ourselves on a relational level of description.


Generativemechanism

Interactions Class of systems

Order fromdisorder

Statistical Non-positional(betweengiven components)

Physico.chemicalsystems

Order fromorder

Relationalspatial(between com-ponents)

Positional (between givencomponents)

Artificialsystems(machines)

Order in thenothing

Relationalabstract(betweenprocesses)

Non-positional(betweencomponents that are pro-duced by the system itself)

Living systems

The order in the nothing is different from the first kind of order becauseit is independent from material parts. But it is also different from the ma-chine kind because of its fluidity and because the geometry of relations isnot spatial but abstract. It doesn’t connect solid components but processes,characterizing itself as a meta level of relations which emerges when theseones assume a shape that allows self-production and self-maintenance to beinstantiated.

It doesn’t belong to the domain of the material structural interactionsbut to a relational abstract independent but complementary domain: thatof organization.

3. Structure and Organization

The origin of the organizational description of the living, considered ascomplementary to the structural one, can be ascribed to a scientific tra-dition developed between the 30’s and the 70’s. It is a trend of researchconstituted by a variety of lines of the contemporary science which facedthe same problem along some theoretical pathways which, although differ-ent, converged on many aspects. This crucial problem was to structurize acategorical approach pertinent to the specificity of the biological domain.

In spite of the different disciplinary origins, these lines shared the sametheoretical assumption. It consists in the idea that the distinctive propertyof the biological domain is autonomy: a relative independence from theenvironment which cannot be referred to the physico-chemical components,but to the integrated totality in which these components are supposed to bedynamically connected. The preliminary definition of autonomy providedby this trends of research refers to an endogenous determination, a self-determination, which reveals itself primarily in the scientific exploration


of the biological behaviours of self-stabilization, that is, of active reactionto exogenous perturbations, consisting in compensative movements whichcannot be ascribed to the individual perturbed components. Typically theyappear as global regulative processes, distributed on the whole biologicalunities to which the components belong.

The development of the investigations performed by these branches ofthe scientific research is usually gathered under the denomination “scientificgenealogy of the notion of self-organization” (Stengers, 1985; Ceruti, 1989;Damiano, 2007). Its characteristic is to be a theoretical movement which,although plural and differentiated, is strictly coherent and oriented. Bornfrom the attempts to modelise the internal functional scheme which allowstability in living systems, this wide theoretical process developed up tothe production of a general modelistic which has the ambition to provide aspecific conceptual definition of the biological domain. It has been realizedthrough the succession of different descriptive models, which shared somecommon methodological and theoretical characteristics: (a) the assump-tion of a qualitative and relational level of analysis, focused not on thespecific physico-chemical components and processes, but rather on theirorganization, that is the functional interconnection that integrates them inthe biological unities; (b) the characterization of the organizational schemewhich allows the expression of the property of the biological autonomy as arelational network with a circular character. The development of this kindof modelistic has been carried out through the progressive formal specifica-tion of the concept of organizational circularity, which allowed the so-called“tradition of self-organization” to gradually approach the rigorous scientificdefinition of the general dynamical mechanism at the basis of the biolog-ical phenomenology. This theoretical movement, accomplished through aprogressive distinction of the organization from the structure of biologicalsystems, can be traced back to three main phases:

(1) the production of the first theories of self-organization, performed bythree independent lines of scientific research which shared a close at-tention to the biological level of nature: (a) the wienerian cybernetics(Wiener, 1948) and its derivations, such as second order cybernetics(von Foerster, Zopf, 1962) and French neo-connectionism (Atlan, 1972);(b) the organicistic embryology (Weiss, 1974); (c) the thermodynamicsof dissipative structures (Prigogine, Stengers, 1979);


(2) the rigorous synthesis of the first conceptualizations on self-organization, operated by Piaget (Piaget, 1967) through the elaborationof a general theory of the biological organization based on the conceptof “organizational closure” (Ceruti, 1989);

(3) the critical revision of the theories of the biological self-organizationperformed by Maturana and Varela, which led them to an original the-oretical definition of the dynamical mechanism proper of the biologicalsystems expressed through the concept of “autopoietic organization”.

3.1. The Origins of the Concept of Self-Organization:

Biological Autonomy and Organizational Circularity

The concept of biological self-organization has its origin in the problem ofthe modelization of one of the most relevant and evident properties of theliving. Biology at first called it homeostasis and defined it as the capabilityproper to the organisms to maintain their internal environment relativelystable against the external perturbations (Cannon, 1932). It is a biologicalproperty that the tradition of self-organization re-conceptualized as auton-omy, conceiving it as the active control performed by the organism on itssame processes: the capability to react to the external destabilizing pertur-bations through self-determined variations which tend to cancel the effectsof the exogenous destabilizations.

This interpretation of the homoeostasis of living beings is at the coreof the fundamental theoretical hypothesis of the biological modelistic pro-vided by the tradition of self-organization: the assumption which tracesthe autonomous behaviour of self-stabilization back to the organizationalcircularity.

A schematic reconstruction of the development of this axis of researchcan be briefly outlined showing the main contributions of the three direc-tions of investigation that characterize the first period of the research onself-organization: (a) the model of the “feedback circuit” or “retroaction”elaborated by Norbert Wiener (Wiener, 1948); (b) the theoretical schemeof the “organized hierarchic system” produced by the embryologist PaulWeiss (Weiss, 1969; 1974); (c) the characterization of the “dissipative sys-tems” proposed by Ilya Prigogine (Prigogine and Stengers, 1979; Nicolisand Prigogine, 1989).

The first of these theoretical models already shows the methodologi-cal and theoretical definitory elements of the descriptive hypothesis whichassociates autonomy to a circular scheme of organization. The model ofthe feedback circuit was developed by Wiener coherently with the general


approach of cybernetic, a discipline based on the recognition of the conver-gence between the domain of the technological research and that of biologywith respect to the problem of stability. The common referring by thesetwo scientific areas to objects exhibiting the capability of self-stabilization(servo-mechanisms and organisms), led Wiener to the elaboration of cyber-netics as the locus of bidirectional theoretical exchanges between technologyand biology. The idea was to use the analysis of the technological mecha-nisms of stabilization to advance hypotheses on the functioning of biologicalones and vice versa to exploit the knowledge of the latter for the implemen-tation of the formers. Such a procedure of reciprocal exchange of knowledgehas an important consequence. It requires a theorization able to set asideproperties of the specific components of the two kind of objects explored. Itforces the research to move to an explorative level susceptible to give an ac-ces to the general functional relations of the object investigated, focalizingnot on the structure but on the organization.

This methodological option is at the origin of the model of the feed-back circuit elaborated by Wiener along the axis that goes from technologyto biology: defining the organizational scheme of the servomechanisms andproposing it as the hypothetical model of the general functional organiza-tion that allows the living systems to carry out self-stabilizing behaviours.As it is well known, the peculiarity of the wienerian model is to delineatea circular causal relation between a sensor and an effector, mediated bya regulator able to make the sensorial recording of the perturbative devia-tion and to act on the devices which perform compensative actions. Onceapplied to living systems, this scheme traces their general functional orga-nization as a ring : a functional circle which, interconnecting the sensorialand effectorial apparatus, implies that the effects of perturbations triggercompensative activities expressed by motorial interactions which are effec-tive in the environmental context. This model, associated by Wiener to theidea of “self-regulation” of biological systems, became the cybernetic proto-type of the notion of “self-organizing system” developed in the modelizationof biological autonomy by the heterodox lines of research of the BiologicalComputer Laboratory (Von Foerster and Zopf, 1962; von Foerster, 1980)and of the French neo-connectionism (Atlan, 1972).

The fundamental methodological and theoretical lines of Wiener’s cy-bernetic modelization of the autonomy of the living, were independentlyintroduced in the context of the organicistic embryology, where they wereimmediately associated to the qualitative description of biological sys-tems as “self-organizing systems”. An exemplary model of the conceptual


production characteristic of this line of research is the weissian theoreticalscheme of the hierarchical organized system which develops the theoreticalline opened by the wienerian modelistic.

This descriptive scheme was elaborated by Paul Weiss (Weiss, 1969;1974) from the experimental investigations on the self-stabilizing behaviourof biological systems, that led him to identify their characteristic dynamics.Typically the presence of local alterations in the internal dynamics of a liv-ing system do not induce the activation of a localized and specific stabilizingcentre, but triggers a series of correlated modifications in the elementaryprocesses. If the destabilisation is lower that the stability threshold of thesystem, it causes a compensation.

In Weiss’s exploration this remark becomes primarily a methodologicaloption. It consists in the refusal of the approaches to the study of biologicalstability focalized on the individual physico-chemical components and theirprocesses, in favour of the assumption of an explorative procedure centredon the functional relations that dynamically correlate the elements in thebiological global unities constituted by the living systems.

With the adoption of this methodological perspective of an organiza-tional character, Weiss aligned his modellistic production to the wienerianone not only from the procedural point of view but also from the theoreticalone. The hypothesis he developed to modelise the organization allowing thebiological stability proposes the idea of an organizational circle. From theseminal insights of the founders of the organicistic embryology,∗ Weiss con-ceptualized it as a strict functional correlation with the shape of a “closednetwork”: every element of the system is functionally correlated to anotherone so tightly that a behavioural deviation of it entails a compensativedistributed reaction in the whole network.

This characterization of the biological circularity of organization distin-guishes weissian modelistic from the wienerian one. It leads to the explici-tation of the idea that in the living systems the compensative reaction tothe perturbations is a collective action. The capability to self-stabilize doesnot belong to the individual components of a biological system, but to theirorganizational reticular correlation—to the totality.

On the basis of this remark Weiss produced a general descriptive schemethat distinguishes in the living systems two qualitatively different levels:that of the individual parts, subjected to continuous alterations, and that

∗The reference is to the research team which founded the organicistic embriology, thatis the so called Group of Cambridge (Stengers, 1985).


of the unity which comprehend them, strongly and actively conservative:“The variability of the complete system V is much lower then the sum ofvariabilities v of its components.

V<<[v(a)+ v(b) + v(c) + ... + v(n)]This formula contains the essential aspect of systems dynamics with

respect to the composition of elements. It could not be respected if the com-ponents were free and independent.” (Weiss, 1969).

This is the model of a biological hierarchical organization. It describesa relational global unity which interconnects its subunities to one anotherthrough functional reticular links. By so doing the totality subjects its com-ponents to its own global dynamics—a collective and coordinated dynamicsof its elements—and reacts to the elementary local deviations by regulat-ing the components’ singular behaviours to achieve its own conservation.The theoretical idea of “self-organizing system” was associated by Weissto this concept of biological system: the notion of a totality that, throughthe organizational reticular constraints that constitute it, produces its ownconstitutive dynamics, imposing it to its own elements and stabilizes itselfin presence of exogenous perturbations, acting on its individual elementaryprocesses.

By doing this, the weissian model leads to a decisive enrichment ofthe descriptive hypothesis that associates the autonomous behaviour to theorganizational circularity. First of all it extends the meaning of the notion ofliving autonomy, connecting it not only to the property of self-stabilizationbut to a more fundamental one, that of self-production with respect towhich the first is characterized as derived. Secondly, he connects the thesisof the circular organizational scheme to that of the stratification of theliving systems into al least two interdependent and qualitatively differentlevels—the individual parts and the totality.

Both these developments are now acquired by the scientific theory ofself-organization. The last one was conceptualized by Weiss through thecontroversial notion of “emergence”, which describes the capability of theorganizational circle to generate a level of reality characterized by qualitiesthat are not present in the lower level. It is the hypothesis of a qualitativedifference between the parts and the totality , that Weiss made intelligiblethrough an argumentation widespread today: the reticular organizationalconnections, constraining the components to each other in a circular way,inhibite the expression of some of the properties of the individual compo-nents and make possible the expression of global properties.


From this notion, inseparably associated to the model of the hierarchi-cal organized system, Weiss built a complex theoretical characterizationof the living. It consists of a perspective which identifies biological sys-tems as “molecular ecologies”, describing them as structurized in a plu-rality of levels of “molecular organization” of increasing complexity andstability—cells, groups of cells, tissues, organs, apparatus etc. It is a theo-retical idea that played a decisive role in the development of the traditionof self-organization. It gave a significant contribution to the constructionof the third of the explorative direction of the research on natural self-organization (Stengers, 1985).

The thermodynamics of dissipative structures was founded by Prigoginein the context of the development of a program of research finalized to dealexperimentally with the crucial problem of the relation between physicsand biology: the antagonism between the cosmological scenario pointed outby the principle of the entropic growth and the evidence of the biologicalevolution towards complexity. The ambition to provide a solution to thisissue led Prigogine to develop the weissian hypothesis of the “molecularecologies”: the idea that a molecular population with a high degree of free-dom can generate a level of stable integration and able to evolve towardshigher levels of complexity.

The assumption of this thesis oriented the prigoginian research programto the exploration of the behaviour of thermodynamically open molecularsystems, an investigation characterized by relevant results. It put into evi-dence that these systems, under certain conditions (Prigogine and Stengers,1979; Nicolis and Prigogine, 1989), are subject to phenomena of “super-molecular aggregation” which generate persistent macroscopic dynamicalstructures-ordered patterns.

They are irreversible physico-chemical processes that topologically orga-nize themselves: collective coordinated movements of the molecular popula-tion which constitute active and organized global unities that self-generateand self-maintain exploiting the exogenous flux of matter and energy and bydoing so producing entropy which is released in the environment. Prigoginecalled them “dissipative structures” to make explicit the compatibility withthe second principle of thermodynamics. He provided a characterization ofthem in terms of self-organizing systems, taking the weissian theoreticalframework of the circular and stratified organizational scheme. He explainedthese natural forms as emergent organized unities. He attributed to themthe minimal and incomplete forms—“ancestral” (Prigogine and Stengers,1979)—of some of the properties shown at the biological level. Among these


properties he included autonomy: the capability of these systems to self-produce their own constitutive dynamics, to stabilize against a wide rangeof perturbations and to exploit external perturbations to evolve—by an en-dogenous re-organizational control, that is by positive retroaction (Nicolisand Prigogine, 1989)—towards higher levels of complexity and stability.

Such a theoretical characterization puts the biological autonomy insidean evolutive scenario which grounds it in the physical domain. It proposesan image of the prebiotic evolution that generates more and more stableand complex molecular ecologies, which plausibly are able to develop untiltranscending the physico-chemical domain and proceeding into the biolog-ical one. It thus opens the possibility of a connection between the physicaland biological domain, not a reductionistic, but an emergentistic one. Itoffers the opportunity of an exchange of knowledge between these two do-mains of science which is not unidirectional and reductive, but founded ona bilateral “dialogue” (Prigogine and Stengers, 1979; Stengers, 2003).

3.2. The Piagetian Synthesis: The Notion of

“Organizational Closure”

Besides these first lines of research, the scientific genealogy of the notions ofautonomy and self-organization includes some trends of investigation defin-able as of first derivation. They are scientific orientations generally relatedto programs of research developed in the context of the human sciences anddirected to bridge the cartesian cut: to overcome the theoretical gap thatseparates the natural and the human sciences, by producing a naturalisticnotion of human being which avoids to reduce him to physical and bio-logical aspects. The peculiarity of these lines of investigation it that theyacknowledged in the evolutive scenario opened by the first research on self-organization the possibility to re-articulate the anthropo-social sphere onthe biological and the physical ones, according to a non-reductionistic butemergentistic theoretical approach. The aim of this operation was realizedthrough the investigation of the first research on self-organization, in orderto perform theoretical syntheses able to produce a complex and multidi-mensional notion of the human being. (Morin, 1973; Morin and Piattelli-Palmarini, 1974; Jantcsh, 1980; Dumouchel and Dupuy, 1983).

Among these programs of research the piagetian genetic epistemologyassumes a particular relevance, due to its decisive contribution to the sci-entific characterization of the biological organization (Ceruti, 1989). Thisline of the contemporary epistemology was directed by Jean Piaget to-wards the construction of a natural science of cognition based on a specific


interpretation of the general assumption of cognitive sciences that identifylife with cognition. It consists in a minority theoretical option that refusesthe computationalist modelization of living systems in favour of the iden-tification of autonomy as their definitory property.

