ITP-UU-14/15SPIN-14/13

The Cellular Automaton Interpretationof Quantum Mechanics

A View on the Quantum Nature of our Universe,

Compulsory or Impossible?

Gerard ’t Hooft

Institute for Theoretical PhysicsUtrecht University

and

Spinoza InstitutePostbox 80.195

3508 TD Utrecht, the Netherlands

e-mail: g.thooft@uu.nlinternet: http://www.staff.science.uu.nl/~hooft101/

When investigating theories at the tiniest conceivable scales in nature, “quantum logic” is

taking over from “classical logic” in the minds of almost all researchers today. Dissatisfied, the

author investigated how one can look at things differently. This report is an overview of older

material, but also contains many new observations and calculations. Quantum mechanics is

looked upon as a tool, not as a theory. Examples are displayed of models that are classical

in essence, but can be analysed by the use of quantum techniques, and we argue that even

the Standard Model, together with gravitational interactions, may be viewed as a quantum

mechanical approach to analyse a system that could be classical at its core. We then explain

how these apparently heretic thoughts can be reconciled with Bell’s theorem and the usual

objections voiced against the notion of ‘superdeterminism’. Our proposal would eradicate the

collapse problem and the measurement problem.

Version June 10, 2014

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Contents

1 Introduction 6

1.1 A 19th century philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Dirac’s notation for quantum mechanics . . . . . . . . . . . . . . . . . . . 15

2 Deterministic models in quantum notation 18

2.1 The basic structure of deterministic models . . . . . . . . . . . . . . . . . . 18

2.1.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 The Cogwheel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Generalisations of the cogwheel model: cogwheels with N teeth . . . . . . 22

2.3.1 The most general deterministic, time reversible, finite model . . . . 23

2.4 Infinite, discrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 More general continuous ontological models . . . . . . . . . . . . . . . . . 26

2.6 The continuum limit of a cogwheel: the harmonic oscillator . . . . . . . . . 29

2.6.1 The operator ϕop in the harmonic oscillator . . . . . . . . . . . . . 31

2.6.2 The operators x and p in the ϕ basis . . . . . . . . . . . . . . . . 31

2.6.3 The transformation matrix relating the harmonic oscillator with theperiodic ϕ system . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.4 Appendix to Section 2.6: Explicitly evaluating the transformationmatrix 〈ϕ |x 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Other physical models 37

3.1 Time reversible cellular automata . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 ‘Neutrinos’ in three space dimensions . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Algebra of the beable ‘neutrino’ operators . . . . . . . . . . . . . . 49

3.3.2 Orthonormality and transformations of the ‘neutrino’ beable states 53

3.3.3 Second quantisation of the ‘neutrinos’ . . . . . . . . . . . . . . . . . 55

3.4 The ‘neutrino’ vacuum correlations . . . . . . . . . . . . . . . . . . . . . . 57

4 PQ theory 59

4.1 The algebra of finite displacements . . . . . . . . . . . . . . . . . . . . . . 60

4.1.1 From the one-dimensional infinite line to the two-dimensional torus 60

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4.1.2 The states |Q,P 〉 in the q basis . . . . . . . . . . . . . . . . . . . 63

4.2 Appendix 4A Transformations in the PQ theory . . . . . . . . . . . . . . 64

4.3 Appendix 4B. Resume of the quasi-periodic phase function φ(κ, ξ) . . . . 67

4.4 Appendix 4C. The wave function of the state |0, 0〉 . . . . . . . . . . . . . 68

5 Models in two space-time dimensions without interactions 69

5.1 Two dimensional model of massless bosons . . . . . . . . . . . . . . . . . . 69

5.1.1 Second-quantised massless bosons in two dimensions . . . . . . . . 70

5.1.2 The cellular automaton with integers in 2 dimensions . . . . . . . . 74

5.1.3 The mapping between the boson theory and the automaton . . . . 76

5.1.4 An alternative ontological basis: the compactified model . . . . . . 79

5.1.5 The quantum ground state . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Bosonic theories in higher dimensions? . . . . . . . . . . . . . . . . . . . . 81

5.2.1 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.2 Abstract formalism for the multidimensional harmonic oscillator . . 84

5.3 (Super)strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.1 String basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.2 Strings on a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3.3 The lowest string excitations . . . . . . . . . . . . . . . . . . . . . . 93

5.3.4 The Superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.5 Deterministic strings and the longitudinal modes . . . . . . . . . . 96

5.3.6 Some brief remarks on (super) string interactions . . . . . . . . . . 99

6 Symmetries 100

6.1 Classical and quantum symmetries . . . . . . . . . . . . . . . . . . . . . . 101

6.2 Continuous transformations on a lattice . . . . . . . . . . . . . . . . . . . . 101

6.2.1 Continuous translations . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Continuous rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1 Covering the Brillouin zone with circular regions . . . . . . . . . . . 104

6.3.2 Rotations using the P, Q formalism . . . . . . . . . . . . . . . . . 106

6.3.3 Quantum symmetries and Noether charges . . . . . . . . . . . . . . 107

6.3.4 Quantum symmetries and classical evolution . . . . . . . . . . . . 109

6.4 Large symmetry groups in the CAI . . . . . . . . . . . . . . . . . . . . . . 111

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7 The discretised hamiltonian formalism in PQ theory 111

7.1 Conserved classical energy in PQ theory . . . . . . . . . . . . . . . . . . . 112

7.1.1 Multi-dimensional harmonic oscillator . . . . . . . . . . . . . . . . . 114

7.2 More general, integer-valued hamiltonian models with interactions . . . . . 115

7.2.1 One-dimensional system: a single Q, P pair . . . . . . . . . . . . . 117

7.2.2 The multi dimensional case . . . . . . . . . . . . . . . . . . . . . . 121

7.2.3 Discrete field theories . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3 From the integer valued to the quantum hamiltonian . . . . . . . . . . . . 123

8 Standard interpretations of quantum mechanics 126

8.1 The Copenhagen interpretation . . . . . . . . . . . . . . . . . . . . . . . . 126

8.2 Einstein’s view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.3 The Many Worlds interpretation . . . . . . . . . . . . . . . . . . . . . . . . 130

8.4 The de Broglie Bohm pilot waves . . . . . . . . . . . . . . . . . . . . . . . 132

8.5 Schrodinger’s cat. The collapse of the wave function . . . . . . . . . . . . . 134

8.6 Born’s probability axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.7 Bell’s theorem, Bell’s inequalities and the CHSH inequalities. . . . . . . . . 136

8.7.1 The mouse dropping function . . . . . . . . . . . . . . . . . . . . . 140

9 Deterministic quantum mechanics 143

9.1 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.1.1 The double role of the hamiltonian . . . . . . . . . . . . . . . . . . 145

9.2 The classical limit revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.3 Born’s probability rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.3.1 The use of templates . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.3.2 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

10 Quantum Field Theory 151

10.1 General continuum theories – the bosonic case . . . . . . . . . . . . . . . . 152

10.2 Fermionic field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.3 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.4 Vacuum fluctuations, correlations and commutators . . . . . . . . . . . . . 156

10.5 Commutators and signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.6 The renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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11 The cellular automaton 161

11.1 Local reversability by switching from even to odd sites and back . . . . . . 161

11.1.1 The Margolus rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

11.1.2 The discrete classical hamiltonian model . . . . . . . . . . . . . . . 163

11.2 The Baker Campbell Hausdorff expansion . . . . . . . . . . . . . . . . . . 164

11.3 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11.4 An other approach to obtain a local hamiltonian density . . . . . . . . . . 167