The piagetian work of exploration and integration of the scientific pro-duction on the natural self-organization led the genetic epistemology to theelaboration of an innovative theoretical element, quickly acquired by thescientific research and still at the core of the investigations on biologicalautonomy. It is a concept that rigorizes the previous achievements on theorganizational circularity of living systems: the notion of “organizationalclosure”, proposed by Piaget in Biologie et Connaissance as a concept com-plementary to that of thermodynamical openness, earlier emphasized byvon Bertalanffy’s systemic biology (Piaget, 1969).

“The central ambiguity is that of the ‘open system’, for, if system exist,then something like a closure intervenes, which has to be reconcilied with the‘opening’. The opening is certainly justified and is founded on the basic ideathat ‘in biology there is not rigid organic form carrying out vital processesbut a stream of processes which are revealed as forms of a seemly persistentkind’ (von Bertalanffy). The opening then is the system of exchanges withenvironment, but this in no way excludes a closure, in the sense of a cyclicrather than a linear order. This cyclic closure and the opening of exchangesare, therefore, not on the same plan, and they are reconcilied in the followingway, which may be entirely abstract but will suffice for a analysis of a verygeneral kind.

(AxA’)→ (BxB’)→(CxC’)→...→ (ZxZ’)→(AxA’)→ ecc.

A,B,C ...: the material or dynamic elements of a structure

with cyclical order

A’, B’, C’...: the material or dynamic elements necessary for

their maintenance: the interaction of the terms of the first range

with those of the second→: the end points of these interactions

In a case like this we are confronted by a closed cycle, which expressesthe permanent reconstitutions of the elements A,B,C ... Z, A, and whichis characteristic of the organism; but each interaction (AxA’), (BxB’),etc., at the same time represents an opening into the environment as asource of aliment.” (Piaget, 1967).

The piagetian notion of closure can be considered as a developmentof the weissian formulation of the theoretical assumption that associatesthe biological autonomy to the organizational circularity. The conceptformulated by Piaget presents biological autonomy as an endogenous


determination of the living systems that has the character of the self-production and correlates it to the idea of an organizational scheme realizedby the circular functional interconnection of the components. The specificcontribution of the piagetian notion of closure consists in the idea thatthe organizational circularity corresponds to a concatenation of processeswhich continuously re-constitute the components of the living systems. Thetheoretical picture is that of a closed chain of elementary transformativeoperations which, by realizing themselves, trigger each other, giving rise toa self-determined recursive and cyclical process that, producing the com-ponents, produces the organism itself.

Piaget associated this conceptualization to the theoretical explicitationof the difference between organization and structure (Ceruti, 1989). Hedefines the first as the general relational scheme of all the living systemsand the second as its materialization in specific processes and components.Piaget underlined that the two aspects are distinct because, as the ideaof organizational closure points out, the peculiarity of the living systemsis that their materialization keeps changing: what persists in them are thefunctional relations that integrates the components in the global unity,while the specific components are permanently in flux.

By this theoretical distinction Piaget laid the basis for what can be con-sidered the most mature development of the genealogy of self-organization:the formulation of the autopoietic biology, oriented by Maturana and Varelatowards the definition of the dynamical mechanism of living systems.

3.3. The Autopoietic Biology: The Duality of Structure and

Organization in the Scientific General Definition of

Living Systems

The basic theoretical project of the autopoietic biology was to provide a newkind of solution to the problem of the identification of the living. Maturanaand Varela wanted to develop a criterion of identification which does notconsist in the classical enumeration of properties we can recognise in theliving from our external point of view. Instead of undertaking an analyticdescriptive procedure, doomed to go on indefinitely, Maturana and Varelaconceived a criterion which identify the living by specifying a mechanismable to produce these properties, or, more accurately, a mechanism able toproduce all the living phenomenology.

“Our aim is to proceed scientifically: if it is not possible to produce aninventory which characterizes a living being,


why not conceiving a system which, while operating, produces all itsphenomenology?” (Maturana and Varela, 1987)

This can be considered the main aim of the autopoietic biology: definingthe living from the inside, by specifying its deep “dynamical mechanism”,the inner mechanism able to generate the living phenomenology.

The development of the theory of autopoiesis relies on two fundamentalthesis:

(1) the most basic property of the living is autonomy, understood as thecapability that such natural systems have to produce and maintain,all by themselves, their own identity, by a triple endogenous action:a) self-production; b) permanent topological self-distinction in an en-vironment; c) self-stabilisation against endogenous and exogenous per-turbations;

(2) autonomy is a property which does not belong to the individual physico-chemical components of the living, but to their organizational correla-tion, that is the functional “organization” which integrates them in therelational unities usually called “organisms” (Maturana and Varela,1973).

The adoption of these hypotheses coincided with the alignment of the au-topoietic biology with the tradition of self-organization, a connection whichremained implicit and offered the ground for a transformative develop-ment of the production of this tradition. Here is what allows us to con-sider autopoietic biology as a critical reform of the first research on self-organization: although it shared with the latter some theoretical presuppo-sitions and some conceptual elements, this biological school spent more timestressing the flaws of the tradition of self-organization than acknowledgingits theoretical debts towards it.

The main shortcoming that autopoietic biology attribute to this tra-dition of research consists in a insufficiency. The genealogy of self-organization has postulated that the biological level of reality is populatedby relational unities of elements whose primarily property is autonomy. Ithas postulated biological self-organization, but it didn’t try to scientificallyexplain it, that is to define the mechanism able to generate these unitiesand their phenomenology (Maturana and Varela 1973).

In order to do it, one essential consideration, substantially missed by theresearch on self-organization, has to be taken in account. It consists in theidea that the organization which functionally supports the biological au-tonomy constitutes the invariant of the biological phenomenology—both at


the ontogenetic and at the phylogenetic levels. The relational unity of com-ponents is what maintain itself in the permanent flux of physico-chemicalelements peculiar to the organism’s life. This unity is what is permanentwithin the strong transformations that can make the individual living beingun-recognisable from one observation to another. The relational unity is thebiological element which is transmitted through reproduction: it is the fea-ture of the living that remains unchanged generation after generation. Theglobal unity is what is constant in the evolutive differentiation, to such anextent it constitutes the feature shared by all the living: being a relationalunity of components.

It is from this consideration that Maturana and Varela infer the gen-erative hypothesis of the theoretical conceptual structure of their au-topoietic biology, according to which the organization is the identity thatthe living permanently produces and keeps, thanks to a permanent andself-determined change the physico-chemical components (Maturana andVarela, 1973).

It is a thesis that implies the refuse of the term “self-organization”,which can seem to express the idea of a self-determined change of the livingorganization. It can seem to miss the specificity of the dynamics of the liv-ing, namely, the conservation of the organizational invariance coupled withthe permanent flux of physico-chemical elements. Here relies the need fora new term, namely, “autopoiesis”, which can avoid the confusion betweenthe invariant and the variant aspects of the living dynamics.

This term expresses the acknowledgment that biological systems con-stitute dynamical systems in which the relation between invariance andchange has a peculiar aspect: in these systems what is conserved is therelational unity of components, which, in themselves and in their specificfunctional relations, permanently change.

Maturana and Varela elaborated this point through the implicit recov-ering and strict scientific treatment of the conceptual dyad introduced afew years before by Piaget: the duality of organization and structure, whichwithin the autopoietic biology coincides with the duality between the in-variant element of biological phenomenology and the variant one. Here “or-ganization” refers to the stable and permanent relations which define a bi-ological individual as a unity, while “structure” refers to the particular andtransient materializations of a living unity. These two notions are distinctbut inseparable for they express two complementary aspects of the living.In this kind of systems the relational unity of the components –the organi-zation- cannot be without a concrete and transient realization into specifics


elements and relations between them –a structure. Conversely there cannotbe a concrete and transient unity of elements in flux—a structure withoutthe stable relations—the organization which integrates them into a persis-tent unity.

The formulation of such a conceptual complementarity has constitutedthe core of the development of autopoietic biology’s program, aiming atdefining the dynamical mechanisms of living systems through the individu-ation of the interplay between the invariant element of their dynamics andthe variant one. Maturana and Varela carried out this theoretical opera-tion on the basis of a specific hypothesis concerning the relation betweenorganization and structure: the continuous structural change, due to thetransformative interactions of the components, produces and maintains theorganization which, in turns, enables the structural change.

This thesis allowed autopoietic biology to provide the definition of aplausible mechanism for the living dynamics, elaborated by Maturana andVarela in relation to the minimal and “fundamental” living system: thecellular system, present in all living forms, evolutively anteceding them,and therefore able to generate them.

The dynamical mechanism of the cellular unity has been explained byMaturana and Varela through the definition of cellular organization, calledby them “autopoietic organization”. It is a theoretical elaboration thatimplicitly recovers the piagetian concept of organizational closure. It cor-responds to the notion of a circular dynamic mechanism: a close chain ofoperations of elements transformations in which the realization of one op-eration triggers and integrates another one, in such a way that the globalcyclical process that emerges is essentially characterized by the propertyto determinate and regenerate itself. It is easy to recognize the influence ofthe piagetian idea in the definition of autopoietic organization, accordingto which:

“ [The autopoetic organization] (...) is a network of production pro-cesses (transformation and destruction) of components which produces thecomponents which :

(1) Through their interactions and transformations, permanently regener-ate and realise the network of processes (relations) which produces thecomponents ; and

(2) Constitute a concrete unity in space, within which they (the compo-nents) exist by specifying the topological domain of its realisation inthat network. ” (Maturana and Varela, 1973)


The more relevant aspect of the autopoietic model is that this theoreti-cal scheme considers relations between processes that involve components.It opens a large range of possible materializations of the autopoietic or-ganization. It does not impose conditions on the specific basic elementaryconstitution of autopoietic systems (Bich and Damiano, 2007). It pointsout the crucial aspect of elements not in their intrinsic properties, but inthey interactive specificity: the relational properties that define the formsof functional correlation that the elements can develop (Bich, 2005).

The constitution of the elementary level is allowed to change, but withinin a well defined range of variability: the space of all the elementary com-positions able to generate a recursive chain of functional relations of recip-rocal production. It is by imposing this only relational constraint onto thelevel of components that the descriptive scheme is able to generate the twodimensions that constitute the dynamics of the living unity: it poses the or-ganization as the invariant element and the structure as the variable one. Indoing so it succeeds in expressing theoretically the idea that in this kind ofsystem the conservation of organization is obtained through the continuousstructural variation.

4. Relational Biology and Rosen’s (M,R)-Systems

A parallel line of thought that goes in the same direction inaugurated by thestudies of cybernetic and continued by genetic epistemology and autopoiesisis the one opened by Nicholas Rashevsky and carried on by Robert Rosen.It has the merit not only to focus on the integrated organization of theliving systemic unity but also to develop specific formal tools in order todescribe it.

Rashevsky developed mathematical biology since the early 30’s (Abra-ham, 2004), giving a great contribution to the mathematical study of thebiological processes of self-organization which led to Alan Turing’s and IlyaPrigogine’s models of pattern generation. His first approaches dealt withseparate biological phenomena from a physical point of view.

In his seminal paper “Topology and Life” in 1954 Rashevsky shifted hisapproach to the study of living systems from the construction of structuralmodels of the biological processes to the modelization of the relational prop-erties of organism, that is, to the study or their organization. His purposewas to develop a mathematical theory able to treat the integrated activityof the organism as a whole: “we must look for a principle which connectsthe different physical phenomena involved and express the biological unityof the organism and of the organic world as a whole” (Rashevsky, 1954).


Following the idea, he reflected upon the difference between physicaland biological phenomena. According to him, the formers are characterizedby metrical aspects and the latter ones by relations. Also, while the struc-tural physical description focuses on the differences between systems, therelational one considers their similarity, embedded in the common minimalorganization that defines their class, in this case the organization of the min-imal living system, shared by all the other more complex organisms. Theidentity is the starting point of his research. The differences are secondaryto the theoretical characterization of the organisms, which is primary withregards to the investigation of the variety of their phenomenology. In look-ing for what makes biology special we notice in fact not only quantitativeaspects but, above all, the presence of complex relations. These ones, ac-cording to Rashevsky, do not concern shape, like in the approaches a laD’Arcy Thompson, but consist in relations between processes or proper-ties.

His new line of research, called relational biology, focused on the abstractdescription of the relations that connect the fundamental properties of or-ganisms up to letting us recognize that network of relations as characteristicof the minimal living being. He opened the way to the development of amathematical theory able to treat not the metric aspect of these systemsbut the integrated properties realizing the organism.

His work in relational biology thus, was based on two related theoreticalassumptions:

(1) the first concerns the importance of relations in identifying and char-acterizing the living: “we postulate that the highly complex biologicalstructures are due to relational forces, that is, that they are formed be-cause as a result of this formation certain qualitative relations appear,relations which make us recognise the structure as a living organism”(Rashevsky, 1972).

(2) The second concerns the possibility to study living systems startingfrom their common organization. It is the Principle of Bio-topologicalmapping, according to which “all organisms can be mapped on eachother in such a manner that certain basic relations are preserved inthis mapping” (Rashevsky, 1960).

Rashevsky’s approach, nevertheless, was characterized by the attempt toconnect the different structural processes and properties observed in livingsystems. It was an additive procedure which had the purpose to build therelational schemes of biological functions of different organisms. What was


conserved in the transformation from one scheme to the other was thehypothetical relational structure of the minimal organization common toall living systems.

Rosen started from Rashevsky’s insights but followed a different path.On a higher level of abstraction, he extended the approach of relational bi-ology to the form that the fundamental relations should assume to expressthe basic autonomy of living beings. He was looking for the mechanism thatcauses the biological properties to be instantiated. In doing so, he conceptu-ally started from the integrated unity of the living organism, characterisedby the capability to self-produce and self-maintain.

In the late 50’s Rosen proposed a formal model based on the mathematicof Category Theory, the (M,R)-System, which described the organizationof the living as the integration of the functioning and the fabrication of thesystem (Rosen, 1958a; 1958b; 1959). His early works are quite obscure andhe clarified some of their mathematical aspects after many years (Rosen,1972), but he fully understood the theoretical meaning of his model onlylater (Rosen, 1991). A recent and clear explanation of the mathematicalformulation of (M,R)-Systems together with some attempts to relate it tobiochemical processes, was provided by some researchers belonging to theautopoietic school (Letelier et al., 2006).

Rosen’s basic assumption in the construction of his model is that everycomponent must be produced, or replaced after its degradation, inside thesystem. He uses some instrument of Control Theory, considering every pro-cess like an input-output one, but in the end he realizes a formal structurevery different from those characteristic of machines.

The first step is to consider the “metabolic” process that usually takesplace in a cell, where some catalysts transform into components the sub-strates coming from outside the system. It is expressed formally by a setof mappings H(A,B) which transform a set of substrates A into a set ofcomponents B.f : A → Bf(a) = b with f ∈ H(A, B)The catalyst f , which has a limited lifespan, must also be produced or re-placed by some processes inside the system. Living organism in fact, unlikemachines, are characterized by a continuous turnover of components. SoRosen introduces another process, called “repair” in his original papers (or“replacement” in Letelier et al., 2006), that produces the metabolic catalystf out of the components belonging to the set B.φ: B → H(A, B)


Fig. 1.

φ(b) = f with φ∈ H(B,H(A,B))

Fig. 2.

Now we have the concatenation of two processes:A → B→ H(A, B)Where:f(a) = bφ(b) = f

But again the mapping φ, which plays the role of catalyst in the productionof f is not produced inside the system. So another process, another map-ping, is needed. It has to be placed inside the system too, but this procedurecan lead to an infinite regress. In order to avoid it, Rosen introduces themapping that makes its model interesting. In fact he looks for it in the com-ponents inside the system, in such a way that the models folds onto itselfand every mapping becomes the output of another mapping of the system.He calls this third process “replication”, but Letelier et al. prefer to call it“organizational invariance” because it is the process that closes the modelonto itself and makes it a self-producing and self-maintaining system underthe continuous turnover of components, while replication reminds more of


biological reproduction (Letelier et al., 2006).The third mapping is β that produces φ out of f :

β: H (A, B) → H (B, H(A, B))β (f) = φ with β∈ H(H(A,B), H(B, H(A,B)))How can we obtain it from inside the system itself? The answer is to identifyβ with b ∈ B. The procedure in order to achieve this result is to consideran evaluation map Evb: H(B,H(A,B)) → H(A,B) that evaluates all the pos-sible choices of φ at b. It is the mapping that correspond to an element b∈ B.Evb: H(B,H(A,B)) → H(A,B)Evb(φ) = φ(b) with Evb ∈ H(B,H(A,B),H(A,B)In order to obtain β from b, the evaluation map must be invertible, thatis, it must be injective.with Evb(φ) = φ(b)Evb(φ)= Evb(φ’ ) implies φ = φ’By the definition of evaluation maps Evb(φ) = φ(b) means also thatφ(b) = φ’ (b) implies φ = φ’Consequently we can obtain β: H (A, B) → H (B, H(A, B)),as the inverseevaluation map Ev−1

b corresponding to b, such that the system closes ontoitself. It gives rise to a hierarchical folded chain of processes:A → B→ H(A, B) → H (B, H(A, B))Where:f(a) = bφ (b) = f

β (f) = φ with β corresponding to b.

Fig. 3.

By this procedure Rosen achieved a circularity inside its model, a closurevery similar to the organizational one proposed by autopoietic theory. Hecalled it “closure under efficient causation” to mean that all the mappingsare entailed by other mappings in the system itself. In fact f entails b,


that corresponds to β which entails φ which entails f so realizing a causalloop. The only element that comes from the environment is the substrateA, coherently with the physical hypothesis of thermodynamical openness.This model expresses the basic circularity of the organization that charac-terizes the continuous flux of processes of production of components insidethe living systems, together with its maintenance made possible by theorganizational invariance embedded by the mapping β.

This model has a particular characteristic, in that its formalism allowsmappings acting on other mappings, and thus self-referential functions. Andalso, it assumes the shape of a closed loop. This introduction of circularityinto a formal model of a minimal living system is of extreme interest be-cause it has usually been avoided in mathematics and in natural sciences.Instead Rosen, who starts from the biological point of view on the organis-mic unity, puts it at the core of Biology. This kind of model marks a deepdifferentiation from the study of artificial machines. Even if Rosen startsfrom mappings that remind of input-output functions, its (M,R)-Systemsare characterized by a qualitatively different approach. In fact as alreadypointed out above, the mappings which represent the active components,can be entailed by other mappings. On the contrary, in artificial systemsthe mechanic components are not produced inside the system, and conse-quently their functions are not entailed in the same way as it happens here.In the homeostatic models of the retroaction mechanisms, the loops concernonly the variables, which re-enter the functions. In this model instead, thesame functions act on each other, so that the system acts not only on someof its own parameters but on its same production rules.

The (M/R)-System is also very different from physical models, becauseof its formal characterization that opens the way to the study of circularity.Also it is placed on a different epistemological level, that of organization.Biology has always lacked an integrated mathematical approach similar tothat characteristic of physics (Bailly and Longo, 2006). The reason is inthe complexity of its phenomenology, that allows only the construction ofstructural models with a limited range of application. The purpose of Rela-tional Biology is to find a top-down approach that focuses on the relationalcharacter of the organizational order proper of the living, that makes us asobservers to identify its unitary identity in spite of its continuous turnoverof components and of the variety of its different realizations. Along thisline of research, Rosen’s (M,R)-Systems are an interesting attempt to catchformally the order in the nothing, on the abstract level of relations be-tween processes. The purpose in fact is explicitly to answer to Schrodinger’s


question on its proper level of analysis. “The graph looks very much like anaperiodic solid, and indeed it possesses many of the properties Schrodingerascribed to that concept. The novel thing is that it is not a “real” solid. Itis, rather, a pattern of causal organization; it is a prototype of a relationalmodel” (Rosen, 2000).

The limit of this approach consists in its abstractness, which makesproblematic to connect these formal mappings to the biochemical processesobserved in the cell, even if it can provide conceptual tools in order todevelop a theory of the cell. In fact the mappings, which represent theproduction rules in the metabolism of the living system, can be conceptuallyequated with enzymes, that determine the reactions that the metaboliccomponents can undergo. Interpreted along this way, the closure underefficient causation means that all the catalysts necessary for an organismto be alive, are produced by the organisms itself (Cornish-Bowden, 2006).

(M,R)-Systems can be considered an explicitation of the concept of or-ganizational closure proposed by the autopoietic theory, especially of thefirst part of the definition of the living provided by Maturana and Varela(Maturana and Varela, 1973). Thanks to its formal nature it can be veryuseful in order to understand the epistemological and theoretical conse-quences of closure. An effort in the direction of an integration of the twotheories has already been started (Letelier et al., 2003).

5. Conclusive Remarks

Starting from the theoretical perspective we tried to present here in order todeal with the complex domain of biological systems, we can assert that sci-ence needs something more than physical structural models. Understandingwhat makes a living being to be alive and different from physical systemsand machines has thus significant consequences for scientific explanation.In fact it can widen the range of the possible descriptions, forcing us toassume different points of view and develop new tools along a path whichcan lead to the construction of a new class of models.

To briefly summarize the conceptual scheme we introduced here, weshowed how physical, artificial and living systems can be classified accord-ing to three different kinds of order, due to the properties of the differentdomains where their specific properties are realized. The third kind, theorder in the nothing, was shown to require a specific modellistic to be de-veloped, which is the result of a long tradition of transdisciplinary scientificresearch. The constructivist concept of organization in fact requires a mod-elization placed on a level of abstraction detached from the properties of


material components and characterized by the acknowledgment the circu-larity between the processes that realize the living. The paramount impor-tance of this basic circularity is shown in Rosen’s proto-relational modelwhose analysis is still only in an initial phase.

As a consequence of the remarks expressed in this paper, we can pointout that the concept of organization as it has been developed by the lineof research that we outlined here, can be put in the middle between twodifferent positions. The first considers organization as an epiphenomenon,as a merely phenomenal consequence of the statistical behaviour of the ma-terial component of the system observed. This is the case of the theoreticalapproach developed by Kupiec and Sonigo presented above under the classof the order from disorder (Kupiec and Sonigo, 2000). The second positionconsiders organization as self-sufficient in itself and puts it on an indepen-dent level. This theoretical position gives rise to a strong duality betweenstructure and organization, that could lead to an ontological contraposition.

The approach whose most rigorous expression is achieved by the au-topoietic theory considers the relation between organization and struc-ture as a descriptive complementarity: “the [organization] of living systemsand their actual (material) components are complementary yet distinct as-pects of any biological explanation: they complement each other reciprocally”(Varela and Maturana, 1972; see also Varela, 1979). They are mutuallydefining concepts, because structure is what changes while organization re-mains invariant. The distinction thus, is an epistemological operation, auseful tool in order to focus on different aspects of the same system. It isalso important to underline again that the organizational scheme charac-terizing a system has important consequences as it allows to make sense ofthe presence of global processes in the system considered. The basic exam-ple is the self-stabilization achieved when a circular relation is realized, likein homeostatic machine. At an higher level of complexity typical of livingsystem the role of organization is crucial in order to understand not onlyself-stabilization but also self-production.

A new step in the development of a Relational Biology that recognizesthe mutual interaction between the structural and organizational descrip-tion can be the attempt to an integration of the thermodynamical andrelational description. The line of development that goes from the insightand issues coming from biology to the development of physics is character-ized by the attempt to widen thermodynamics in order to make sense ofthe peculiarity of the living systems (Prigogine and Stengers, 1979; Kauff-mann, 2000). A possible line of research, coherent with the approach that we


outlined here, is that of a relational thermodynamics (Mikulecky, 2001b).An interesting result in this direction, if we assume the point of view ofautopoietic theory, is Tellegen’s theorem, based on the circular scheme ofclosure in electronic circuits (Tellegen, 1952; for the possible connectionswith the autopoietic theory see Letelier et al., 2005). This approach is stillrelated to the description of artificial machines, based on a positional re-lational order, but it can be a starting point towards the understandingof the consequences of the relational circularity in the thermodynamics ofliving systems.

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371

Anticipation in Biological and Cognitive Systems:The Need for a Physical Theory of Biological Organization

Graziano Terenzi

Universita degli Studi di Pavia, [email protected]

This paper deals with the problem of understanding anticipation in biologicaland cognitive systems. It is argued that a physical theory can be consideredas biologically plausible only if it incorporates the ability to describe systemswhich exhibit anticipatory behaviours. The paper introduces a cognitive leveldescription of anticipation and provides a simple characterization of anticipa-tory systems on this level. Specifically, a simple model of a formal anticipatoryneuron and a model of an anticipatory neural network which is based on theformer are introduced and discussed. Some learning algorithms are also dis-cussed together with related experimental tasks and possible integrations. Theoutcome of the paper is a formal characterization of anticipation in cognitivesystems which aims at being incorporated in a comprehensive and more generalphysical theory.

Keywords: Biological Systems; Cognitive Systems; Biophysics; Complex Sys-temsPACS(2006): 87.10.+e; 87.16.Ac; 89.75.Fb

1. Introduction

The ability to anticipate specific kinds of events is one of the most amazingfeatures of biological organization. Mainly, it is related to the functioning ofcomplex biological systems which perform cognitive functions. An anticipa-tory system is a system which decides its behaviour by taking into account amodel of itself and/or of its environment; i.e. it determines its current stateas a function of the prediction of the state made by an internal model for afuture instant of time (Rosen 1985). A system endowed with anticipatoryqualities is also named a proactive system. Proactivity and anticipation,thus, are two important features of biological organization.

As a matter of fact, anticipatory properties play an important role in themechanisms which rule learning and control of complex motor behaviours.Indeed, as it stands, motion is the only means biological organism have to

372 Graziano Terenzi

interact both with their environment and with other organisms (Wolpert,Gharamani & Flanagan 2001). Recent empirical studies have demonstratedthe involvement of the motor system in processes ranging from the observa-tion and anticipation of action, to imitation and social interaction (Gallese,Fadiga, Fogassi & Rizzolatti 1996; Rizzolatti & Arbib 1998; Liberman &Whalen 2000; Wolpert, Doya & Kawato 2003). In this context the problemof anticipation is essentially the problem of understanding how cognitivesystems of complex organisms can take into account the future evolution ofevents which take place in the interaction with their environment in orderto take decisions and determine their behaviour (Sutton 1990; Stolzmann1998; Baldassarre 2002, 2003).

The study of anticipation in the context of sensory-motor learning andcoordination has been carried out by developing an interesting ensemble ofcomputational models which aim to take into account the anticipatory prop-erties of biological neural networks. Among these models we must mentionthe Forward Models by Jordan e Rumelhart (1992), the architectures basedon Feedback-Error Learning procedure (Kawato 1990) and on the Adap-tive Mixture of Local Experts (Jacobs, Jordan, Nowlan & Hinton 1991; Ja-cobs 1999), such as the MOSAIC model (Wolpert & Kawato 1998; Haruno,Wolpert & Kawato 2001). Essentially, on the basis of computational stud-ies it has been hypothesized that the central nervous system is able tosimulate internally many aspects of the sensory-motor loop; specifically, ithas been suggested that a dedicated neural circuitry is responsible of suchprocessing and that it implements suitable “internal models” which repre-sent specific aspects of the sensory-motor loop. Internal models predict thesensory consequences of motor commands and, for this reason, are calledforward models, as they model forward causal relations (i.e. to a future in-stant of time) between actions and their sensory consequences. A forwardmodel is employed to predict how the state of the motor system will changein response to a given motor command (motor command ♦ state of the mo-tor system). A forward model is therefore a predictor or a simulator of theconsequences of an action. The inverse transformation from the state to themotor command (state ♦ motor command), which is needed to determinethe motor command required to reach a given goal, is performed by whatis called an inverse model.

On the other hand, it is also clear that biological systems are first of allphysical systems. For this reason understanding anticipation is a problemfor any physical theory which aims at explaining the emergence of biolog-ical organization. For this same reason, the quest for such an explanation

Anticipation in Biological and Cognitive Systems 373

requires both the clarification of the levels of description of the systemsunder study, and their integration in a unitary and more comprehensivetheoretical framework.

The main goal of this paper, then, is to identify a possible characteriza-tion of anticipation in biological systems which perform cognitive process-ing, in such a way as to bring to light some properties that any biologicallyplausible physical theory must incorporate.

2. Anticipatory Neurons

Essentially, a formal neuron is function F which maps an input space Ito a corresponding output space A. A non-anticipatory formal neuron is afunction which has the form

at = F (it),

where it and at are respectively input and activation of the neuron at timet.An anticipatory formal neuron instead can have the following form

at = F (it, it+τ ),

it+τ = G(it),

where it+τ is computed by a suitable function G, named a predictor.If the activation at depends both on the input at time t and on activation

of the neuron predicted at time t+τ the neuron has the following form

at = F (it, at+τ ),

at+τ = G(at).

Fig. 1. Three different models of a neuron: (a) a standard non-anticipatory formal neu-ron; (b) an anticipatory formal neuron which anticipates over input; (c) an anticipatoryformal neuron which anticipates over its own output.

This amounts to saying that the activation of a neuron at time t does notdepend only on its input at time t, but also depends on its input (or in case


on its output) at time t+τ , as it is computed by a suitable function G. Here,τ represents a “local time” parameter strictly related to predictor G. In thecontext of this model, we have to distinguish indeed between the global timet and the local time t’= t+τ . Whereas global time t describes the dynamicsof function F, local time t′ describes both the activation dynamics and thelearning dynamics of the predictor function G. Parameter τ identifies atemporal window over which the dynamics of the anticipatory neuron isdefined. It sets up the limit of an internal counter which is used, step bystep, to fix values of function G during the learning phase (approximation).In this sense τ represents a measure for the quantity of information thata neuron can store while executing its computation. This fact also meansthat the neuron is endowed with a kind of internal memory. This memorymakes the neuron take into account more information than the informationwhich is already available instantaneously.

3. An Example of Dynamical Evolution of a SingleAnticipatory Neuron

The following example illustrates the dynamics of a single anticipatory neu-ron trained to associate input and output by means of the Widrow–Hoffrule (Widrow and Hoff 1960). The predictor also is trained by means of theWidrow–Hoff rule. Let us consider a neuron with the generic form like theone illustrated in Figure 2.

Fig. 2. An anticipatory formal neuron with suitable connections weights.

Let us suppose that F is a sigmoidal function, that is it has the form

at =1

1 + e−Pi

where P is the activation potential of the neuron and it is computed as,

Pi =∑

j

wijij


and where wij are connection coefficients of the input links to the i-th unit,and ij are its input values. Let us suppose that G has the same form as Fexcept for computing the quantity, i’t+τ , devoted to approximate its inputat time t+τ instead of computing output at that is:

i′t+τ =

11 + e−Pi

where P, i.e. the potential of G, is computed as the potential of F.As it can be seen, the anticipatory neuron is a system which includes

two different subsystems, the anticipator F and the predictor G. Whereas Fcomputes its activation on the global time scale t, G computes its activationon the local time scale t’; and the activation is computed by storing inmemory the input to the anticipator F both at time t and at time t+τ .Function G, on the other hand, is approximated by resorting to these values.

Table 1 illustrates synthetically a generic dynamics for the previouslyintroduced anticipatory formal neuron.

Table 1- t = 0- τ =1- t’ = (t + τ)=1- it = i0

- t = 1- τ =1- t’ = (t + τ)=2- it = i1

- t = 2- τ =1- t’ = (t + τ)=3- it = i2

- t = . . .- τ =1- t’ =. . . .- it = . . .