11.4.1 Locality in fractional time evolution . . . . . . . . . . . . . . . . . . 169

11.4.2 Taking the continuous time limit. A local hamiltonian? . . . . . . . 170

12 Concise description of the CA interpretation 172

12.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

12.1.1 The wave function of the universe . . . . . . . . . . . . . . . . . . . 175

12.2 The rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

12.3 Features of the Cellular Automaton Interpretation (CAI) . . . . . . . . . . 178

12.3.1 Beables, changeables and superimposables . . . . . . . . . . . . . . 179

12.3.2 Observers and the observed . . . . . . . . . . . . . . . . . . . . . . 180

12.3.3 Inner products of physical states . . . . . . . . . . . . . . . . . . . . 181

12.3.4 The energy basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

12.3.5 The Earth–Mars interchange operator . . . . . . . . . . . . . . . . . 182

12.3.6 Rejecting local counterfactual realism and free will . . . . . . . . . 184

12.3.7 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

12.3.8 The superposition principle in Quantum Mechanics . . . . . . . . . 186

13 Alleys to be further investigated, and open questions 187

13.1 Positivity of the hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 187

13.2 Information loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

13.3 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

13.4 More about edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

13.5 Invisible hidden variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

14 Conclusions 194

14.1 The CAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

14.2 Counterfactual reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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14.3 Superdeterminism and Conspiracy . . . . . . . . . . . . . . . . . . . . . . . 197

A Some remarks on gravity in 2+1 dimensions 198

A.1 Discreteness of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

A note in advance

This work contains basic material as well as advanced subjects, with rather extendednarratives in between. It was originally intended to be a book, but ended up being morelike a review paper. Our hope is that it will bring home some important observations onecan make when studying the quantum mechanical nature of the universe we live in.

1. Introduction

The discovery of quantum mechanics may well have been the most important scientificrevolution of the 20th century. Not only the world of atoms and subatomic particlesappears to be completely controlled by the rules of quantum mechanics, but also theworld of chemistry, the world of thermodynamics, and all radiation phenomena can onlybe understood by observing the laws of the quanta. The successes of quantum mechanicsare phenomenal, and furthermore, the theory appears to be reigned by marvellous andimpeccable internal mathematical logic.

Not very surprisingly, this great scientific achievement also caught the attention ofscientists from other fields, and from philosophers, as well as the public in general. It istherefore perhaps somewhat curious that, even after nearly a full century, physicists stilldo not quite agree what the theory tells us – and what it does not tell us – about reality.

The reason why the theory works so well is that, in practically all areas of its applica-tions, exactly what reality means turns out to be immaterial. All that this theory says,and that needs to be said, is about the reality of the outcomes of an experiment. Thetheory tells us exactly what one should expect, how these outcomes may be distributedstatistically, and how these can be used to deduce details of the internal parameters of atheory. Elementary particles are the prime examples here. A theory has been arrived at,the so-called Standard Model, that requires the specification of some 30 internal constantsof nature, parameters that cannot be predicted using present knowledge. Most of theseparameters could be determined from the experimental results, with varied accuracies.Quantum mechanics works flawlessly every time.

So, quantum mechanics with all its peculiarities is rightfully regarded as one of the

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most profound discoveries in the field of physics, revolutionising our understanding ofmany features of the atomic and sub-atomic world. But physics is not finished. In spiteof some over-enthusiastic proclamations just before the turn of the century, the Theoryof Everything has not yet been discovered, and there are other open questions remindingus that physicists have not yet done their job completely. Therefore, encouraged by thegreat achievements we witnessed in the past, scientists continue along the path that hasbeen so successful. New experiments are being designed, and new theories are developed,each with ever increasing ingenuity and imagination. Of course, what we have learned todo is to incorporate every piece of knowledge gained in the past, in our new theories andeven in our wilder ideas.

There is a question of strategy here. Which roads should we follow if we wish to putthe last pieces of our jig-saw puzzle in place? Or even more to the point: what do weexpect those last jig-saw pieces to look like? And in particular: should we expect theultimate theory of the future to be quantum mechanical?

It is at this point that opinions among researchers vary, which is how it should be inscience, so we do not complain about this. On the contrary, we are inspired to searchwith utter concentration precisely at those spots where no-one else has taken the troubleto look before. The subject of this report is the ‘reality’ behind quantum mechanics. Oursuspicion is that it may be very different from what can be read in most text books. Weactually advocate the notion that it might be simpler than anything that can be readin the text books. If this is really so, this might greatly facilitate our search for bettertheoretical understanding.

Many of the ideas expressed and worked out in this treatise are very basic. Clearly,we are not the first to advocate these ideas. The reason why one rarely hears aboutthe obvious and simple observations that we will make, is that they have been mademany times, in the recent and the more ancient past, and were subsequently categoricallydismissed.

The reason why they have been dismissed is that they were unsuccessful; classical,deterministic models that produce the same results as quantum mechanics were devised,adapted and modified, but whatever was attempted ended up looking much uglier than theoriginal theory, which was plain quantum mechanics with no further questions asked. Thequantum mechanical theory describing relativistic, subatomic particles is called quantumfield theory, and it obeys fundamental conditions such as causality, locality and unitarity.Demanding all of these desirable properties was the core of the successes of quantum fieldtheory, and that eventually gave us the Standard Model of the sub-atomic particles. Ifwe try to reproduce the results of quantum field theory in terms of some deterministicunderlying theory, it seems that one has to abandon at least one of these demands, whichwould remove much of the beauty of the generally accepted theory; it is much simpler notto do so, and therefore one usually drops the existence of a classical underlying theory asa basic requirement.

Not only does it seem to be unnecessary to assume the existence of a classical worldunderlying quantum mechanics, it seems to be impossible also. Not very surprisingly,researchers turn their heads in disdain, but just before doing so, there was one more

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thing to do: if, invariably, deterministic models that were intended to reproduce typicallyquantum mechanical effects, appear to get stranded in contradictions, maybe one canprove that such models are impossible. This may look like the more noble alley: close thedoor for good.

Thus, no-go theorems were produced. The first such attempts was the famous Einstein-Podolsky-Rosen Gedanken experiment[4]. The most notorious, and most basic, studyof this example led to Bell’s theorem. J. S. Bell[12] studied the correlations betweenmeasurements of entangled particles, and found that, if the initial state for these particlesis chosen to be sufficiently generic, the correlations found at the end of the experiment,as predicted by quantum mechanics, can never be reproduced by information carriersthat transport classical information. He expressed this in terms of the so-called Bellinequalities, later extended as CHSH inequality[13] . It is obeyed by any classical systembut violated by quantum mechanics. His conclusion was that we have to give up producingclassical, local, realistic theories. They don’t exist.

So why the present treatise? Almost every day, we receive mail from amateur physi-cists telling us why established science is all wrong, and what they think a “theory ofeverything” should look like. Now it may seem that I am treading in their foot steps. AmI suggesting that nearly one hundred years of investigations of quantum mechanics havebeen wasted? Not at all. I insist that the last century of research lead to magnificentresults, and that only one thing missing so-far was a more radical description of what hasbeen found. Not the equations were wrong, not the technology, but only the wording ofwhat is often referred to as “quantum logic” should be replaced. It is my intention toremove every single bit of mysticism from quantum theory.

Maybe a reader will simply conclude that I lost my mind, but I do find that localdeterministic models reproducing quantum mechanics, do exist; they can easily be con-structed. The difficulty signalled by Bell and his followers, is actually quite a subtleone. The question we do address is: where exactly is the discrepancy? If we take oneof our classical models, what goes wrong in a Bell experiment with entangled particles?Were assumptions made that do not hold? Or do particles in our models refuse to getentangled?

The aim of the present study is to work out some fundamental physical systems.Some of them are nearly as general as the fundamental, canonical theory of classicalmechanics. The way we deviate from standard methods is that, more frequently thanusual, we introduce discrete kinetic variables. We demonstrate that such models appearto have much in common with quantum mechanics. In many cases, they are quantummechanical, but also classical at the same time. Some of our models occupy a domain inbetween classical and quantum mechanical, a domain always thought to be empty.

Will this lead to a revolutionary alternative view on what quantum mechanics is?There are some difficulties with our theories that have not yet been settled. A recurringmystery is that, more often than not, we get quantum mechanics alright, but a hamiltonianemerges that is not bounded from below. In the real world there is a lower bound, so thatthere is a vacuum state. A theory without such a lower bound not only has no vacuumstate, but it also does not allow a description of thermodynamics using statistical physics.