-INITIALIZEVARIABLES

-COMPUTE G

i′1 = G(i0)

-PROPAGATE AC-TIVATIONa0 = F (i0, i

′1)

- APPROXIMATE“F”

-COMPUTEERROREF = 1

2 (a0 − aT0 )

-WEIGHTSUPDATE∆w12 = −η

∂EF∂w12

∆w11 = −η∂EF∂w11

APPROXIMATE “G”

- COMPUTE ERROR

EG = 12 (i

′1 − i1)

- WEIGHTS UPDATE∆w21 = −η

∂EG∂w21

=

= −η(i′1− i1) ·G′(PG) · i0

with PG = (w21i)0 − sG

-COMPUTEACTIVATION Gi′2 = G(i1)

- PROPAGATE ACTI-VATIONa1 = F (i1, i

′2)

APPROXIMATE “F”- COMPUTE ERROREF = 1

2 (a1 − aT1 )


∂EF∂w12

∆w11 = −η∂EF∂w11

APPROXIMATE “G”

- COMPUTE ERROR

EG = 12 (i

′2 − i2)


∂EG∂w21

=

= −η(i′2−i2)·G′(PG)·i1

with PG = (w21i)1 − sG

- COMPUTE ACTIVA-TION Gi′3 = G(i2)

- PROPAGATE ACTI-VATIONa2 = F (i2, i

′3)

APPROXIMATE “F”- COMPUTE ERROREF = 1

2 (a2 − aT2 )


∂EF∂w12

∆w11 = −η∂EF∂w11

. . . . . .

. . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .


Essentially, in the course of the dynamical evolution of the neuron, foreach couple of patterns presented to the neuron, a propagation cycle and alearning cycle are first executed for the predictor G, and then a propagationcycle and a learning cycle are executed for the anticipator F. It must beunderlined that the choice of Widrow-Hoff rule for the training cycle ofthe neuron has been taken only for simplicity and for illustrative purposes;other algorithms could suit well.

4. Networks of Anticipatory Neurons

By taking inspiration from the simple model of an anticipatory neuronintroduced in the previous section, networks of anticipatory neurons canbe designed that carry out complex tasks which involve the employment ofsuitable “internal models” (be they implicit or explicit) of their external orinternal environment in order to take decisions.

The scheme in Figure 3 describes a simple example of a neural networkconstructed by resorting to anticipatory neurons as its building blocks,which calculate their activation as a function of input values predicted attime t+τ . Input units within the scheme are represented by an array i of

Fig. 3. A simple neural network constructed by resorting to anticipatory neurons as itsbuilding blocks.

input values. Hidden units are represented by a composition of sigmoidalfunctions, i.e. g and f respectively. Whereas the functions g within thehidden layer give rise to specific activation values, xg, represented here by anarray xg, functions fgive rise to another set of activation values. Similarly,for the output layer, the activation of output units is computed by means of


composition of two suitable functions g and f , and their activation valuesare collected respectively in the arrays ug and uf . It must be stressed thateach function g is endowed with a stack A, i.e. a memory which storesinput vectors spread across a suitable temporal window t+ τ ; it can berepresented by a τ x N dimensional matrix M, where N is the numberof storable patterns, and τ is the time step when they are given as inputto g. For each layer of the network there are two kinds of connectionswhich play different roles: connections to functions f(f -connections) andconnections to functions g(g-connections). Indeed, f -connections betweeninput and hidden layers are characterized by the connection weights w

(which in the figure are represented without indices); g−connections, on theother hand, are represented by connection weights w′. Similarly, on the onehand, f -connections between hidden and output layers are characterized byconnection weights c; on the other hand, g-connections are characterized byconnection weights c′. The case reported in figure shows a network in whicheach anticipatory neuron is endowed with a neuron-specific stack A whichlets the neuron store and compare patterns given as input to the neuronin its temporal window (i.e. Input or output patterns, depending on thenature of the device). With regards to the latter, the parameter τ can beapproximated by suitable learning algorithms, thus giving rise to networkswhich are able to perform tasks which require indefinite long-range timecorrelations for their execution.

Training in such a network can be made in many ways. A basic optionis to employ a back-propagation algorithm (Rumelhart, Hinton & Williams1986) in order to learn weights w and c of the functions f within the layersof the network, and to use Widrow-Hoff rule for the weights w′and c′of thecorresponding functions g.

In the case of an “on-line” learning procedure, a learning step for thewhole network can be implemented, for example, through the followingsuccession of phases:

• Initialize Variables• Give a pattern as Input and Propagate Pattern• For each layer,

– Modify weights of functions g according to their temporal win-dow

– (Widrow-Hoff)– Compute g

– Compute f


– Modify weights of functions f according to the correspondingtarget pattern (Backpropagation)

In table 2, a list of the basic variables of the model, i.e. a set of constructsthat can be used to implement and simulate the model under study.

Table 2Constructs Descriptioni Input vectorxf Vector of the activation of functions f within the

hidden layerxg Vector of the activation of functions g within the

hidden layeruf Vector of the activation of functions f within the

output layerug Vector of the activation of functions g within the

output layerW Weight matrix of the f -connections between input

and hidden layersW’ Weight matrix of the g-connections between input

and hidden layersC Weight matrix of the f -connections between hidden

and output layersC’ Weight matrix of the g-connections between hidden

and output layersAw τ x N matrix wich stores N input to functions g of

input-hidden layer within temporal window τ

Ac τ x N matrix wich stores N input to functions g ofhidden-output layer within temporal window τ

5. τ -Mirror Networks: Architectures

In the previous sections we have introduced both single anticipatory neu-rons and a simple network architecture. In the example of the last sectioneach function g was connected to one and only one function f within itslayer, and, moreover, it was characterized by a neuron-specific time win-dow. This fact entails the employment of a specific stack for each functiong. According to this description, the stack can be represented as a variable-


structure quadrimensional matrix which comprises 1) the parameter τ ofthe temporal window, 2) the number N of storable patterns, 3) a dimensionwhich represents neurons, and 4) a dimension which represents the value ofτ for those neurons.

Another possibility is to consider network architectures where functionsg are connected in output not only to a single corresponding function f butto all of the other functions f belonging to the same layer. Moreover we canconsider a generalized window t+ τ for every g of the same layer or of thewhole network. The set of all functions g of the network constitutes a real“temporal mirror” for the corresponding functions f . In a sense, we can saythat a part of the network learns to represent the behaviour of the otherpart from the point of view of the specific task at hand, by constructingan implicit model of its own functioning. Briefly, the netwok is capable ofself-representation. For this reason the units that approximate functions g

of the network within the temporal window t + τ (i.e. the ones we havepreviously dubbed “predictors”) are called “τ -mirror units” of the network.The schemes presented in Figure 4 and 5 represent this idea synthetically.

Fig. 4. An anticipatory neural network which anticipates over its input space.

Figure 4 illustrates an anticipatory network which predicts over its inputspaces. As anticipated, if each function g is connected to every function fwithin the same layer, and not only to the corresponding ones, then they canbe grouped together in a suitable layer, called a “mirror layer”. Moreover,if the same temporal window t+τ is generalized to all of the mirror unitswithin the same layer, then the complexity of both the representation andthe implementation of the model can be further reduced.


Fig. 5. An anticipatory neural network which anticipates over its output space.

Figure 5 represents an anticipatory neural network which predict overits output spaces.

6. τ -Mirror Networks: Learning Algorithms

We have previously introduced a learning scheme for the training of theneural network model under study. And this scheme is a supervised learningscheme which is based on 1) the employment of the Backpropagation rulefor the adjustment of the weights of the functions f(W and C) and 2)the employment of the Widrow-Hoff rule for the adjustment of the weightsof the functions g. By the way, there are other possibilities that can beexplored.

6.1. Reinforcement Learning

A straightforward solution is given by the employment of a reinforcementlearning algorithm for approximating the weights of the functions f . Typ-ically, reinforcement learning methods are used to approximate sequentialdynamics in the context of the online exploration of the effects of actionsperformed by an agent within its environment (Sutton & Barto 1998; Bal-dassarre 2003). At each discrete time step t the agent perceives the state ofthe world st and it selects an action At according to the latter perception.As a consequence of each action the world produces a reward rt+1 and anew state st+1. An “action selection policy” is defined as a mapping fromstates to action selection probabilities π: S×A → [0, 1]. For each state s inS, a “state-evaluation function” is also defined, Vπ[s], which depends on the


policy π and is computed as the expected future discounted reward basedon s:

V π[s] = E[rt+1+γrt+2+γ2rt+3+..] =∑

a∈A

[π[a, s]∑

s′∈S

[pass′ (rt+1 + γV π[s

′])]],

where π[a, s] is the probability that the policy π selects an action a giventhe state s, E is the mean operator and γ is a discount coefficient between 0and 1. The goal of the agent is to find an optimal policy to maximize Vπ[s]for each state sin S. In a reinforcement learning setting, the estimationof expected values of states s, V(s), is computed and updated tipically byresorting to Temporal Differences Methods (TD methods); this amountsto saying that the estimation of V(s) is modified by a quantity that isproportonal to the TD-error, defined as follows

δ = r + γ V (s′)− V (s).

An effective way to build a neural netowrk which is based on reinforcementlearning is to employ a so called Actor-Critic Architecture. Actor-Critic

Fig. 6. An Actor-Critic architecture.

Architectures are based on the employment of distinct data structures foraction-selection policy management (“actor”) and for the evaluation func-tion (“critic”). The “actor” selects actions according to its input. In orderto select an action, the activation of each output unit of the actor is given


as input to a stocastic action selector which implements a winner-take-allselection strategy. The probability for an action ag to become the selectedaction aw is given by

P [ag = aw] =mg[yt]∑k mk[yt]

,

where mg[yt] is the activation of output units as a function of input yt.On the one hand, the “critic” is constituted by both an Evaluator and

a TD-critic. The Evaluator is a network which estimates the mean futurerewards which can be obtained in a given perceived state of the worldyt = st. The Evaluator can be implemented as a feed-forward two-layernetwork with a single output unit which estimates V’[s] of Vπ[s], where thelatter is defined as above. On the other hand, the TD-critic is a neural im-plementation of the function that computes the TD-error et as a differencebetween the estimations of Vπ[s] at time t + 1 and time t:

et = ((V π[yt]))t+1 − ((V π[yt]))t = (rt+1 + γ V π[yt+1])− V π[yt] ,

where rt+1 is the observed reward at time t + 1, V’π[yt+1] is the estimatedvalue of the expected future reward starting from state yt+1 and γ is,as usual, a suitable discount parameter. According to such a TD-error,the “critic” is trained by means of Widrow-Hoff algorithm; specifically itsweights wj are updated by means of the following rule

∆wj = η etyj = η((rt+1 + γ V π[yt+1])− V π[yt] )yj .

Also the “actor” is trained by resorting to the Widrow–Hoff algorithm andonly the weights which correspond the the winning unit are updated by therule

∆wwj = ζ et(4mj(1−mj))yj

= ζ((rt+1 + γV ′π[yt+1])− V ′π[yt])(4mj(1−mj))yj ,

where ζ is a suitable learning parameter between 0 e 1 and (4 mj (1- mj))is the derivative of the sigmoid function corresponding to the j-th outputunit which wins the competition, multiplied by 4 in order to omogeneizethe learning rates of the two networks.

6.2. Reinforcement Learning For Anticipatory Networks

Extending a reinforcement learning algorithm to the anticipatory neuralnetworks described above is quite straightforward. Let us consider an an-ticipatory network like the one described in section 4., but with two-layers


only. For this reason it will have a single layer of τ -mirror units. Then, letus consider an “Actor” constructed by resorting to the latter network; inthis case, input units code for the state of the world at time t, whereasoutput units code for possible actions that can be executed at time t. The“Critic” can be considered as analogous to the one just discussed in theprevious section.

In this context, the approximation of weights for the functions f withinthe output layer of the network goes exactly as described in the previoussection, that is by modifying the weights of the f -connections accordingto the TD-error computed by the Critic. But how can the weight over theg−connections be updated? Also in this case, the extension is straight-forward if we consider the fact that the adjustment of the weights of theseunits is based on the Widrow–Hoff rule. The idea is to employ the TD-errorscorresponding to the reference temporal window rather than the quadraticerror corresponding to input (or output) patterns. This essentially meansthat the stack of the g units has to store, beside the input patterns to g

within the temporal window τ , also the corresponding TD-errors. Therefore,the error signal is computed not by considering the quadratic error withrespect to suitable targets, but by resorting to the TD-error correspondingto the reference time step.

6.3. Other Learning Algorithms

Beside the ones discussed in the preceding sections, there is a wider set oflearning methods that could be employed for the training of this kind of net-works. However, their application must be suitably assessed and analyzedfirstly by discussing their theoretical implications. Even if it is not the goalof this paper to discuss this topic, we recognize that it is a very interestingpossibility to extend the application of non-supervised learning algorithms(such as Kohonen-like-methods) to this training setting. Another possibilityis to employ genetic algorithms.

7. Experimental Tasks for Simulation Assessment

How can the viability of such an approach to the study and design of neuralarchitectures be assessed? If, on the one hand, the assessment of the viabilityof a theoretical model essentially amounts to test its predictive and/orexplicative power with respect to natural phenomena, on the other hand,the assessment of its utility from a practical point of view requires it tosatisfy suitable constraints and goals within a design setting; in this sense


it must solve specific scientific and design problems.Scientific evaluation, on its side, calls for the construction of a model

whose behaviour can be confronted with real neuroscientific data. In thissense, the τ -mirror model can be employed in the context of problemswhich entail time-series processing. Problems of this kind are fundamentalin the Neurosciences, but the domain which can mostly provide the basisfor assessing the anticipatory architecture introduced above is the one thatrelates to the understanding and reconstruction of the neural mechanismswhich perform sensory-motor learning and control.

For example, a possible experimental task is based on an simulation setup which includes a simulated artificial environment interacting with anartificial organism endowed with suitable sensors and effectors; specificallyit is characterized by a robotic arm. In this context the agent learns to per-form a so called “pointing” task which consist of following a trajectory ofan object which moves in the environment. Moreover it must learn to antic-ipate the trajectory of the object. In order to do so, the artificial organismmust be endowed with suitable sensors and actuators; specifically it mustbe endowed with an artificial retina, a two-joint arm (whose position canbe represented in a polar coordinate reference system ) and an anticipatorycognitive system.

Fig. 7. A possible setting regarding the simulation of a pointing task by a robotic arm.

A simpler simulation task, which is similar to the one just described,is to follow and anticipate a trajectory of an object in a more abstractmanner, that is by employing the whole body of the simulated agent andnot the simulated arm.


Other experimental settings can be conceived that involve the executionof both planning and navigation tasks in simulated environments. More ab-stract simulation tasks can involve learning and anticipation of things suchas numeric series, grammars and automata, which do not possess necessarilya material counterpart.

8. Conclusions and Future Work

The construction of neural network architectures endowed with self-representation abilities as well as with the proactive ability to show be-haviours which do not depend only on their state at time t but also ontheir predicted state at a future time step is a strategic lever for under-standing phenomena ranging from sensory-motor control to thought andconsciousness. In this explorative paper we have introduced a possible formof both anticipatory neurons and networks which have been named “τ -mirror networks”. We have proposed some learning algorithms for this kindof networks and have put forward the possibility to extend other algorithmsin order to train them.

Among the future developments, a possibility (which has to be evaluatedin the detail) is to change the form the anticipatory networks in order tomake them more biologically and physically plausible. For example, Dubois(1998) has introduced an anticipatory form of the neuron of McCullochand Pitts which is based on the notion of hyperincursion and that couldbe employed in the construction of another kind of anticipatory network.The hyperincursive computing systems are demonstrated to show stabilityproperties much relevant if compared to the non-anticipatory counterparts.Therefore, a future development could be the integration of hyperincursivitywithin the theroetical framework presented in this paper.

Notwithstanding the aforementioned theoretical developments, the mostimportant and critical step is to develop a more comprehensive physical the-ory which will be able to describe anticipatory systems including those ofthe kind introduced in this paper. We strongly believe that such an inte-gration is necessary to understand this very important aspect of biologicalorganization. Without such a step, indeed, it would be impossibile to ac-count for a wide range of phenomena occurring in both the biological andcognitive realms.

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1. Baldassarre (2002) Planning with Neural Networks and ReinforcementLearning. PhD Thesis. Colchester-UK: Computer Science Departmente,


University of Essex.2. Baldassarre (2003) “Forward and bi-directional planning based on reinforce-

ment learning and neural networks in a simulated robot” in Butz, Sigaud& Gerard (Eds), Adaptive Behaviour in Anticipatory Learning Systems,Springer, Berlin, pp 179–200.

3. Dubois (1998) “Computing anticipatory systems with incursion and hyper-incursion”. AIP Conference Proceedings 437, pp 3–29.

4. Gallese, Fadiga, Fogassi & Rizzolatti (1996) “Action recognition in the pre-motor cortex”. Brain 119, pp 593–609.

5. Haruno, Wolpert & Kawato (2001) “Mosaic model for sensorimotor learningand control”. Neural Computation, vol. 13, 2201-2220.

6. Jacobs, Jordan, Nowlan & Hinton (1991) “Adaptive mixtures of local ex-perts”. Neural Computation, Vol. 3, 79-87.

7. Jacobs (1999) “Computational studies of the development of functionallyspecialized neural modules”. TRENDS in Cognitive Sciences, Vol. 3, No. 1,pp 31–38.

8. Jordan & Rumelhart (1992) “Forward models: supervised learning with adistal teacher”. Cognitive Science, Vol. 16, pp 307–354.

9. Kawato (1990) “Computational schemes and neural networks for formationand control of multijoint arm trajectory”, in Miller III, Sutton & Werbos(Eds) Neural Networks for Control, MIT Press, Cambridge, MA.

10. Liberman & Whalen (2000) “On the relation of speech to language”.TRENDS in Cognitive Science, Vol. 4, pp 187–196.

11. Luppino & Rizzolatti (2000) “The organization of the frontal motor cortex”.News in Physiological Science, Vol 15, pp 219–224.

12. McClelland & Rumelhart (Eds) (1986) Parallel Distributed Processing. Ex-ploration in the Microstructure of Cognition, MIT Press, Cambridge, MA.

13. Miller III, Sutton & Werbos (Eds) (1990) Neural Networks for Control, MITPress, Cambridge, MA.

14. Rizzolatti & Arbib (1998) “Language within our grasp”. Trends in Neuro-sciences, Vol. 21, pp 188–194.

15. Rizzolatti, Fogassi & Gallese (2001) “Nerophysiological mechanisms under-lying the understanding and imitation of action”. Nature Reviews Neuro-science, Vol 2, pp 661–670.

16. Rizzolatti & Craighero (2004) “The mirror-neuron system”. Annual Reviewof Neuroscience, Vol 27, pp 169–192.

17. Rosen (1985) Anticipatory Systems, Pergamon Press.18. Rumelhart, Hinton & Williams (1986) “Learning Internal Representations

by Error Propagation”, in McClelland & Rumelhart (1986).19. Stolzmann (1998) “Anticipatory classifier systems”. Genetic Programming

1998: Proceedings of theThird Annual Conference, pp 658–664.


20. Sutton (1990) “Integrated architectures for learning, planning and reactingbased on approximating dynamic programmino”. Proceedings of the SeventhInternational Conference on Machine Learning, San Mateo, CA: MorganKaufman, pp 216–224.

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389

How Uncertain is Uncertainty?

Tibor Vamos

Computer and Automation Research Institute,Hungarian Academy of Sciences, Hungary

[email protected]

The gist of the paper is the fundamental uncertain nature of all kinds of un-certainties and consequently a critical epistemic review of historical and recentapproaches, computational methods, algorithms. The review follows the devel-opment of the notion from the beginnings of thinking, via the Aristotelianand Skeptic view, the medieval nominalism and the influential pioneeringmetaphors of ancient India and Persia to the birth of modern mathematicaldisciplinary reasoning. Discussing the models of uncertainty, e.g. the statisti-cal, other physical and psychological background we reach a pragmatic modelrelated estimation perspective, a balanced application orientation for differentproblem areas. Data mining, game theories and recent advances in approxima-tion algorithms are discussed in this spirit of modest reasoning.

Keywords: Uncertainty; Probability; Randomness; Epistemics; History of Sci-ence; Computational PragmaticsPACS(2006): 01.90.+g; 02.50.Cw; 01.65.+g; 02.50.r; 01.70.+w

1. They have No Science?

We are unable to get deep into the minds of our ancestors. Neverthelesswe can draw some hypotheses on their beliefs, on their view about theworld order mirrored in their fantasy. With some certainty, we can statethat most of the events, circumstances were uncertain for them, especiallyfrom our point of view. The difference in definition is relevant for the wholeessay. What I take as certain, can be uncertain for any contemporary andeven more for somebody living in the future and possessing a much deeperknowledge about the phenomena.

The world was nearly totally uncertain for them who had to transfer allcauses to imagined beings. Even the most certain event, death was fuzzifiedby several rites into a life transition and continuation in somehow similarmanner. That, for us, primitive looking manner of burial cults survives stillin more sublime forms of current high brow civilizations.

390 Tibor Vamos

Tales, sculptural figures, preserved folklores testify the hypothesis thatdue to a general feeling of uncertainty, being exposed to epidemics, dis-asters of natural environment, famine, the ancient man was much morehysteric than the everyman of today. That was true as long as the MiddleAges, Huisinga [21], a great scholar of the period, described that with livelydocumentation.

The divorce of uncertainty and certainty started with science. Scienceinterpreted in our sense, that is with the Greek enlightenment. Generallyspeaking, until the height of the Greek philosophy, people of the Mediter-ranean and Eastern civilizations collected plenty of remarkable phenomena,and some can be still in our eyes considered to remain of scientific char-acter; but they looked at these phenomenological facts as naturally givenrealities (all kinds of spirits, gods included) and had no curiosity feeling tosearch for further causes, interactions, generalizations, especially not in theframes of an organized, culturally adopted search discipline.

This means that, for the advanced Greeks, what could be included intothe frame of their contemporary science was structured and disciplined ina new edifice of science; and all that remained, formed the uncertain realmof non-science, beliefs, religions, curiosities. In that evolving rationalism,combined with the mythical foggy environment of the Pythagorean age,even science and arts were not separated. According to a witty remark ofBertrand Russel [35], Pythagoras, could have been some kind of a mixtureof Einstein and Mary Eddy Baker, the 19th century founder of the AmericanChurch, Christ Scientist. May we add Schonberg?

Aristotle, in his unique effort in creating a rational, causal conceptualworld, adhering to the sensible and conceivable world of the observable, sep-arated the basic, causal and eternal looking primitives from the secondary,accidental features of those. He tried to do this separation with a referenceof the accidental co-occurrence of the causal.

That view is still surviving in the philosophy of strong causality, thoughwhat we consider to be causal now, after experimental demonstration ofquantum physical entanglement is even more open than for the geniusesof the last two and a half millennia. The discussion of the current state ofcausal, accidental and other intermediate philosophical considerations, laymuch beyond of the extent of this paper.

We can only admire those pioneers of scientific thinking and thinkingin general who tried to understand the world, first without prejudices andwithout any of the fantastic experimental observation devices of modernages. The Aristotelian analysis of the accidental secondary looking features


[3] conclude in a definition very close to our information theory view aboutsignal and noise. And, a wonderful scientific manifesto added to all those: Ido not dwell with these secondary (συµβεβηκoς accidental) phenomena be-cause they have no science The inclusion of these, mostly affections relatedphenomena, into the scope of science, as we understand now, is a rather re-cent advance as modeling of human behavior in economy and social sciencesbecame possible subjects of computational efforts.

Aristotle was “only” a beginning of modern scientific thinking but along row of Greek philosophers added still remarkable thoughts to thatbeginning, some contradicting to the rather fundamentalist looking causal-logical teaching of Aristotle. The dogma of the excluded middle was andremained the cornerstone of that way of thinking. Yes or not, true or false isalways a pragmatic, sometimes necessary simplification of problems, thoughthe reality covers the whole space between the two.

2. The Skeptics, Forerunners of Modern Science Philosophy

From the few but in remains admirable and admirably rich treasury ofGreek philosophy we give a prominence to the Skeptics. The unclear originsgo back to the fourth-third century BC, to Purrhon (Πυρρω′ν), and contin-ued under heavy criticism and several later additions to the texts of SextusEmpiricus [37] and Diogenes Laertios [13] in the second and third centuryAD reporting especially the modalities of Aneisidemos (Aνεισιδηµoς), wholived in the first century BC. In these modalities, the tropos (τρoπoς), arecollected ten various arts of relativities in the perception of the reality. Mostof these relativity modalities are parts of our commonsense thinking that,per se, are strikingly modern in our eyes, and provoked well understandableresistance on behalf of the most lucid contemporary Greek minds.

We can recognize rather precise definitions of modern ideas about therelativity of scientific ideas due to social conditions, i.e. the ideas of Bloor[6] and the school of the social-philosophical “strong program”. Even moreinteresting is the tropos about the fuzzy nature or estimations, not differentfrom the current fuzzy concept of Lotfi Zadeh.

The world of uncertainty was alien for the rational, scientific orientedphilosophy that tried to include all phenomena into well understandableand sensible general frames without the instruments of further periods thatcould differentiate much more, based on more sophisticated theories aboutcausal relations.

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3. Nominalists of Middle Ages: de dicto - de re Uncertainty

The philosophy of the Middle Ages, referring to several ideas of the Greekphilosophy, concentrated on the divine organized, overall regulated worldview and, on the other hand the same divine, not understandable creationof mysticism, surprising wonders of the everyday life, i.e. on those, notcomprehended by the contemporary discourse of science.

For us, the most relevant stream of thinking is the growing deepness ofnominalism, the separation of the name from the named, de dicto – de reconnection and antithesis. In the course of developments in thinking aboutlogic Kneale, W. and Kneale, M. [24] proved an excellent orientation.

That line of thinking is one of the most important research areas ofcommunication, the problem of adequate information emission and its per-ception. The novelty is the inclusion of the third constituent, the vehicle ofinformation. Nevertheless, we can trace some past characteristics of the ve-hicle of information history by concerning the destiny of scripts and books.Pro captu lectoris habent sua fata libelli, the booklets have their own fate,how they are accepted by the reader, wrote Terentianus Maurus, a gram-matologist in the second century.

By the ubiquity and instant nature of electronic information world, thesequestions received a highly practical relevance. One of these aspects is theindirect, electronic representation of the individual, of the Self, used forall kinds of substitution of the direct presence. Personal identification is acrucial subject of individual freedom, privacy and security.

The same, relativistic nominalist problem arises in the effects of var-ious mass media, unprecedented manipulation technologies by electroniccommunication. The problem of legal regulation is now an everyday andubiquitous scruple, depending on the supposed weights of communicationperception of the population concerned.

We can return to the Aristotelian dictum: has no science with therelevant complement: has no unified science and a disputable, not provableremark: will not have a unified science. Unified science is addressed, in spiteof their theoretically provable incompleteness and continuous developments,moreover reinterpretations, not similar to other branches of mathematics.

4. Mathematics as the Last and Computable Form of anyScience

Here we have to make a small detour defending that rather awkward def-inition about science, mathematics and computability. We speak about


science, in the Anglo-Saxon sense, i.e. science in the sense of reproducible,experimentally strongly provable phenomena, those having firm definitionsfor models of reproduction under defined conditions; and these model def-initions are firm enough to be translated into disciplinary formulae andmathematics. The last step is more obvious for the present everyday view:based on the mathematical formulae the model can be programmed andthe outcome, the computational reproduction computable.

The last statement clarifies the way of reasoning. First in physics, butnow every practically manageable problem of biology related medicine andagriculture, statistics and finance related economy followed the overall dom-inating modern production processes of technology: computer aided design,computer aided scheduling and manufacturing. All additional problems (re-member the Aristotelian συµβεβηκoς, i.e. additional, secondary, acciden-tal) are related to some kinds of uncertainty, not well computable incidents.

5. Ars Coniectandi, Statistics

Ian Hacking′s excellent book, The Emergence of Probability [15] ries toanalyze the other reasons of the late emergence and the relevance of Pascal’s[32] and Jakob Bernoulli’s [5] contribution to the ars coniectandi, the artof dicing (and we add, the same Latin word is for reasoning based on aguess) and soon later to the fast growing interest on problems, related tostatistics in economy, social conditions, meteorology and measurement inexperiments on physical, chemical and biological phenomena.

Observation of returning frequencies and estimation of future events runconnected and independent; the second more on the Bayesian conditionalprobability lines. The detailed history of mathematical ideas about proba-bility and the roles of great mathematicians are well detailed in that citedbook of Hacking [15] and many others e.g. recently by Halpern [17].

Models of uncertainty defined the progress of calculations. For a ratherlong time dicing and similar games served as models, the way of observa-tion developed statistics and the epistemic curiosity was the contradictionbetween the rather good approximations given for large numbers of exper-iments and the complete uncertainty of a single roll.

Statistical models followed the hypotheses about the nature of the inves-tigation. Typical are distributions around a certain mean value, variationslike the biological data of height, weight, life expectancy, etc. symmetric andnot symmetric. E.g. life expectancy at a certain advanced age is typicallyasymmetric.

Another model, used later for many purposes, is the reliability of

394 Tibor Vamos

content estimation in beer bottles, the Student distribution, measuredon a limited number samples. The problem is similar to the Nala – Kalistory of mythology from India, that will be cited a little later, but now itreceived a firm scientific analysis and a conjecture, where to use, how touse? This remark put light to the question, where does science start andwhere sophisticated observation ends.

6. The Models of Random Motion

A series of application rich models started with the collision problem ofrigid and elastic objects, first, the Brownian motion but expanded towardsthe more and more sophisticated variants of different border conditions,objects, elasticity, not only in physics but in move of human and other hys-teric masses. For masses, events where the motion is not primarily definedby collisions the Monte Carlo random walk model is applied best, as oneof the main patterns of data mining or any other search in a completelyunknown distribution field. A further constraint of independence from pre-vious events, or remembering only a fixed certain number of preliminaries,is the Markovian model, applied for instance in the great search machines.These methods work mostly in search on the internet, i.e. on huge amountsof information.

7. Models as Hypotheses or Prejudices

The models serve as hypotheses and prejudices, consciously or hidden. Thebest example of an ideal model is the duly celebrated Kolmogorov [26]model of probability, with the condition of the closed world assumptionbased on the excluded third logic and the stationary temporal behavior,ergodic, i.e. equal chance for experimentation with one sample many timesor many samples.

The classic probability hypothesis works well in those worlds, ensem-bles of events that can be approximately modeled by these constraints,on the other hand not appropriate applications can be totally misleading,especially those related to relatively fast changing conditions, not with con-stant frequency changing events. Typical examples are the stock exchangeprices in hectic, crisis periods or application of experience with mental andphysical capabilities of significantly different ages, environments.

The circle looks to be really vicious: having a number of observationswe start to calculate evidence on the basis of a certain theoretical, or expe-rience suggested model and get results that should be checked on the same


hypothetical model or supposed similar circumstances. The whole historyof science can be recounted on that scheme and the only consolation whatwe can attribute as virtue, practicality and power of science is the continu-ous mass refinement of model approximations and getting more and moreproofs for better forecasts. The story of all more or less sophisticated tech-nologies, mass production and reproduction is one of that proofs, the sameis more or less valid for progress in medicine, much less in pedagogy andeconomy. Science can be considered as an infinite story of model approxima-tions with check of experimental improvement, receiving more justifications,more standing forecasts.

8. Mystic of Uncertainty, Mystic of Life: The EarlyMessage from India

Uncertainty starts to be nonexisting if it can be modeled in the way andreliability of scientific certainty. That is the reason why we cannot speakabout uncertainty in general, as about a subject of science and not as aconcept of philosophy and general discourse.

This mystic of uncertainty could be one of the reasons why Europeanscience alienated itself from these irrational looking problems. Indian math-ematics was put in a different framework, may be due to the more naturalattitude to nature. That was first admired by Europeans in respect of love,after this continent started to liberate itself from the ideological shacklesby the fresh air of the Renaissance and of the Age of Reason. One of themost famous love story about a success, related to statistical estimation,was the tale of Nala and Kali, who latter was also a demigod of dicing! Nalacould give a perfect estimation about the number of leaves and fruits on abig tree by calculating the relation between two, typical singular branchesand of the whole. The original story was more ancient than the last referredversion from about 400 A.D.