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Such a theory would not be suitable for describing our world. How serious do we haveto take this difficulty? We suspect that there will be several ways to overcome it, thetheory is not yet complete, but a reader strongly opposed to what we are trying to dohere, may well be able to find a stick that seems suitable to destroy our ideas. Others, Ihope, will be inspired to continue along this path. There are many things to be furtherinvestigated, one of them being superstring theory. This theory seems to be ideally suitedfor the approach we are advocating.

The complete and unquestionable answers to many questions are not given in thistreatise, but we are homing in to some important observations. As has happened inother examples of “no-go theorems”, Bell and his followers did make assumptions, and intheir case also, the assumptions appeared to be utterly reasonable. Nevertheless we nowsuspect that some of the premises made by Bell may have to be relaxed. This discussionis touched upon at various levels in Chapters 8, 9, 12 and 13.

We invite the reader to draw his or her own conclusions. What we do intend to achieveis that questions concerning the deeper meanings of quantum mechanics are illuminatedfrom a new perspective. This we do by setting up models and by doing calculations inthese models. Now this has been done before, but most models I have seen appear tobe too contrived, either requiring the existence of infinitely many universes all interferingwith one another, or modifying the equations of quantum mechanics, while the originalequations seem to be so beautifully coherent and functional.

Our models, discussed in Chapters 2–5 and 7, have in common that they are local,realistic, and based on conventional procedures in physics. They also have in commonthat they are limited in scope, they do not capture all features known to exist in thereal world, such as all particle species, all symmetry groups, and in particular special andgeneral relativity. The models should be utterly transparent, they indicate directions thatone should look at, and finally, they suggest a great approach towards interpreting thequantum mechanical laws that are all so familiar to us.

Our models suggest that Einstein may still have been right, when he objected againstthe conclusions drawn by Bohr and Heisenberg. It may well be that, at its most basiclevel, there is no randomness in nature, no fundamentally statistical aspect to the lawsof evolution. Everything, up to the most minute detail, is controlled by invariable laws.Every significant event in our universe takes place for a reason, it was caused by the actionof physical law, not just by chance. This is the general picture conveyed by this report.We know that it looks as if Bell’s inequalities have refuted this possibility, so yes, theyraise interesting and important questions that we shall address (Chapters 8, 9, 12 and13).

We set up a systematic study of the Cellular Automaton Interpretation of quantummechanics. We hope to inspire more physicists to do so, to consider seriously the possibil-ity that quantum mechanics as we know it is not a fundamental, mysterious, impenetrablefeature of our physical world, but rather an instrument to statistically describe a worldwhere the physical laws, at their most basic roots, are not quantum mechanical at all.Sure, we do not know how to formulate the most basic laws at present, but we are col-lecting indications that a classical world underlying quantum mechanics does exist. Our

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models show how to put quantum mechanics on hold when we are constructing mod-els such as string theory and “quantum” gravity, and this may lead to much improvedunderstanding of our world at the Planck scale.

We begin by stating our general philosophy in its most basic, non-technical form(Section 1.1). But we quickly stop philosophising. It’s time to shut up and calculate. Forthe notations we use, see Section 1.2.

Many parts of this report are reasonably self sustained; one may choose either to godirectly to the parts where the basic features of the Cellular Automaton Interpretationare exposed, Chapters 9 and 12, or look at the explicit calculations dome in other chap-ters. These explicit calculations are displayed in order to develop and demonstrate ourcalculational tools; the results themselves are hardly used in the more general discussions.

I hope to catch the attention of string theorists in Chapter 5, Section 5.3. It mayseem that I am employing rather long arguments to make my point. The most essentialelements of our reasoning will show to be short and simple, but just because I want partsof this report to be self-sustained, well readable and understandable, there will be somerepetitions of arguments here and there, for which I apologise. I also apologise for thefact that some parts of the calculations are at a very basic level; the hope is that this willalso make this work accessible for a larger class of scientists and students.

The most elegant way to handle quantum mechanics in all its generality is Dirac’sbra-ket formalism (Section 1.3). We stress that Hilbert space is a central tool for physics,not only for quantum mechanics. It can be applied in much more general systems thanthe standard quantum models such as the hydrogen atom, and it will be used also in com-pletely deterministic models (we can even use it in Newton’s description of the planetarysystem, see Section 12.3.5).

In any description of a model, one first chooses a basis in Hilbert space. Then, what isneeded is a hamiltonian, in order to describe dynamics. A very special feature of Hilbertspace is that one can use any basis one likes. The transformation from one basis to anotheris a unitary transformation, and we shall frequently make use of such transformations.Everything written about this in Section 1.3 is completely standard.

Chapter 2 will be easy to read, until we arrive at some technical calculations in thelast part, Section 2.6. Like other technical calculations elsewhere in this report, they weredone on order to check the internal consistency of the systems under study. It was funto do these calculations, but they are not intended to discourage the reader. Just skipthem, if you are more interested in the general picture.

Chapter 3 also begins at an easy pace but ends up in lengthy derivations. Here also,the reader is invited to enjoy the intricate features of the ‘neutrino’ model, but they canjust as well be skipped.

PQ theory, Chapter 4 is a first attempt to understand links between theories based onreal numbers and theories based on integer, or discrete, numbers. The idea is to set up aclean formalism connecting the two, so that it can be used in many instances. Eventually,we hit upon a more powerful and generic procedure, in Chapter 7; my ideas were evolvingwhile writing this report. Chapter 4 also shows some nice mathematical features, with

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good use of the elliptic theta functions. The calculations look more complicated thanthey should be, just because we searched for an elegant mechanism relating the real lineto pairs on integers on the one hand and the torus on the other, keeping the symmetrybetween coordinates and momenta.

In Chapter 5, we found some other interesting extensions of what was done in Chap-ter 4. A very straightforward argument drew our interest to String Theory and Super-string Theory. We are not strongly advocating the idea that the only way to do interestingphysics at the Planck scale is to believe what string theoreticians tell you. It is not clearfrom our work that this theory is the way to go, but we do notice that our program showsremarkable links with string theory. In the absence of interactions, the local equations ofstring- and superstring theory appear to allow the construction of beables, exactly alongthe route that we advocate.

Symmetries, discussed in Chapter 6, are difficult to understand in the CA Interpre-tation of quantum mechanics. However, in the CAI, symmetry considerations are asimportant as anywhere else in physics. Most of our symmetries are discrete, but in somecase, notably in string theory, continuous symmetries such as the Poincare group, can berecovered.

We do not want to run away from conceivable problems in our approach, and one ofthese is the positivity of the hamiltonian. This is why we wrote Chapter 7, where theusual Hamilton formalism is extended to include discrete variables, evolving in discretetime. When we first tried to study this, it seemed like a nightmare, but it so happensthat the ‘discrete Hamilton formalism’ comes out to be almost as elegant as the usualdifferential form. And indeed here, the hamiltonian can easily be chosen to be boundedfrom below.

In Chapters 8 and 9, we finally begin to deal with the real subject of this research:the question of the interpretation of quantum mechanics. First we deal with the standardapproach(es), emphasising those points where we have something to say, after which weformulate as clearly as possible what we mean with deterministic quantum mechanics. Itmay sound as a blasphemy to some quantum physicists, but this is because I do not goalong with some of the assumptions usually made. Most notably, it is the assumption thatspace-like correlations in the beables of this world cannot possibly generate the ‘conspiracy’that seems to be required to violate Bell’s inequalities. We derive the existence of suchcorrelations. Even if it is still unclear whether or not the results of these correlationshave a conspiratory nature, one can base a useful and functional interpretation doctrinefrom the assumption that the only conspiracy the equations perform is to fool some oftoday’s physicists, while they act in complete harmony with credible sets of physical laws.The measurement process and the collapse of the wave function are two riddles that arecompletely solved by this assumption, as will be indicated.