9. Where does Science Start and where SophisticatedObservation Ends, a Message from Persia: Serendipity

The other story from a rich civilization is the Persian fairy tale about thethree princes of Serendip. The narrative is a metaphor of making fortunateand unexpected discoveries by accident. The story was made popular byHorace Walpole but the beautiful origins and witty metaphors are bestin the original tales, and the reader is advised to look after the excellentreviews to be found in details on the internet.

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The answer from the European scientist after the blossoming Age ofReason, Pasteur: In the field of observation chance favors the prepared mind.The question is open for several interpretations as any, related to the generalsubject of uncertainty.

10. An Uncertain Classification of Uncertainty

Returning to the different arts of uncertainty, a rough and somehow arbi-trary classification can be given:

• objective, physical uncertainty of the quantum world;• uncertainty due to some existing but not yet discovered phenom-

ena, e.g. causes and medicament of some diseases;• uncertainty due to non computable complexity, e.g. long range,

exact meteorological forecast;• uncertainty due to lack of knowledge, non availability of data, es-

pecially of historical events;• uncertainty due to unknown, unexpected interactions, especially

human behavior.

These different kinds of uncertainties can reach a more definite classi-fication after a clarification of the problem, i.e. after scattering the fog ofuncertainty. The explanation of human behavior can be ordered into dis-covery of a neural disorder, due to anticipation of a meteorological change,uncertainty about a not yet disclosed verdict and of the not yet discov-ered interactions in the brain. Looking at this enumeration it is clear thatin spite of the uncertainty of typology the different cases require differ-ent approximation models. The model matching is sometimes more an artof serendipity than science of Nala and Kali, an art of sensitivity for theopportunities and pitfalls of analogous thinking.

All these are good reasons why scientists hated or abhorred uncertaintyand why it is the great challenge of science now. The epistemic loop of ob-servation, creating hypotheses based on observations analogies, affirmationthese hypotheses by new observations and consolidating that knowledge bytheories, is the progress rout of current science. The difference to the past isthe availability of immense observation data, possibility of review their fea-tures by renewed, moreover methodically revised observations. That loopis strengthened by the critical norms of unbiased science, its possible inde-pendence from ideological biases.


11. The Pragmatic View of Methodologies

From our point of view we emphasize two relevant features of the process:the critical and pragmatic view of the methodologies and the computerscience support of data management, especially data mining.

Methodologies are enumerated here only. They are well discussed inexcellent textbooks and university courses. The enumeration serves theoverview of practical but not general devices as a set of tools for manu-facturing different shaped surfaces:

• statistics, with its temporal, conditional, dependence constraintsand evaluation methods;

• classic probability, with its definitional constraints and develop-ments for calculation the effects of disturbing these constraints;

• Bayesian methods for conditional probabilities, estimations withcombination of statistical, probabilistic data;

• possibility methods (like Dempster-Shafer, certainty factor, etc.)with attention to combinations of statistics, probability conceptsand anomalies related to cumulating of marginal estimations;

• fuzzy methods for estimation of memberships within hypotheticalor real groups if instances, major method for using human estima-tions and neglecting uncertain relations of distributions. In its nonprobabilistic view the method has a different course in acceptanceand applications.

All these methods have some natural roots in set theory as dealing withhypothetical sets of data, instances, qualifications; in algebra and logic dueto the tasks of combination uncertainty data. All these methods, sooner orlater, have less or more attention to the problem of evidence based on mea-surement data and on human estimation. Even the theoretically strongestclassic probability theory had a rather early branch called subjective prob-ability.

We [38] and the literature use the words probability, vagueness, pos-sibility, evidence, estimation in different contexts and textbooks in similarmanner and in strictly defined separate meaning with reference of the originof data, the calculation method, the state of reliability, measure of confi-dence. This variety of definitions’ naming reflects the intrinsic nature ofuncertainty.

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12. Where are we Now?

The greatest difference to earlier situation of science was underlined:

• the availability of never before reliable, wide range instrumentationincreasing the possibilities of human observation by about thirtymagnitudes;

• the possession of never before imaginable amount of observationdata and computational power, mathematical methods for man-agement of those;

• a continuously developing epistemic discipline for critical revisionsof hypotheses and theories, freedom of since achieved in permanentstruggle against interests, alien to science.

13. Data Mining

Data mining is a collective designation of all modern methods for discov-ery connections, structures among seemingly unstructured data. The firstanswer was the application of classic methods in statistics: significance, fac-tors, etc. Soon turned out that in case huge amounts of data that is mostlynot enough, the relations are more hidden than according to the popu-lar saying, needle in a haystack. The escape route returned to the modelhypotheses but in a conscious way of hypothesis control.

The challenge mobilized a wide variety of mathematical disciplines. Themost frequently applied methods are the hidden markovian procedures,search for returning phrases, expressions, contexts. From this short refer-ence it is visible that sophisticated connection searches should apply severaltricks, hypothetical schemes and a possibly comprehensive expertise withinthe special subject concerned.

Data mining is in most cases equivalent with the search for hiddenstructures. Mathematics in some sense is the theory of structures, searchfor structures and building structures. All disciplines of mathematics canbe interpreted in that way and this generalization helps the continuous in-terplay within the realm of mathematics, especially between geometry, itsfurther abstractions, like topology and all others, algebra, algebraic foun-dations of logic, analysis, etc.

The search for structures in data is a search within the certain branchof science and practice, related to technology or any phenomena of na-ture, the human being included. If we accept this interpretation, the searchcan be understood as application of knowledge about abstracted structuresto the investigated discipline and in between the method and the special


discipline, the hypothetical model. These selected models are the guessesof the solution hypotheses and further some guiding hints to the possiblyindependent methods of hypothesis control. Possibly independent refers tothe model itself and the computational procedure, too.

The search applies the old usual methods, too i.e. analysis, matchingdata to some basic functions of temporal behavior and frequency analysisbased on the classical methods and on different waveforms, like wavelets,fractals. These methods are in some sense closest to the model view, tothe evolutionary exponential and logarithmic decay, to powers of energyrelations and further, well known actions and interactions of complicatedmotion or frequencies of all kinds of oscillations, returning phenomena.

Functional analysis is in some sense a search for a model, for the bestlooking class of functions and by that a reverse reasoning to the nature ofthe data scheme, the model. The application of functional analysis approx-imation search is a recent attempt in mining economy and psychology data[1].

14. Spaces of Events and Search

The first model and even linguistic metaphor is geometry. The fantasy richdevelopment of Riemann-surfaces, non-Euclidean geometry, topology [14],the new world of related metric is the most promising present and futurefield of model building and search procedures on these strange, multidimen-sional spaces. These spaces are strange for a traditional view of a humanculture, first attached only to the two-dimensional surface of a flat look-ing earth and slowly accustomed to a similar view in the third dimension.Modern science has shown that nature is not as unimaginative as we were.The strange multidimensional, topological space is really the dynamic rep-resentation of nature, not only of the far cosmological one but representingall kinds of internal and external phenomena, those not understood untilnow.

Not understood is a highly relative attribute. The models and so thespaces of present science are rather strictly bound to our early visual im-pressions; and though they are beautiful and admirable further abstrac-tions, but “only” expressions of relations in our language. Language meansalso abstractions of natural language towards any means of representation.

The vicious circle has pragmatic solutions. The models are understoodin that working sense, they are accepted as long as they are in conformancewith the data, the observations. Data mining is the search for a good model,good in adjustment to the data, in coherence with other related knowledge

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and with possibility of computational application. Model means in thisrespect the relative understanding, the place of the model within the generalworld of current knowledge. Nothing more in the view of final totality andnothing less than an admirable instrument of everyday life, of orientationin the complexity of the practical world.

15. Restitution of Approximations, the Sorcerer′s Hulls

The next remark concerns the approximation itself. The origins of math-ematics were for a long time the art of approximations, approximationsin the sense of calculation everyday phenomena. Especially after the rev-olutionary achievements of science in the 19th century, approximation formathematics was a kind of dirty, fuzzy practice, not the world of elegantmathematical theory. That view changed once more in the past very fewdecades.

The change is due to the dramatic development of computer powerand in connection with that a new horizon of the earlier only theoreticallyconsidered limits of computability. As stated before, one of the arts of un-certainty is the computability limit, because of the exponential growth ofcomplexities. That art of uncertainty promoted to be a major obstacle,possible, the major obstacle. All others, related to the unknown, not dis-covered, not understood phenomena are or can be the subjects of furtherresearch, new experimental and theoretical ideas, they are open challengesfor research. Even the most quoted example of objective uncertainty, theHeisenberg state-measurement puzzle is or can be an emerging problemfrom the realm of mystery.

High complexity, problems beyond the current physical computabilitylimits have a different nature. The mathematical background, the nature ofproblems where the models are unavoidably beyond the limits of polynomialtime solutions, the classes of these hard algorithmic problems is now thesubject of excellent university textbooks, not to be detailed here [20].

The efforts for model transformations are until now partly limited forseveral important real cases, how to interpret some non polynomial modelsas polynomials related to the algorithmic operation requirements. The cel-ebrated traveling salesman problem′s broad analogy class belongs to those.A great many of environmental and mental subjects lie certainly beyondthose, in the infinities of computational steps.

If we accept the theses about the modeling role of mathematics forany scientifically treatable problem and the conclusion of manageability ofthose all, by computation, than we find the new limit of knowledge in the


computability problem. Short: if not computable, not manageable by thefirm reproducible methods of rational science.

Not too much satisfactory for any kind of rational, goal oriented humanintellect. That is the main reason why approximation reached a new sta-tus in rigorous science and by that and as a natural feedback, great newtheories, mathematical instruments of approximation calculi and their ofthe confidence of those. The best example is the development of topologyrelated methods, creation of possibly convex hulls around the feasible solu-tions, reduction, transformation of these hulls into manageable dimensionsand patterns. The manageable mean realistic conditions of computationand of pattern matching. The last refers to the analogy thinking, findingsimilarities, measures of similarity and dissimilarity. All these are, on onehand by highly sophisticated and until now only perceivable for very dis-tinct mathematical brains, and on the other hand, in fast progress, practicaltools for routine engineering design. All problems of nonlinear and stochas-tic nature, and that is the practical case for most engineering tasks, aresomehow covered by these Sorcerer′s hulls.

Similar to that trend was accepted the idea of demonstrating math-ematical proofs by computational experimentation. That has also a veryrelevant practical meaning, not only for the development of mathematics,but in the mirror of the above applications, a new instrument for validity,confidence estimations, a key problem for engineering design.

Not completely independent is the renaissance of number theory as anew, stimulating subject in modern cryptography and in general search pro-cedures after structures in huge data masses. An apparent connection pointis the ancient problems of prime numbers, as somehow mystic elementarybuilding blocks of all numbers, and by that for hidden keys and hiddenstructures. The secrecy of the keys is a routine requirement of the wholesociety, the discovery of hidden structures is an objective of most researchproblems, e.g. pharmaceutical chemistry and genetics.

16. Compromise: Inclusion of Human Estimates andConclusion about the Ubiquity of Research

The third novelty is also an unorthodox development: for calculation ofhuman influenced processes, that is economy, other types of negotiations,planning strategies in complex action tasks the guess of subjectivities is un-avoidable. Analyzing the human decision process two hypotheses and someother won Nobel- prizes. The first group is characterized by the moderncontinuation off classic, Adam Smith founded rational choice school, later

402 Tibor Vamos

enriched by Max Weber, Talcott Parsons, Buchanan and Tullock and fur-ther, the other by the psychology based groups, characterized by the worksof Kahnemann, Slovic, Tversky and their followers. All these is flavored byhe Condorcet-Arrow line of proof for the not solvability of optimal choicebetween more different, well founded interests and not even the possibilityof a definitely acceptable voting system. These developments were appliedin the frames of the Neumann initialized game theories and the 18th centuryhistorical beginnings of optimization continued now by theories of reason-able compromises, balances, started by Pareto and in our age, Neumann,Nash, Harsanyi, Black and Scholes, Aumann and Schelling, only to mentionthe Nobel-class of economy.

The main gist was and is the balance among different attitudes, objec-tives, environments by combinations of rational strategies and motivationsof individuals and various masses, explainable by present circumstances andby the several hundred million years evolutionary heritage of the humananimal. That immense task of present human existence and coexistencedemonstrate the ubiquity of uncertainty, of its research and the intrinsiccomplexity, moreover coherence of all related scientific disciplines.

References

1. Aczel, J., Falmagne, J. C., Luce, D. (2000) Functional equations in thebehavioral sciences, Math. Japonica 52 3, pp 469-512.

2. Andreka, H., Madarasz, J., Nemeti, I. (2006) Logic of space-time and rela-tivity theory, Book manuscript.

3. Aristotle, Metaphysics, Dir William David Ross ed., Oxford, ClarendonPress 1924, repr. 1981, Book. K8, 1065a line 38–39.

4. Bayes, T.: An essay towards solving a problem in the doctrine of chances.Philosophical Transactions of the Royal Society of London, 53, pp 37–418,1763

5. Jacob Bernoulli (1744) Opera Omnia 2. Vol. Geneva, repr. Bruxelles, 1975.6. Bloor, D. (1976) Knowledge and Social Imaginary, Routledge and Kevin

Paul, London.7. Bollobas, B., Varopoulos, N. T. (1975) Representation of Systems of Mea-

surable Sets, Math. Proc. Cambridge Philos. Soc. 78, 2, 323-3258. McCarthy, J.:Epistemological problems of artificial intelligence, Proc. IJ-

CAI’pp 0-1038-1044,Cambridge, MA, MIT Press9. Chazell, B. (Dec 2006) Proof at a roll of the dice, Nature, 28, pp. 1018-1019

10. Chomsky, N. (1955) Logical Structure of Linguistic Theory. MIT HumanitiesLibrary. Microfilm.

11. Darwin, C. (1979) The Origin of Species by Means of Natural Selection,1859-1872 Random House Value Publishing, New York.

12. Dennett, D. (1995) Darwin’s Dangerous Idea, Simon & Schuster.


13. Diogenes, L. (1925) Lives of Eminent Philosophers, tr. by Hicks, R.D., Lon-don, 2 Vols., Loeb Classical Library, Harvard University Press, Cambridge,MA.

14. Gromov, M. (1998) Metric Structures for Riemannian and Non-RiemannianSpaces, Birkhauser, Boston.

15. Hacking, I. (1975) The Emergence of Probability. A philosophical study ofearly ideas about probability, induction and statistical inference, CambridgeUniversity Press, Cambridge.

16. Haenni, R., Hatmann, S. (2006) Special Issue of Minds and Machines onCausality, Uncertainty and Ignorance, Mind ann Machines 16, 3.

17. Halpern, J. (2003)Reasoning About Uncertainty, MIT Press.18. Halpern, J., Pearl, J. (2001) ”Causes and explanations: A structural-model

approach. In Proceedings of the Seventeenth Conference on Uncertainty inArtificial Intelligence, San Francisco, CA: Morgan Kaufmann, pp 194–202.

19. Hardy, G. H. (1940) A Mathematician’s Apology, Cambridge UniversityPress.

20. Hromkovic, J. (2002) Algorithmics for Hard Problems, Springer.21. Huisinga, J. (1937) The Waning of the Middle Ages, London.22. Jaeger, G. (2006) Quantum Information: An Overview, Springer, Berlin.23. Jones, S. (1999) Almost Like a Whale: The Origin of Species Updated, Dou-

bleday.24. Kneale, W., Kneale, M. (1962) The Development of Logic, Oxford University

Press, Oxford.25. Kornai, A. (2006) Mathematical Linguistics, Book manuscript.26. Kolmogorov, A. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung,

Springer, Berlin.27. Kuhn, T. S. (1970) The Structure of Scientific Revolutions, International

Encyclopedia of Unified Science, University of Chicago Press, Chicago.28. Lakatos, I.: Falsification and the Methodology of Scientific Research Pro-

grammes, in I. Lakatos and A. Musgrave (eds) Criticism and the Growth ofKnowledge, Cambridge, Cambridge University Press, 970, pp 91–196

29. Minsky, M. (1985) The Society of Mind, Simon and Schuster, N.Y.30. Minsky, M. (2006) The Emotion Machine 9/6/2006, draft book.31. Popper, K. (1972) Conjectures and Refutations, Routledge and Kevin Paul,

London.32. Pascal, B. (1654) his first ideas about the new science of probability in:

Oeuvres de FermatT, Vol. 2, (Eds.: P. Tannery and C. Henry) Gauthier-Villars, Paris 1894.