Eventually, we wish to reproduce effective laws of Nature that should take the form oftoday’s quantum field theories. This is still quite difficult. It was the reason for setting upour procedures in a formal way, so that we will keep the flexibility to adapt our systemsto what Nature seems to be telling us through the numerous ingenuous experiments thathave been performed. We explain some of the most important features of quantum field

11

theory in Chapter 10. Most notably: in quantum field theories, no signal can carryuseful information faster than the speed of light, and probabilities always add up to one.Quantum field theory is entirely local, in its own inimitable quantum way. These featureswe would like to reproduce in a deterministic quantum theory.

To set up the Cellular Automaton Interpretation, we first elaborate some technicalissues in cellular automata in general (Chapter 11). These are not the technicalitiesencountered when computer programs are written for such systems; software experts willnot understand much of our analysis. This is because we are aiming at understanding howsuch systems may generate quantum mechanics at the very large time and distance limit,and we wish to connect to elementary particle physics. Not all issues will be resolved,and the suspicion is aired concerning the source of our difficulties: quantum gravitationaleffects may be of crucial importance, while it is exactly these effects that are still notunderstood as well as is needed here.

Then, in Chapter 12, we display the rules of the game. Readers might want to jump tothis chapter directly, but might then be mystified by some of our assertions if one has notyet been exposed to the general working philosophy developed in the previous chapters.Well, you don’t have to take everything for granted; there are still problems unsolved,and further alleys to be investigated. They are in our last chapter.

1.1. A 19th century philosophy

Let us go back to the 19th century. Imagine that mathematics were at a very advancedlevel, but nothing of the 20th century physics was known. Suppose someone had phrased adetailed hypothesis about their world being a cellular automaton. The cellular automatonwill be precisely defined in Section 3.1 and in Chapter 11; for now, it suffices to characteriseit by the requirement that the states Nature can be in are given by sequences of integers.The evolution law is a classical algorithm that tells unambiguously how these integersevolve in time. Quantum mechanics does not enter; it is unheard of. The evolution law issufficiently non-trivial to make our cellular automaton behave as a universal computer [25].This means that, at its tiniest time and distance scale, initial states could be chosensuch that any mathematical equation can be solved with it. This means that it will beimpossible to derive exactly how the automaton will behave at large time intervals; it willbe far too complex.

Mathematicians will realise that one should not even try to deduce exactly what thelarge-time and large-distance properties of this theory will be, but they may decide to trysomething else. Can one, perhaps, make some statistical statements about the large scalebehaviour?

In first approximation, just white noise may be seen to emerge, but on closer inspection,the system may develop non-trivial correlations in its series of integers; some of thecorrelation functions may be calculable, just the way these may be calculated in a Vander Waals gas. We cannot rigorously compute the trajectories of individual molecules inthis gas, but we can derive free energy and pressure of the gas as a function of densityand temperature, we can derive its viscosity and other bulk properties. Clearly, this is

12

what our 19th century mathematicians should do with their cellular automaton modelof their world. In this report we will indicate how physicists and mathematicians of the20th century can do even more: they have a tool called quantum mechanics to give evenmore sophisticated details, but even they will have to admit that exact calculations areimpossible. The only effective, large scale laws that they can ever expect to derive arestatistical ones. The average outcomes of experiments can be predicted, but not theoutcomes of individual experiments; for doing that, the evolution equations are far toodifficult to handle.

In short, our imaginary 19th century world will be controlled by effective laws witha large stochastic element in them. This means that, in addition to an effective deter-ministic law, random number generators may seem to be at work that are fundamentallyunpredictable. On the face of it, these effective laws together may look quite a bit likethe quantum mechanical laws we have today for the sub-atomic particles.

The above metaphor is of course not perfect. The Van der Waals gas does obey generalequations of state, one can understand how sound waves behave in such a gas, but it isnot quantum mechanical. One could suspect that this is because the microscopic lawsassumed to be at the basis of a Van der Waals gas are very different from a cellularautomaton, but it is not known whether this might be sufficient to explain why the Vander Waals gas is clearly not quantum mechanical.

What we do wish to deduce from this reasoning is that one feature of our world isnot mysterious: the fact that we have effective laws that require a stochastic elementin the form of an apparently perfect random number generator, is something we shouldnot be surprised about. Our 19th century physicists would be happy with what theirmathematicians give them, and they would have been totally prepared for the findings of20th century physicists, which implied that indeed the effective laws controlling hydrogenatoms contain a stochastic element, for instance when an excited atom decides to emit aphoton.

This may be the deeper philosophical reason why we have quantum mechanics: notall features of the cellular automaton at the basis of our world allow to be extrapolated tolarge scales. Clearly, the exposition of this chapter is entirely non-technical and it may bea bad representation of all the subtleties of the theory we call quantum mechanics today.Yet we think it already captures some of the elements of the story we want to tell. Ifthey had access to the mathematics known today, we may be led to the conclusion thatour 19th century physicists could have been able to derive an effective quantum theoryfor their automaton model of the world. Would the 19th century physicists be able todo experiments with entangled photons? This question we postpone to Section 8.7 andonwards.

1.2. Notation

It is difficult keep our notation completely unambiguous. In Section 4, we are dealingwith many different types of variables and operators. When a dynamical variable is aninteger there, we shall use capitals A, B, · · · , P, Q, · · · . Variables that are periodic with

13

period 2π , or at least constrained to lie in an interval such as (−π, π] , are angles, mostlydenoted by Greek lower case letters α, β, · · · , κ, θ, · · · , whereas real variables will mostoften be denoted by lower case Latin letters a, b, · · · , x, y, · · · . Yet sometimes we run outof symbols and deviate from this scheme, if it seems to be harmless to do so. For instance,indices will still be i, j, · · · for space-like vector components, α, β, · · · for spinors andµ, ν, · · · for Lorentz indices. The Greek letters ψ and ϕ will be used for wave functionsas well.

Variables will sometimes be just numbers, and sometimes operators in Hilbert space.If the distinction should be made or if simply clarity can demand it, operators will bedenoted as such. We decided to do this simply by adding a super- or subscript “op” to thesymbol in question.1 The caret ( ) will be reserved for vectors with length one, the arrowfor more general vectors, not necessarily restricted to three dimensional space. Only inchapter 10, where norms of vectors do not arise, we use the caret for the Fourier transformof a function.

Dirac’s constant ~ and the velocity of light c will nearly always be defined to be onein the units chosen. In previous work, we used a spacial symbol to denote e2π as analternative basis for exponential functions. This would indeed sometimes be useful forcalculations, when we use fractions that lie between 0 and 1, rather than angles, and itwould require that we normalise Planck’s original constant h rather then ~ to one, butin the present monograph we return to the more usual notation.

Concepts frequently discussed are the following:

- discrete variables are variables such as the integer numbers, that can be counted.Opposed to continuous variables, which are typically represented by real or complexnumbers.

- fractional variables are variables that take values in a finite interval or on a circle.the interval may be [0, 1) , [0, 2π) , (−1

2, 1

2] , or (−π, π] . Here, the square bracket

indicates a bound whose value itself may be included, a round bracket excludesthat value. A real number can always be decomposed into an integer (or discrete)number and a fractional one.

- a theory is ontological, or ‘ontic’, if it only describes ‘really existing’ objects; it issimply a classical theory such as the planetary system, in the absence of quantummechanics. The theory does not require the introduction of Hilbert space, although,as will be explained, Hilbert space might be very useful. But then, the theory isformulated in terms of observables that are commuting at all times.

- a feature is counterfactual when it is assumed to exist even if, for fundamentalreasons, it cannot actually be observed; if one would try to observe it, some otherfeature might no longer be observable and hence become counterfactual. This sit-uation typically occurs if one considers the measurement of two or more operators

1Doing this absolutely everywhere for all operators was a bit too much to ask. When an operatorjust amounts to multiplication by a function we often omit the super- or subscript op , and in some otherplaces we just mention clearly the fact that we are discussing an operator.

14

that do not commute. More often, in our models, we shall encounter features thatare not allowed to be counterfactual.

- We talk of templates when we describe particles and fields as solutions of Schro-dinger’s equation in an ontological model. This will be explained further in Subsec-tion 9.3.1.