33. Pearl, Judea (July 2004) ”Robustness of Causal Claims” In Proceedings ofthe 20th Conference on Uncertainty in Artificial Intelligence, AUAI Press:Arlington, VA, pp 446–453.

34. Ramsey, F. P. (1931) Truth and Probability, The Foundations of Mathemat-ics and other Logical Essays, ed. R.B. Braithwaite, London.

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36. Savage, L. J. (1954)The Foundations of Statistics, John Wiley and Sons,New York.

37. Sextus Empiricus, 4 Vols., tr. by Bury, R. G., Loeb Classical Library, Har-vard University Press, Cambridge, MA, pp 1933–1949

38. Vamos, T. (1991) Computer Epistemology, World Scientific, Singapore.39. Venn, J. (2006) The Logic of Chance, London, 1866, Dover Publications.40. Weinberg, S. (2003) Facing up and Its Cultural Adversaries, Harvard Uni-

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Kevin Paul, London.

405

Archetypes, Causal Description and Creativity inNatural World

Leonardo Chiatti

Laboratorio di Fisica ASL VTVia S. Lorenzo 101 01100 Viterbo, Italy

[email protected]

The idea, formulated for the first time by Pauli, of a “creativity” of naturalprocesses on a quantum scale is briefly investigated, with particular referenceto the phenomena, common throughout the biological world, involved in theamplification of microscopic “creative” events at macroscopic level. The in-volvement of non-locality is also discussed with reference to the synordering ofevents, a concept introduced for the first time by Bohm. Some convergences areproposed between the metamorphic process envisaged by Bohm and that en-visaged by Goethe, and some possible applications concerning known biologicalphenomena are briefly discussed.

Keywords: Metamorphosis; Synordering; Quantum Leap; ArchetypePACS(2006): 91.65.Pj; 03.65.Fd; 87.10.+e; 77.84.Dy; 77.80.-e

1. Introduction

Not many researchers today would be convinced that the Naturphilosophieapproach to the knowledge of nature would return. Even though excellentarguments have been provided to support this return (Card [1996]), there isno doubt that it meets a great deal of resistance. In fact it is vital that theassumptions of the Natural Philosophy are reformulated in order to har-monize them with contemporary natural sciences. It is necessary to recoverthe contrast between the causal, local and mechanistic descriptions whichthese sciences have chosen and the holistic, morphological and qualitativeinclination that has always characterised the philosophy of nature. Is thispossible? Does an authentic form of “creativity” in natural world exist,one that is not a mere epiphenomenon of basic causal and deterministicprocesses? And how should these unpredictable “creative” phenomena bedescribed, since they cannot be represented by the predictive models formu-lated in the context of the mechanical causality? Could these descriptions

406 Leonardo Chiatti

utilise the same concepts familiar to the once professed Naturphilosophiesuch as “archetype” (Goethe’s Urbild) or “metamorphosis”?

In the attempt to address such difficult questions, and without any pre-tence of resolving them, we will take into consideration not the specificaspects of the naturalistic realm, but those of microphysics. If we wishto reformulate these questions in really innovative terms we must beginour investigation at molecule level, from the atoms and from the subatomicparticles. Subsequently, we will be in a better position to discuss the macro-scopic world of daily experience and its environment, the biosphere.

2. Quantum Level: The Physical World as a Network ofEvents

One of the fundamental aspects of the structure of matter is atomism:each physical object is an aggregate of homogenous entities subdivided inclasses of elements identical to each other. Among these we can enumeratethe atoms and molecules: all the atoms of iron in a determined quantumstate are identical, and the same applies to water molecules. The set of theinternal states of each of these micro-entities is generally discrete, not con-tinuous; this characteristic is not common in macroscopic objects, whoseconfigurations generally can be varied continuously.

Molecules and atoms are not elementary: clashing against each other atsufficiently high energy, these entities are broken into smaller entities, suchas electrons and nuclei, thus they are no longer atoms nor molecules. Enti-ties are considered elementary when, however high the energy involved inthe collision process, the products obtained still belong to the same class ofreagents. This occurs to those micro-entities called “elementary particles”,such as electrons, protons, neutrons, etc.

The term particle is derived from the Latin word particula, i.e. smallpart. It reminds us of that particular entity that in classical physics iscalled “material point”: a point provided with mass that moves describ-ing a trajectory in space and persisting for a determined time. This im-age has generated an infinite number of controversies over the real na-ture of elementary particles, which are not the material points of classicalphysics. In fact the material point is an abstraction that does not have agenuine counterpart in nature other than as an idealization that can bemapped on certain classical physical systems in some circumstances (i.e.“small” object of negligible size), and dynamics of micro-entities such asatoms, subatomic particles, nuclei is in effect completely different to thatof the classical macroscopic objects. Micro-dynamics is described, at a level


today considered fundamental, by the quantum field theory, while a moreapproximate description is given by quantum mechanics. Since the aspectsof micro-dynamics which we wish to focus on briefly are qualitatively thesame in both descriptions, we will limit our discussion to quantum mechan-ics (Caldirola [1974]).

In order to adequately understand the significance of what we intend todemonstrate, it is necessary first of all to eliminate some classical prejudices,whose persistence has contributed in creating the myth of incomprehensi-bility of the quantum level of reality; a myth which is deeply rooted in manyphysicists.

In the macroscopic world of daily life the distinction of an entity andits properties is obvious: the entity is, in a certain sense, the support of itsproperties. For example, a leaf is a different thing compared to its colour,its form, its dimensions, etc. The group of properties of the leaf is not theleaf itself, intended as a material substance endowed with those properties.Therefore an object is considered to be a material support persistent intime and to which given characteristics can be attributed, independent ofpossible interactions with other objects.

The situation of a micro-entity such as an elementary particle, an atomor a molecule, is completely different: a micro-entity is nothing other thanthe connection of two particular events, one following the other in time.The first event is the creation (that is the physical manifestation) of a de-termined set of characteristics (A), while the second event—chronologicallysuccessive to the first—is the destruction (physical de-manifestation) of asecond set of characteristics (B). Set A is the “initial state” of the micro-entity, while set B is its ”final state”. These events are the effect of theinteraction of the micro-entity with other micro-entities; for example butnot necessarily, with those macroscopic aggregates of atoms which are mea-surement apparatuses (Fock [1978]).

One should pay particular attention to the fact that during the intervalbetween the first and the second event the micro-entity does not exist atall; in particular, it cannot be conceived as a corpuscle that follows a tra-jectory or as a wave that is propagated in space, the two modes of matterand energy transfer known in classical physics. Nor can it be conceived asa combination of these two modes, as various theorists of the 20th century(De Broglie, Vigier and many others) hoped, elaborating many different(and often internally inconsistent) models that have not been supported byexperiment.

Reducing the existence of the micro-entity to only two separate instants


means that the micro-entity is not a persistent material object. Such con-dition is only apparently paradoxical: an elementary entity is by definitionwithout an internal structure, therefore its “structure” can be defined onlyby its behaviour during the interaction with other similar entities. But theevents “creation of A” and “destruction of B” are precisely interactions ofthe micro-entity with other micro-entities or aggregates of micro-entities,therefore it is only in these events that the “structure” of the micro-entitycan be and is manifested: it is manifested as the initial or final state ofthat micro-entity. To ask ourselves what is the state of the micro-entity inan intermediate instant has no sense, unless an interaction of the micro-entity with other micro-entities occurs at that instant; but in this case anew and completely different interaction scheme is defined, in which themicro-entity created in state A during the first event is destroyed in stateC at the intermediate instant, with the creation of a new micro-entity inthe same state C; a micro-entity which is subsequently destroyed in thefinal state B during the last event.

In other words, the fact that the structure of the micro-entity is defin-able only during external interactions limits the possibility of defining itsphysical state only at those instants in correspondence to events of inter-action; in all other instants the physical state of the micro-entity is notdefinable. For example, the position of the micro-entity is not defined, thusthe micro-entity cannot be localised in space. An important consequenceis thus derived: the micro-entity cannot be viewed as a material substancethat persists in time and is extended in space, support of its own charac-teristics.

Therefore, the micro-entity is nothing other than the combination ofcharacteristics A and B: it is not different from its initial and final proper-ties evaluated during the only instants of its existence. And what types ofproperties are the elements of A and B? They are physical quantities suchas energy, angular momentum and linear momentum, etc. that describe thebehaviour of the micro-entity during the interactions when it is created ordestroyed. In this way, the properties created or destroyed during the in-teractions are in effect the labels of the interactions themselves.

These ideas can be illustrated considering the particular formulation ofquantum mechanics known as wave mechanics. In such formulation, prop-erties A and B are represented by the initial and final wavefunctions ofthe micro-entity, indicated by ψA and ψB respectively. Two distinct deter-ministic and causal laws of the evolution of these wavefunctions are postu-lated, one towards the future or “forward” and the other towards the past


or “backward”. In the most simple case of nonrelativistic systems with-out spin, these two laws are represented by the forward and the backwardSchrodinger equations. These laws define respectively the value of the func-tion ψforward(t) for each instant t following instant tA when properties Aare created, and the value of the function ψbackward(t) for each instant tbefore instant tB when the properties B are destroyed. We can imply that:

ψforward(tA) = ψA, ψbackward(tB) = ψB ;

but generally there is no direct relation between ψforward(tB) and ψB ,nor between ψbackward(tA) and ψA . Consequently, we are not dealing withthe time evolution of a single entity from the past towards the future, butwith two distinct time evolutions of two distinct wavefunctions, one fromthe past towards the future, and the other from the future towards the past.

Let’s investigate now the physical significance of the functionsψforward(t) and ψbackward(t). First of all, we must say that from their valuesat times tA, tB respectively [that is from the functions ψA, ψB respectively]it is possible to mathematically derive the sets of observables A and B. Suchpossibility has lead physicists to incorrectly conclude that these functionsrepresent the state of the micro-entity at time t intermediate between tA,tB , while no micro-entity exists at that instant at all.

In effect, the conditions represented by the two relations illustratedabove are the two initial conditions under which the two distinctSchrodinger equations can be solved, therefore they cannot be derived start-ing from such equations. If we admit to specify as “casual” everything thatis not predictable through causal laws, these two initial conditions wouldtherefore in effect be considered casual events. More precisely, we shouldbe aware that the second condition, relative to the evolution backward intime, is in effect a final condition. If the genuinely initial condition (i.e.the first one) is determined, for example as an effect of the preparationof the state of the micro-entity in a given experiment, the second remainsundetermined. Experience has taught us that this condition is manifested,at an arbitrary time tB , with a probability (understood as the frequencyof its recurrence on an ensemble of trials with identical initial preparation)equal to:Probability (result of B at time tB given the preparation A at time tA)

= | ∫ ψ∗Bψforward(tB)|2,[where, for simplicity, the variables of integration and the constant of nor-malisation have been omitted]. This is the well known Born’s postulate,that affirms the casual and probabilistic character of quantum events such


as the creation of A or the destruction of B. The existence of a conditionedprobability expressed in Born’s postulate should not surprise us: result Bappears in an event of interaction between a determined micro-entity andother micro-entities and is therefore conditioned by the structure of theinteraction and the initial preparation of that micro-entity. Therefore, theappearance of this result can not be a completely random event : it will bemanifested with a definite and computable conditioned probability. How-ever, it is not a deterministic event. Function ψB becomes the new functionψforward(t) for t = tB , a fact known as “quantum jump” or “quantumleap”.

In quantum mechanics we have therefore the association of two distinctevolutions: the deterministic and causal evolution of wavefunctions, and theprobabilistic and acausal evolution of real quantum events. An entire pe-riod of research was aimed at identifying a supposed “sub-quantum” levelof reality, governed by causal, deterministic and local laws; the hope wasto derive the Born’s postulate from these laws as a statistical approxima-tion. This period ended with the EPR experiments of 1980-90, which finallydemonstrated the impossibility of the project (D’ Espagnat [1980], Licata[2003]).

A final note. The description of the micro-entity and the temporal evo-lution of wavefunctions mentioned above are completely symmetric respectto the direction of time: they do not distinguish the past-future directionfrom the opposite direction. However the physical quantities such as en-ergy can be demonstrated to propagate from the past to the future, re-producing the usual chronological order of physical phenomena (Cramer[1980],[1986],[1988]; Chiatti [1994],[1995],[2005]). In any case, the existenceof the final conditions on quantum processes introduces a sort of finality,if not teleology, in these processes. Thus, we ask: what influence does allthis have on macroscopic aggregates composed of an enormous number ofmicro-entities like, for example, living systems?

Such aggregates are constituted of volumes in space where billions of bil-lions of billions of quantum creation-destruction events occur every nanosec-ond. In each event in which certain micro-entities are destroyed other micro-entities are created and vice versa; therefore we are dealing with a verydense network of interactions in time and in space.

Any “object” is really an aggregate of relatively constant aspects in thisnever-ending flow of events. Therefore the “essence” of the object, whichsupports these aspects, is nothing else but this very flow of events. Onlyin the so called classical limit, that is in the case of an enormous amount


of mutually interacting micro-entities, can the distinction between the ma-terial support and its properties be observed. It is at this level that theobjects appear, and when we can talk about the “form” of the object andthe “morphogenesis” in naturalistic terms.

In the classical limit, the probabilistic laws of quantum mechanics be-come the usual deterministic laws of classical physics. The acausal evolutiondisappears and the causal evolution of the state of the system, described bythe laws of Newton, Lagrange or Hamilton, remains. We thus ask ourselves:does the acausal evolution really disappear or is it merely concealed by thelanguage of the classical description ?

3. Acausality and the “Creativity of Nature” According toPauli

The laws of classical physics determine the temporal evolution of the stateof an object. In the definition of this state, all references to “flow of events”(i.e. the essence of the object) is eliminated: references are made only to theinvariable, or slowly variable, properties of this macroscopic flow. Havingadopted this level of description, the acausal component of events of theflow is concealed, in that it is hidden in the language used for the descrip-tion.

At this point let us make some considerations, before proceeding further.In all evidence it appears that the search of an authentically creative∗ or in-novative activity of Nature cannot be performed at the purely macroscopiclevel of natural phenomena, because this is governed by the inflexible ne-cessity expressed in the local, deterministic and causal structure of classicallaws. For example it is useless to search for a creativity in the hypotheticalviolation of the Newton laws in a phenomenon such as the mechanics of theopening of a flower: such phenomenon is correctly and rigorously describedby these laws. The attempt made by many, and in which many researchworkers still insist on, to identify “biological laws” that would describe thebiological phenomena, giving predictions different from those obtained fromthe application of known physical laws to the same phenomena, has everbeen inconclusive and seems destined to total failure if aimed at introduc-ing a sort of creativity of Nature.