1.3. Dirac’s notation for quantum mechanics

In most parts of this report, quantum mechanics will be used as a tool kit, not a theory.Our theory may be anything; one of our tools will be Hilbert space and the mathematicalmanipulations that can be done in that space. Although we do assume the reader to befamiliar with these concepts, we briefly recapitulate what a Hilbert space is.

Hilbert space H is a complex vector space, whose number of dimensions is usuallyinfinite, but sometimes we allow that to be a finite number. Its elements are called states,denoted as |ψ〉, |ϕ〉 , or any other “ket”.

We have linearity : whenever |ψ1〉 and ψ2〉 are states in our Hilbert space, then

|ϕ〉 = λ|ψ1〉+ µ|ψ2〉 , (1.1)

where λ and µ are complex numbers, is also a state in this Hilbert space. Furthermore,for every ket-state |ψ〉 we have a ‘conjugated bra-state’, 〈ψ| , spanning a conjugatedvector space, 〈ψ|, 〈ϕ| . This means that, if Eq. (1.1) holds, then

〈ϕ| = λ∗〈ψ1|+ µ∗〈ψ2| . (1.2)

Furthermore, we have an inner product, or inproduct: if we have a bra, 〈ϕ| , and a ket,|ψ〉 , then a complex number is defined, the inner product denoted by 〈ϕ|ψ〉 , obeying

〈ϕ|(λ|ψ1〉+ µ|ψ2〉) = λ〈ϕ|ψ1〉+ µ〈ϕ|ψ2〉 ; 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗ . (1.3)

The inner product of a state |ψ〉 with its own bra is real and positive:

‖ψ‖2 ≡ 〈ψ|ψ〉 = real and ≥ 0 , (1.4)

while 〈ψ|ψ〉 = 0 ↔ |ψ〉 = 0 . (1.5)

Therefore, the inner product can be used to define a norm. A state |ψ〉 is called a physicalstate, or normalised state, if

‖ψ‖2 = 〈ψ|ψ〉 = 1 . (1.6)

A denumerable set of states |ei〉 is called an orthonormal basis of H if every state|ψ〉 ∈ H can be approximated by a linear combination of a finite number of states |ei〉up to any required precision:

|ψ〉 =∑i

λi|ei〉+ |ε〉 , ‖ε‖2 = 〈ε|ε〉 < ε , for any ε > 0 . (1.7)

15

(a property called ‘completeness’), while

〈ei|ej〉 = δij (1.8)

(called ‘orthonormality’). From Eqs. (1.7) and (1.8), one derives

λi = 〈ei|ψ〉 ,∑i

|ei〉〈ei| = I , (1.9)

where I is the identity operator: I|ψ〉 = |ψ〉 for all |ψ〉 .In many cases, the discrete sum in Eqs. (1.7) and (1.9) will be replaced by an integral,

and the Kronecker delta δij in Eq. (1.8) by a Dirac delta function, δ(x1 − x2) . We shallstill call the states |e(x)〉 a basis, although it is not denumerable.

A typical example is the set of wave functions ψ(~x) describing a particle in positionspace. They are regarded as vectors in Hilbert space where the set of delta peak wavefunctions |~x〉 is chosen to be the basis:

ψ(~x) ≡ 〈~x|ψ〉 , 〈~x|~x ′〉 = δ3(~x− ~x ′) . (1.10)

The Fourier transformation is now a simple rotation in Hilbert space, or a transition tothe momentum basis :

〈~x|ψ〉 =

∫d3~p 〈~x|~p 〉〈~p |ψ〉 ; 〈~x|~p 〉 = 1

(2π)3/2ei~p·~x . (1.11)

Many special functions, such as the Hermite, Laguerre, Legendre, and Bessel functions,may be seen as forming different sets of basis elements of Hilbert space.

Often, we use product Hilbert spaces: H1 ⊗ H2 = H3 , which means that states |φ〉in H3 can be seen as normal products of states |ψ(1)〉 in H1 and |ψ(2)〉 in H2 :

|φ〉 = |ψ(1)〉|ψ(2)〉 , (1.12)

and a basis for H3 can be obtained by combining a basis in H1 with one in H2 :

|e(3)ij 〉 = |e(1)

i 〉|e(2)j 〉 . (1.13)

In particular these factor Hilbert spaces are often finite-dimensional vector spaces,which of course also allow all the above manipulations2. We have, for example, the 2-dimensional vector space spanned by spin 1

2fields. A basis is formed by the two states

| ↑ 〉 and | ↓ 〉 . In this basis, the Pauli matrices σop1,2,3 are

σop1 =

(0 11 0

), σop

2 =

(0 −ii 0

)σop

3 =

(1 00 −1

). (1.14)

2The term Hilbert space is often restricted to apply to infinite dimensional vector spaces only; here wewill also include the finite dimensional cases.

16

The states

| → 〉 = 1√2

(11

), | ← 〉 = 1√

2

(1−1

), (1.15)

form the basis where the operator σ1 is diagonal: σop1 =

(1 00 −1

).

Dirac derived the words ‘bra’ and ‘ket’ from the fact that the expectation value for anoperator Oop can be written as the operator between brackets, or

〈Oop〉 = 〈ψ| Oop|ψ〉 . (1.16)

More generally, we will often need the matrix elements of the operator in a basis |ei〉 :

Oopij = 〈ei|Oop|ej〉 . (1.17)

The transformation from one basis |ei〉 to another, |e′i〉 is a unitary operator Uopij :

|e′i〉 =∑j

Uopij |ej〉 , Uop

ij = 〈ej|e′i〉 ;∑k

Uop ∗ik Uop

jk =∑k

〈e′i|ek〉〈ek|e′j〉 = δij . (1.18)

This will be used frequently. For instance, the Fourier transform is unitary:∫d3~p 〈~x|~p 〉〈~p |~x′〉 = 1

(2π)3

∫d3~p ei~p·~x−i~p·~x

′= δ3(~x− ~x′) . (1.19)

The Schrodinger equation will be written as:

ddt|ψ(t)〉 = −iHop|ψ(t)〉 , d

dt〈ψ(t)| = 〈ψ(t)|iHop ; |ψ(t)〉 = e−iHt|ψ(0)〉 , (1.20)

where Hop is the hamiltonian, defined by its matrix elements Hopij = 〈ei|Hop|ej〉 .

Dirac’s notation may be used to describe non-relativistic wave functions in three spacedimensions, in position space, in momentum space or in some other basis, such as a partialwave expansion, it can be used for particles with spin, it can be used in many-particlesystems, and also for quantised fields in solid state theory or in elementary particle theory.The transition from a Fock space notation, where the basis elements are spanned bystates containing a fixed number N of particles (in position or in momentum space,possibly having spin as well), to a notation where the states are spanned by the functionsrepresenting the fields of these particles, is simply a rotation in Hilbert space, from onebasis into another.

This point needs to be emphasised. It appears that many physicists are still notquite ready to view Hilbert space as a vector space where any basis should be as goodas any other. Whenever we talk of some physical process, its description is not morereal or less real if we make a transition to a totally different basis. For instance, inelementary particle theory we may tend to view an interaction as an exchange of particles,as pictured in a Feynman diagram. The Feynman diagram then seems to be ‘real’. Butalternatively, the same process can be described in terms of interacting fields. The fields

17

are then ‘real’. Some physicists maintain that photons are ‘merely energy quanta ofthe electromagnetic field’, whereas protons are ‘real particles’. Here we stress that suchdistinctions are meaningless. Neither photons nor protons are entirely real.

In this respect, Hilbert space should be treated as any ordinary vector space, suchas the space in which each of the coordinates of the planets in our planetary system aredefined. It makes no difference which coordinate frame we use. Should the z -axis beorthogonal to the local surface of the earth? Parallel to the earth’s axis of rotation? Theecliptic? Of course, the choice of coordinates is immaterial. Clearly, this is exemplifiedin Dirac’s notation. This beautiful notation is extremely general and as such it is quitesuitable for the discussions presented in this report.

But then, after we declared all sets of basis elements to be as good as any other, in asfar as they describe some familiar physical process or event, we venture into speculatingthat a more special basis choice can be made. A basis could exist in which all super-microscopic physical observables are beables ; these are the observable features at thePlanck scale, they all must be diagonal in this basis. Also, in this basis, the wave functiononly consists of zeros and a single one. It is ontological. This special basis is hidden fromus today but it should be possible to identify such a basis3, in principle. This report isabout the search for such a basis. It is not the basis of the particles, the fields of theparticles, of atoms or molecules, but something considerably more difficult to put ourhands on. As will be argued later, also all classical states, describing stars and planets,automobiles, people, and eventually pointers on detectors, must be diagonal in this basis.

2. Deterministic models in quantum notation

2.1. The basic structure of deterministic models

For deterministic models, we will be using the same Dirac notation. A physical state |A〉 ,where A may stand for any array of numbers, not necessarily an integer or real number,is called an ontological state if it is a state our deterministic system can be in. Thesestates themselves do not form a Hilbert space, since in a deterministic theory we haveno superpositions, but we can declare that they form a basis for a Hilbert space that weherewith define[53][56], by deciding, once and for all, that all ontological states form anorthonormal set:

〈A|B〉 ≡ δAB . (2.1)

We can allow this set to generate a Hilbert space if we declare what we mean when wetalk about superpositions. In Hilbert space, we now introduce the physical states, as beingmore general than the ontological states:

|ψ〉 =∑A

λA|A〉 ,∑A

|λA|2 ≡ 1 . (2.2)

We subsequently decide to define what these states mean:

3There may be more than one, non equivalent choices, as explained later (Section 13.5).

18

A physical state of the form (2.2) describes a situation where the probabilityto find our system to be in the ontological state |A〉 is |λA|2 .

Note, that λA is allowed to be a complex or negative number, whereas the phase of λAplays no role whatsoever. In spite of this, complex numbers will turn out to be quiteuseful here, as we shall see. Using the square in Eq. (2.2) and in our definition above, isa fairly arbitrary choice; in principle, we could have used a different power. Here, we usethe squares because this is by far the most useful choice; different powers would not affectthe physics, but would result in unnecessary mathematical complications. In short, wehave the squares in order to enable the use of unitary matrices in our transformations.

Occasionally, we may allow the indicators A, B, · · · to represent continuous variables,which should be straightforward. In that case, we have a continuous deterministic system;the Kronecker delta in Eq. (2.1) is then replaced by a Dirac delta, and the sums in Eq. (2.2)will be replaced by integrals. For now, to be explicit, we stick to a discrete notation.

There is an imminent danger: we have no direct interpretation, as yet, for the innerproducts 〈ψ1|ψ2〉 if both |ψ1〉 and |ψ2〉 are physical states. Only the absolute squares of〈A|ψ〉 , where 〈A| is the conjugate of an ontological state, denote the probabilities |λA|2 .We briefly return to this in Subsection 12.3.3.

The time evolution of a deterministic model can now be written in operator form:

|A(t)〉 = |P (t)op A(0)〉 , (2.3)

where P(t)op is a permutation operator. We can write P

(t)op as a matrix P

(t)AB containing

only ones and zeros. Then, Eq. (2.3) is written as a matrix equation,

|A(t〉) = UAB(t)|B(0)〉 , U(t)AB = P(t)AB . (2.4)

It is very important, at this stage, that we choose P(t)op to be a genuine permutator, that

is, it should be invertible.4 If the evolution law is time-independent, we have

P (t)op =

(P (δt)

op

)t/δt, Uop(t) =

(Uop(δt)

)t/δt, (2.5)

where the permutator P(δt)op , and its associated matrix Uop(δt) describe the evolution

over the shortest possible time step, δt .

Note, that no harm is done if some of the entries in the matrix U(δt)ab , instead of 1,are chosen to be unimodular complex numbers. Usually, however, we see no reason to doso, since a simple unitary basis transformation can remove these superfluous phases.

We can now state our first important mathematical observation:The physical states |ψ〉 all obey the same evolution equation:

|ψ(t)〉 = Uop(t)|ψ(0)〉 . (2.6)

4One can imagine deterministic models where P(t)op does not have an inverse, which means that two

different ontological states might both evolve into the same state later. We will consider this possibilitylater, see Section 13.2.

19

It is easy to observe that, indeed, the probabilities |λA|2 evolve as expected.5

Much of the work described in this report will be about writing the evolution operatorsUop(t) as exponentials: Find a hermitean operator Hop such that

Uop(δt) = e−iHop δt , so that Uop(t) = e−iHopt . (2.7)

This elevates the time variable t to be continuous, if it originally could only be a multipleof δt . Finding an example of such an operator is actually easy. If there are no ontologicalstates |A〉 with Uop(2δt)|A〉 = |A〉 (which is just a technical restriction), then

Hop =∞∑n=1

(−1)n

in δt(Uop(n δt)− Uop(−n δt)) . (2.8)

is a solution of Eq. (2.7). This equation can be checked by diagonalising Uop . If U(n δt) =e−iω n−ε|n| , where the infinitesimal number ε is inserted to get absolute convergence, onefinds, assuming |ω| < π ,

H =∞∑n=1

(−1)n

in δt((e−iω−ε)n − (eiω−ε)n) =

1

i δt(− log(1 + e−iω−ε) + log(1 + eiω−ε)) =

i ω

i δt. (2.9)

The reason why this is not always the hamiltonian we want is that its eigen values willalways be between −π/δt and π/δt , while sometimes we may want expressions for theenergy that take larger values (see for instance Section 3.1).

We conclude that there is always a hamiltonian. Note, that the ontological states, aswell as all other physical states (2.2) obey the Schrodinger equation,

d

dt|ψ(t)〉 = −iHop|ψ(t)〉 , (2.10)

so, we always reproduce some kind of “quantum” theory!

2.1.1. Operators

We plan to distinguish three types of operators:

(I) beables : these denote a property of the ontological states, so that beables are diag-onal in the ontological basis |A〉, |B〉, · · · of Hilbert space:

Oop|A〉 = O(A)|A〉 , (beable) . (2.11)

5At this stage of the theory, one could have defined probabilities to be given as different functions ofλA , in line with the observation just made after Eq. (2.2). For instance, when taking any other powerof the norm, Eq. (2.6) would also hold. These states, however, would then not stay normalised to one,which would cause considerable complications later.

20

(II) changeables : operators that replace an ontological state by another ontological state,such as a permutation operator:

Oop|A〉 = |B〉 , (changeable) ; (2.12)

These operators act as pure permutations.

(III) superimposables : these map ontological states onto superpositions of ontologicalstates:

Oop|A〉 = λ1|A〉+ λ2|B〉+ · · · . (2.13)

Now, we will construct a number of examples. Later, we shall see more examples ofconstructions of beable operators (e.g. Section 3.3).

2.2. The Cogwheel Model

a)3

21

b)

2

1

0

E

Figure 1: a) Cogwheel model with three states. b) Its three energy levels.