Creativity is something that is not a product of the past, and in the

∗Obviously, when we refer to creative activity of Nature we should not think of an inten-tional activity associated with a self-reflexive consciousness overseeing the phenomena,but of a sort of free self-determination of the phenomena themselves.


language of physical laws this means essential unpredictability. Only at thelevel of single quantum events constituting a macroscopic process can wefind it, in the terms briefly described above. This was one of the last majorideas of Wolfgang Pauli; he believed (Peat [2000]) that Nature exercised afaculty of choice through the discontinuous variation of the quantum state,i.e. exactly the acausal process ψforward(t) → ψB described previously.†

Among the natural, and in particular biological, phenomena there aremany in which a discontinuity of this kind (commonly known as, eventhough inappropriately, “quantum jump”) produces effects that can beamplified—by classical mechanisms within the system or organism—untilthey appear at a macroscopic level. These amplification mechanisms, there-fore, connect the micro-world of quantum jumps (that have an acausal com-ponent) with the macroscopic world governed by classical laws, thus intro-ducing in it a real novelty. Among the biological cases, we are remindedof the genetic point-like mutation, that can produce a mutant phenotypewith macroscopically different characteristics from the wild phenotype; thetransduction of a single optical photon in an electrochemical signal propa-gated along the neurons (culminating in a sensation), after its interactionwith the rods and cones of the retina; in the embryogenesis, the modifiedgrowth of tissue produced from the division of a cell in which the DNA ismutated by a single direct interaction with ionising radiation.

It is necessary to keep in mind that these phenomena do not violate theclassical laws. Simply, the primary events of the causal chains (i.e. quan-tum jumps) are considered, in the classical language, only in their aspect ascause of the following events, such as “random noise” or “external signals”,and not in their essence that includes an acausal aspect. Paying attentionexclusively to the causal relations between the consequences of the initialevent, the simultaneously causal and acausal nature of the event itself (andthen of the entire chain of consequences) is concealed.

The concealment of acausality in turn prepares the concealment of syn-ordering patterns between events, as we discuss in the following section.

4. Levels of the Implicate Order

If we look at the quantum level of reality, we understand that two distinctontological levels are present: one evident and manifest, the other hidden.The manifest level is constituted by quantum jumps (interactions); they are

†Other authors, such as Wheeler and Just, have expressed analogous interpretations; seealso Zeilinger [1990].


manifest because they form the real substance of the physical world, andas such every possible entity “observer” or “observed”, consists of them.Subsequently there is the level of the connection between quantum jumps,formally described by the twofold temporal evolution of the wavefunctionsthat we previously mentioned. We are reminded that during the intervalbetween the event in which the micro-entity is created and the event inwhich it is destroyed, the micro-entity does not exist as a localised objectin space in any form. The link between these two events seems to claimsomething magical, since it remains hidden. The point is that the two eventsare connected by a “nucleus” of physical reality which is beyond space-time.For example if the first event consists of the creation of properties A andthe second event of the destruction of properties B, we will have the scheme:

creation of A ← “nucleus” ← destruction of B.

It seems justifiable to presume that properties B are “reabsorbed” bythe nucleus de-manifesting itself from physical reality, while properties A are“expressed” by the nucleus manifesting itself in physical reality. Since everypossible observer-observed system is constituted by expressions-absorptionsof this kind, the “nucleus” remains forever unobservable. It is the primummobile, the immobile cause of the physical realm; but it is, in itself, physi-cally unobservable. It is unobservable since each conceivable physical obser-vation consists of events projected from it and reabsorbed in it. Peat [2000]therefore proposed to term it as “vacuum”.‡

If we assume the vacuum is unique and universal, all elementary phys-ical events (i.e. quantum jumps) will be connected through it. Therefore,two of these could be generally connected by a couple of wavefunctions inthe previously explained way, but could also be connected in a differentway. In the first case they would be considered causally connected events,in the second case simply synordered. It is obvious that the causal order isonly a particular case of synorder.

The term “synordering” was firstly introduced by Bohm [1980], whodenominated the two ontological levels described at the beginning of thissection respectively as “explicate order” and “implicate order”. At the levelof implicate order, all the quantum jumps that form the entire history of

‡However, the meaning of this term should not be confused with that intended by physi-cists as “vacuum” in field theory. This latter case is simply concerned with the configu-ration of the minimal energy of the field, and therefore with a certain physical property,instead of the source of all the physical properties.


the Universe (past, present and future) are potentially connected,§ result-ing in the entire physical Universe being formed of potentially synorderedevents.

The potential connection of each single elementary physical event withall other past, present and future events, form a sort of significance of theevent itself. We cannot in this paper go into the bohmian analysis of therelationship between implicate and explicate orders; we refer to the originaltexts (Bohm [1980]), which remain unsurpassed in clearness and depth.¶

We limit ourselves to remember that such relationship is expressed gener-ally by non local transforms, representing the whole in one of its parts; justas in a hologram, each fragment of which contains phase-coded informationof the entire represented object. Similarly, the vacuum must correspond toan unobservable level in which all the information concerning the physicalUniverse in its entire past, present and future history is codified in compactform. An essential difference with respect to the hologram is that, while itis static, in the present case the entire temporal domain is codified andtherefore it is more correct to speak of holomovement. Bohm found that anappropriate mathematical description of holomovement could be providedby a certain algebra, which he called holoalgebra. The process of asymp-totic convergence towards the complete identification of this holoalgebra isone of the characteristics of scientific research.‖

§For example, modifying the boundary conditions in a so called “delayed choice” ex-periment of quantum optics, the behaviour of a single photon present in the apparatuscomplies with this change, without any signal informing it of the event (Wheeler [1980]).¶See also the interesting interview of Bohm by Peat and Briggs [1987]. Bohm died in1992.‖On the possible identification between the “vacuum” as interpreted in this paper, the“implicate order” and the Jung’s “unus mundus” see in particular Peat [2000]. Bohmhimself saw in the implicate order the common root of physical matter and psychical phe-nomena (Bohm [1980]). This issue is relevant in order to extend the notion of archetype,that we are introducing here for the physical-biological domain, at the psychologicaldomain, and to compare it with the homonymous Jung’s concept. See also Card [1996]and Mc Farlane [2000]. This fairly controversial argument (Fournier [1997]) will not beapproached in this paper.


5. The Concept of Metamorphosis According to Bohm andGoethe

Even though the bohmian analysis has had a great impact on the metathe-oretical interpretation of quantum formalism (Licata [2003]), its idea ofmetamorphosis (Bohm [1980]) has not been object of much attention bylater academics.

The idea is the following. A given type of explicate order (one that ismanifested, for example, in form or in function of an organ or organism)is generally defined by a certain aggregate E of geometric transforms suchas spatial and temporal translations, axial rotations, etc., that change theconfiguration of the process while remaining within the same explicate or-der.

A metamorphosis M is a transform mapping the considered explicateorder into a more implicate level of order; un optical analogy may be offeredby the holographic imaging of a given object, which we have already seen tobe non-local. Let us indicate with M−1 the inverse operation of unfoldingthe implicate order into a new explicate order; in the optical analogy, it con-sists of the projection of a three-dimensional image of the object throughirradiation of its hologram with coherent light.

Assuming this, the transform E’ = MEM−1then transports explicateorder E into a new explicate order E’, and can be called a similarity meta-morphosis. The reason for this name, instead of the more conventional “sim-ilarity transform” indicated in the standard textbooks of abstract algebra,is the desire to underline the difference between transform that conservethe explicate order and metamorphosis that converts the explicate orderinto a certain implicate order and then eventually reconverts it to a newexplicate order. Transform maintains the causal and local structure of thedynamics of explicate order, while similarity metamorphosis codifies, in agiven explicate order, information regarding the ”rest of the Universe” in anon-local way. And it is in such codification that we can find the synorder-ing of physical events.

It is well known that the structure that Bohm proposes for the holo-movement is the similarity metamorphic flow that alternates the enfoldingof the explicate order into implicate order and unfolding of the implicateorder into the explicate order. The creation of the electronic state ψA attime tA, and the subsequent destruction of the electronic state ψB at timetB , causally connected by the dynamics described in the Schrodinger equa-tions, are in effect interpretable as particular metamorphosis that connect,in a generally non-local way, events which are separated in space-time. The


movement of the electron is not the movement of a persistent material pointin an external space environment, but the alternation of its enfolding (M)into the implicate order and its unfolding (M−1) into the explicate order,within a metamorphic process (Peat e Briggs [1987]). Therefore one can seethat the causal connection is a particular case of similarity metamorphosisor, in other terms, a particular case of synordering.

However, the concept of metamorphosis can be applied, without anysubstantial changes, also to the macroscopic structures of classical objects.

It is possible, for example, that in a biological system specific combi-nations of quantum events (for example an extended group of coordinatedmutations) could occur as the unfolding phase of a similarity metamorpho-sis which, during the enfolding phase, could have codified the state (nononly in the present, but also in the past and in the future) of the entire sys-tem, of the species or of the environment. Such events could then, via thecausal amplification processes discussed above, lead to macroscopic modi-fications within the system, thus to metamorphosis in the biological usualsense of the term (in which we include the mechanisms of differentiationduring ontogenesis and speciation during phylogenesis). Nothing of this willever enter into conflict with the causal nature of the sequence of classicalstates of the system.

Even though the comparison of different concepts matured in differ-ent historical periods and in different cultural contexts constitutes an un-doubtedly hazardous operation, we believe it worth noting the convergencesbetween the concept of metamorphosis developed by Bohm and that elab-orated by Goethe. Regarding this subject the introduction by Stefano Zec-chi to a collection of Goethe’s works, focused on the Metamorphosis ofPlants, (Goethe [2005]) is remarkable. Goethe was aware that an object,bearer of its own form, is in itself flow and movement. As Zecchi eluci-dates: “The metamorphosis of plants emphasizes that the form is immersedin the becoming and we can perceive it during its metamorphic process. Inthe Steigerung (the ascending process of the composition of parts) the formis not determined through an abstraction, or a progressive rarefaction ofits material nature; the mere existence of form requests a continuous re-newal of this nature. It is this metamorphic re-materialisation that keepsthe breath of life within the form. This does not occur in a linear time,but in a cyclic time with a beginning and an ending. The classical conceptof time can be seen in the Goethe’s idea of metamorphosis: the law gov-erning the development of phenomena has nothing to do with history; itstime is a constant present that includes both past and future. The study of


metamorphosis and its rules aims to make explicit what is stable and eter-nal within the contingency. The Steigerung of forms is not therefore linearmovement in time, but circular movement around its own starting point.Birth and death represent the metamorphosis of what is self-identical, stableand eternal, but that also has life”.

And again “ . . . Arber has also suggested the most correct way of scien-tifically interpreting the phenomenon of metamorphosis: it must be intendednot as a visible phenomenon, but as a process of transformation which oper-ates within the same transformative force. Metamorphosis is therefore nota material phenomenon, but an immaterial process in which its effects areperceived in certain particular cases . . .”

The idea of metamorphosis presents itself spontaneously in two impor-tant research areas in biology, that is the ontogenesis and the phylogenesis.And it is well known that Goethe applied it in these areas.

In the last section we attempt to re-propose this approach, that wasdismissed in the 19th Century.

6. Archetypes and Causal Description

An important point to emphasise is that metamorphic transformations aresuperposed on the ordinary causal temporal evolution of living systems,which is in reality none other than a particular aspect of metamorphicflow: the causal relation is only a particular case of synordering. However,in general perspective the synordering is expressed as a sort of “chancestructure”.

Naturally, while all events can be synordered because of their connec-tion through the vacuum, not necessarily is each event synordered with eachother in reality. Actual synordering occurs by means of the (extra-temporal)metamorphic process. The structure of metamorphosis is the pattern of thesynordering to which it is associated.

The actual patterns of synordering of events constitute what we willcall archetypes. Archetypes can be represented mathematically, but suchrepresentation is not generally causal, because they are patterns in the im-plicate order. As we can see, such concept is significantly different from theplatonic concept of “idea”, intended as a sort of “cast” or “blueprint” ofthe manifestation process.∗∗ In particular, it should not be confused with

∗∗In this conception a logical contradiction arises, concerning the relation of inherence.Let’s consider for example an atom of iron of the railway. To which archetype it would beassociated as its partial manifestation? The archetype of “atom of iron”? Of “railway”?


Goethe’s Urbild, of what remains as an important example the Urpflanze(archetypal or model plant), that inspired Goethe during his travel in Italy.

A more complete description of the natural world would attribute thesame importance to the dynamics of the causal-temporal evolution of thesingle systems (typically described by means of differential equations of mo-tion or variational principles) and the extra-temporal patterns of synorder-ing – i.e. the archetypes – often represented as graphs and taxonomies. Asone can easily argue, this involves a new conception of the natural world inwhich contingencies are no longer mere casual accidents, not subsequentlyexaminable and therefore of minor importance compared to the laws. Asa consequence, the usual distinction between prescriptive (such as physics)and descriptive sciences (such as botany) disappears, remaining relativelyvalid only in the operational sense.

In other words, we can claim that the existence of poppies is no longeran incident of evolution. In each instant of its growth, the quantum jumpsconstituting the “poppy” system can express information relative to theoutline of the entire life cycle of the plant, of the outline of the entire lifecycle of the poppy species, or to aspects of the environment in which itlives. The amplification phenomena capable of reverberating these eventsat the macroscopic level, lead to the ontogenesis of the poppy that willno longer be an exclusively local, deterministic and causal process: it willadmit non-local, acausal and authentically creative aspects. An importantpart of scientific inquiry must then aim to isolate these aspects from theother merely mechanical ones: exactly the programme of Goethe.

It is also evident that through the described processes, the environmentcan be codified in mutations, therefore the basic “casuality” of the back-ground mutations could be in effect oriented from the environment; thiscould be the basis of an additional adaptation process other than the pro-cess of natural selection. It may be that the environmental processes thatexercises a selective pressure on the organism and the background muta-tions occurring in it are synordered. If this is the case, we should look atthe phylogenetic tree no longer as a map of evolutionary incidents, but as apartial mathematical representation (in fact it is a graph) of the archetypeinvolved with the evolution of the biosphere on this planet.

It is worth noting that such approach is not necessarily in conflict with

Of “planet Earth” (from which the atom was extracted)? Any object or entity can beseen as an element of distinct aggregates, each one, in platonic terms, associated to an“archetype”. That means the exclusivity of the relation of inherence to only one aggregate[the one which should correspond to the generative process of that entity] fails.


the neo-Darwinian theories: it can represent a completion of them. In factthe basic scheme of the casual mutation subsequently selected by the en-vironment is not denied, but we simply affirm the possibility of an acausalconnection between these two factors. Naturally this does not exclude thefact that the neo-Darwinian model could be extended along multiple direc-tions as, for example, the Barbieri’s semantic theory (Barbieri [1985]).

It would be interesting to consider biological phenomena, fundamen-tally inexplicable from a causal point of view, in the terms proposed inthis paper. An example can be seen in the migratory phenomena of birds.The fact is that the knowledge of the migratory destination cannot be ge-netically transmitted (the information is too detailed!) and that in somecases it is not transmitted by learning. It could be that instead of “knowl-edge” to be “utilised” (possibly an anthropocentric preconception of thebiologist) one should talk about the expression of a complex of reactions,codified at the archetypical level as part of the process consisting of thatspecific species of birds, aimed at determining a flight plan under certainexternal circumstances. These phenomena would therefore be associated tothe spontaneous canalization of the manifestation on selected lines, a sortof “impersonal intentionality” that reminds us of the Chinese doctrine ofChi.

In conclusion, we must mention an issue that is too complex and delicateto be rigorously analysed in this article: what instruments are available todistinguish the causal aspects from acausal ones in a determined dynamicalsystem? There is no simple answer, but the question is unavoidable if onewishes to make scientific and then testable hypotheses.

It is evident that synordering can induce constellations of coordinatedevents which, in its absence, would be extremely improbable. Therefore,wherever it is possible to identify such constellations and statistically es-timate their improbability, the quantification of the level of confidence tofind oneself in front of synordering phenomena is possible. Of course, thisis a rather indirect and a posteriori procedure, but we must keep in mindthat the synordering connects the implicate order to the explicate order,and that our capability of effective control and observation is limited to thelatter.

For instance, it is possible to make conjectures that well known sud-den jumps of biological evolution, involving the simultaneous appearanceof numerous distinct and independent but coordinated mutations, capableof expression in a functional and favourably adapted phenotype, are phe-nomenon of this kind.


Naturally, much theoretical and experimental work is necessary to ar-rive at secure conclusions. But the stakes are a deeper understanding of thenatural world going beyond the supposed absolute character of causality(recently restated through the myth of “intelligent designs”).

References

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