One of the simplest deterministic models is a system that can be in just 3 states, called(1), (2), and (3). The time evolution law is that, at the beat of a clock, (1) evolves into(2), (2) evolves into (3), and state (3) evolves into (1), see Fig. 1 a . Let the clock beatwith time intervals δt . As is explained in the previous section, we associate Dirac ketsto these states, so we have the states |1〉, |2〉, and |3〉 . The evolution operator Uop(δt)is then the matrix

Uop(δt) =

0 0 11 0 00 1 0

. (2.14)

It is now useful to diagonalise this matrix. Its eigen states are |0〉H , |1〉H , and |2〉H ,defined as

|0〉H = 1√3(|1〉+ |2〉+ |3〉) ,

|1〉H = 1√3(|1〉+ e2πi/3|2〉+ e−2πi/3|3〉) ,

|2〉H = 1√3(|1〉+ e−2πi/3|2〉+ e2πi/3|3〉) , (2.15)

for which we have

Uop(δt)

|0〉H|1〉H|2〉h

=

|0〉He−2πi/3|1〉He−4πi/3 |2〉H

. (2.16)

21

In this basis, we can write this as

Uop = e−iHopδt , with Hop = 2π3 δt

diag(0, 1, 2) . (2.17)

At times t that are integer multiples of δt , we have, in this basis,

Uop(t) = e−iHop t , (2.18)

but of course, this equation holds in every basis. In terms of the ontological basis of theoriginal states |1〉, |2〉, and |3〉 , the hamiltonian (2.17) reads

Hop = 2π3 δt

1 α α∗

α∗ 1 αα α∗ 1

, with α = −12

+ i√

36, α∗ = −1

2− i√

36. (2.19)

Thus, we conclude that a physical state |ψ〉 = λ(t)|1〉 + µ(t)|2〉 + ν(t)|3〉 that obeys theSchrodinger equation

d

dt|ψ〉 = −iHop|ψ〉 , (2.20)

with the hamiltonian (2.19), will be in the state described by the cogwheel model at alltimes t that are an integral multiple of δt . This is enough reason to claim that the“quantum” model obeying this Schrodinger equation is mathematically equivalent to ourdeterministic cogwheel model.

The eigen values of the hamiltonian are 2π3 δt

n , with n = 0, 1, 2 , see Fig. 1 b . This isreminiscent of an atom with spin one that undergoes a Zeeman splitting due to a homo-geneous magnetic field. One may conclude that such an atom is actually a deterministicsystem with three states, or, a cogwheel, but only if the ‘proper’ basis has been identified.

2.3. Generalisations of the cogwheel model: cogwheels with N teeth

The first generalisation of the cogwheel model is the system that permutes N ‘ontological’states |n〉ont , with n = 0, · · ·N − 1 , and N some positive integer > 1 . Assume thatthe evolution law is that, at the beat of the clock,

|n〉ont → |n+ 1 mod N〉ont . (2.21)

This model can be regarded as the universal description of any system that is periodicwith a period of N steps in time. The states in this evolution equation are regarded as‘ontological’ states. The model does not say anything about ontological states in betweenthe integer time steps. We call this the simple periodic cogwheel model with period N .

As a generalisation of what was done in the previous section, we perform a discreteFourier transformation on these states:

|k〉Hdef= 1√

N

N−1∑n=0

e2πikn/N |n〉ont , k = 0, · · ·N − 1 ; (2.22)

|n〉ont = 1√N

N−1∑k=0

e−2πikn/N |k〉H . (2.23)

22

Normalising the time step δt to one, we have

Uop(1) |k〉H = 1√N

N−1∑n=0

e2πikn/N |n+ 1 mod N〉ont = e−2πik/N |k〉H , (2.24)

and we can conclude

Uop(1) = e−iHop ; Hop|k〉H = 2πkN|k〉H . (2.25)

This hamiltonian is restricted to have eigen values in the interval [0, 2π) . where thenotation means that 0 is included while 2π is excluded. Actually, its definition impliesthat the hamiltonian is periodic with period 2π , but in most cases we will treat it asbeing defined to be restricted to within the interval. The most interesting physical caseswill be those where the time interval is very small, for instance close to the Planck time,so that the highest eigen values of the hamiltonian will be considered unimportant inpractice.

In the original, ontological basis, the matrix elements of the hamiltonian are

ont〈m|Hop|n〉ont = 2πN2

N−1∑k=1

k e2πik(m−n)/N . (2.26)

So, again, if we impose the Schrodinger equation ddt|ψ〉t = −iHop|ψ〉t and the boundary

condition |ψ〉t=0 = |n0〉ont , then this state obeys the deterministic evolution law (2.21)at integer times t . If we take superpositions of the states |n〉ont with the Born ruleinterpretation of the complex coefficients, then the Schrodinger equation still correctlydescribes the evolution of these Born probabilities.

It is of interest to note that the energy spectrum (2.25) is frequently encountered inphysics: it is the spectrum of an atom with total angular momentum J = 1

2(N − 1)

and magnetic moment µ in a weak magnetic field: the Zeeman atom. We observe that,after the discrete Fourier transformation (2.22), a Zeeman atom may be regarded asthe simplest deterministic system that hops from one state to the next in discrete timeintervals, visiting N states in total.

As in the Zeeman atom, we may consider the option of adding a finite, universalquantity δE to the hamiltonian. It has the effect of rotating all states with the complexamplitude e−i δE after each time step. For a simple cogwheel, this might seem to be aninnocuous modification, with no effect on the physics, but below we shall see that theeffect of such an added constant might become quite significant later.

2.3.1. The most general deterministic, time reversible, finite model

Generalising the finite models discussed earlier in this chapter, we start with any state|n0〉ont . After some finite number, say N0 , of time steps, the system will be back at|n0〉ont . However, not all states |n〉ont may have been reached. So, if we start with anyof the remaining states, say |n1〉ont , then a new series of states will be reached, and the

23

Figure 2: Example of a more generic finite, deterministic, time reversible model

periodicity might be a different number, N1 . Continue until all existing states of themodel have been reached. We conclude that the most general model will be described asa set of simple periodic cogwheel models with varying periodicities, but all working withthe same universal time step δt , which we could normalise to one; see Fig. 2.

a)

E

| 1 ⟩

| 0 ⟩

| N −1 ⟩

0

δE

δE +2π

b)

E

0 δEic)

E

0

Figure 3: a) Energy spectrum of the simple periodic cogwheel model. δE is anarbitrary energy shift. b) Energy spectrum of the model sketched in Fig. 2, whereseveral simple cogwheel models are combined. Each individual cogwheel i may beshifted by an arbitrary amount δEi . c) Taking these energy levels together we getthe spectrum of a generic finite model.

Fig. 3 shows the energy levels of a simple periodic cogwheel model (left), a combinationof simple periodic cogwheel models (middle), and the most general deterministic, timereversible, finite model (right). Note that we now shifted the energy levels of all cogwheelsby different amounts δEi . This is allowed because the index i telling us which cogwheelwe are in is a conserved quantity; therefore these shifts have no physical effect. We do

24

observe the drastic consequences however when we combine the spectra into one, seeFig. 3 c .

Fig. 3 clearly shows that the energy spectrum of a finite discrete deterministic modelcan quickly become quite complex6. It raises the following question: given any kindof quantum system, whose energy spectrum can be computed. Would it be possible toidentify a deterministic model that mimics the quantum model? To what extent wouldone have to sacrifice locality when doing this? Are there classes of deterministic theoriesthat can be mapped on classes of quantum models? Which of these would be potentiallyinteresting?

2.4. Infinite, discrete models

Discrete models with infinitely many states may have the new feature that some orbitsmay not be periodic. They then contain at least one non-periodic ‘rack’. There existsa universal definition of a quantum hamiltonian for this general case, though it is notunique. Defining the time reversible evolution operator over the smallest discrete timestep to be an operator Uop(1) , we now construct the simplest hamiltonian Hop such thatUop(1) = e−iHop . For this, we use the evolution over n steps, where n is positive ornegative:

Uop(n) = Uop(1)n = e−inHop . (2.27)

If we may assume that the eigenvalues ω of this hamiltonian lie between −π and π ,then we can consider the hamiltonian in the basis where both U(1) and H are diagonal.We have then

e−inω = cos(nω)− i sin(nω) , (2.28)

so to find H all we have to do is expand the function ω on the segment (−π, π) in itsFourier modes sin(nω) and cos(nω) . One finds

ω =∞∑n=1

2(−1)n−1

nsin(nω) . (2.29)

And thus we find

Hop =∞∑n=1

(−1)n−1 i

n

(Uop(1)n − Uop(1)−n

). (2.30)

But note that Eq. (2.29) does not converge when there is a mode with period 2, wherex = π (exercise: take ω = π− ε with ε infinitesimal). We will encounter such situations

6It should be self-evident that the models displayed in the figures, and subsequently discussed, arejust simple examples; the real universe will be infinitely more complicated than these. One critic of ourwork was confused: “Why this model with 31 states? What’s so special about the number 31?” Nothing,of course, it is just an example to illustrate how the math works.

25

more frequently; we have to make exceptions for modes where the evolution operatortakes the value −1 . These will be called edge states.

Very often, we will not be content with this hamiltonian, as it has no eigenvaluesbeyond the range (−π, π) . As soon as there are conserved quantities, one can addfunctions of these at will to the hamiltonian, to be compared with what is often done withchemical potentials. Cellular automata in general will exhibit many such conservationlaws. See Fig. 3, where every closed orbit represents something that is conserved: thelabel of the orbit.

In Section 2.6, we consider the other continuum limit, which is the limit δt → 0 forthe cogwheel model. First, we look at continuous theories more generally.

2.5. More general continuous ontological models

In the physical world, we have no indication that time is truly discrete. It is thereforetempting to consider the limit δt → 0 . At first sight, one might think that this limitshould be the same as having a continuous degree of freedom obeying differential equationsin time, but this is not quite so, as will be explained. First, in this section, we consider thestrictly continuous deterministic systems. Then, we compare those with the continuumlimit of discrete systems.

Consider an ontological theory described by having a continuous, multi-dimensionalspace of degrees of freedom qi(t) , depending on one continuous time variable t , and itstime evolution following classical differential equations:

d

dtqi(t) = fi(~q ) , (2.31)

where fi(~q ) may be almost any function of the variables qj .

An example is the description of massive objects obeying classical mechanics in Ndimensions. Let a = 1, · · · , N, and i = 1, · · · , 2N :

i = a, a+N , qa(t) = xa(t) , qa+N(t) = pa(t) ,

fa(~q ) =∂Hc(~x, ~p )

∂pa, fa+N(~q ) = −∂Hc(~x, ~p )

∂xa, (2.32)

where Hc is the classical hamiltonian.

An other example is the quantum wave function of a particle in one dimension:

i = x , qi(t) = ψ(x, t) ; fi(~q ) = −iHS ψ(x, t) , (2.33)

where now HS is the Schrodinger hamiltonian. Note, however, that, in this case, the func-tion ψ(x, t) would be treated as an ontological object, so that the Schrodinger equationand the hamiltonian eventually obtained will be quite different from the Schrodinger equa-tion we start off with; actually it will look more like the corresponding second quantisedsystem.

26

We are now interested in turning Eq. (2.31) into a quantum system by changing thenotation, not the physics. The ontological basis is then the set of states |~q 〉 , obeying theorthogonality property

〈~q |~q ′〉 = δN(~q − ~q ′) , (2.34)

where N is now the dimensionality of the vectors ~q .

If we wrote

d

dtψ(~q )

?= − fi(~q )

∂

∂qiψ(~q )

?= − iHop ψ(~q ) , (2.35)

where the index i is summed over, we would read off that

Hop?= − ifi(~q )

∂

∂qi= fi(~q )pi , pi = −i ∂

∂qi. (2.36)

This, however, is not quite the right hamiltonian because it may violate hermiticity:Hop 6= H†op . The correct hamiltonian is obtained if we impose that probabilities are pre-

served, so that, in case the Jacobian of ~f(~q ) does not vanish, the integral∫

dN~q ψ†(~q )ψ(~q )is still conserved:

d

dtψ(~q ) = −fi(~q )

∂

∂qiψ(~q )− 1

2

(∂fi(~q )

∂qi

)ψ(~q ) = −iHop ψ(~q ) , (2.37)

Hop = −ifi(~q )∂

∂qi− 1

2i(∂fi(~q )

∂qi

)= 1

2(fi(~q )pi + pifi(~q )) ≡ 1

2fi(~q ) pi . (2.38)

The 1/2 in Eq. (2.37) ensures that the product ψ† ψ evolves with the right Jacobian. Notethat this hamiltonian is hermitean, and the evolution equation (2.31) follows immediatelyfrom the commutation rules

[qi, pj] = iδij ;d

dtOop(t) = −i[Oop , Hop ] . (2.39)

Now, however, we encounter a very important difficulty: this hamiltonian has no lowerbound. It therefore cannot be used to stabilise the wave functions. Without lower bound,one cannot do thermodynamics. This feature would turn our model into something veryunlike quantum mechanics as we know it.

If we take ~q space either one-dimensional, or in some cases two-dimensional, we canmake our system periodic. Then let T be the smallest positive number such that

~q (T ) = ~q (0) . (2.40)

We have consequently

e−iHT |~q (0)〉 = |~q(0) 〉 , (2.41)

27

and therefore, on these states,

Hop |~q 〉 =∞∑

n=−∞

2πn

T|n〉HH〈n|~q 〉 . (2.42)

Thus, the spectrum of eigen values of the energy eigen states |n〉H runs over all integersfrom −∞ to ∞ .

In the discrete case, the hamiltonian has a finite number of eigen states, with eigenvalues 2πk/(Nδt)+δE where k = 0, · · · , N−1 , which means that they lie in an interval[δE , 2π/T + δE] , where T is the period, and δE can be freely chosen. So here, wealways have a lower bound, and the state with that energy can be called ‘ground state’or ‘vacuum’.

Depending on how the continuum limit is taken, we may or may not preserve thislower bound. In any case, it is artificial, because all energy eigen values look exactlyalike.

A possible remedy against this problem could be that we demand analyticity whentime is chosen to be complex, and boundedness of the wave functions in the lower half ofthe complex time frame. This would exclude the negative energy states, and still allowus to represent all probability distributions with wave functions. Equivalently, one couldconsider complex values for the variable(s) ~q and demand the absence of singularities inthe complex plane below the real axis. Such analyticity constraints however seem to berather arbitrary and they would certainly have to be handled with caution.

One very promising approach to solve the ground state problem could be Dirac’s greatidea of second quantisation: take an indefinite number of objects ~q , that is, a Hilbertspace spanned by all states |~q (1), ~q (2), · · · ~q (n)〉 , for all particle numbers n , and regardthe negative energy particles as ‘holes’ of antiparticles. This we propose to do in our‘neutrino model’, Section 3.3.

Alternatively, we might consider the continuum limit of a discrete theory more care-fully. This we try first in the next section. In general, however, excising the negativeenergy states is not always a good idea, because any perturbation of the system mightcause transitions to these negative energy states, and leaving these transitions out mayviolate unitarity.

The importance of the ground state of the hamiltonian is further discussed in 9.1.1.The hamiltonian (2.38) is an important expression for fundamental discussions on quan-tum mechanics.

As in the discrete case, also in the continuum case one finds discrete and continuouseigenvalues, depending on whether a system is periodic. In the limit δt→ 0 of the discreteperiodic ontological model, the eigen values are integer multiples of 2π/T , and this isalso the spectrum of the harmonic oscillator with period T , as explained in Section 2.6.The harmonic oscillator is to be regarded as a deterministic system in disguise.

The more general continuous model is then the system obtained first by having a(finite or infinite) number of harmonic oscillators, which means that our system consists

28

of many periodic substructures, and secondly by admitting a (finite or infinite) numberof conserved quantities on which the periods of the oscillators depend. An example isthe field of non-interacting particles; quantum field theory then corresponds to havingan infinite number of oscillating modes of this field. The particles may be fermionic orbosonic; the fermionic case is also a set of oscillators if the fermions are put in a box withperiodic boundary conditions. Interacting quantum particles will be encountered later(Chapter 7 and onwards).

2.6. The continuum limit of a cogwheel: the harmonic oscillator

Let us return to the cogwheel model, considering one orbit. In the N → ∞ limit, thiscogwheel will have an infinite number of states. The hamiltonian will therefore also haveinfinitely many eigenstates. There are two ways to take a continuum limit. As N →∞ ,we can keep the quantised time step fixed, say it is 1. Then, in the hamiltonian (2.25), wehave to allow the quantum number k to increase proportionally to N , keeping κ = k/Nfixed. Since the time step is one, the hamiltonian eigen values, 2πκ , now lie on a circle, or,we can say that the energy takes values in the continuous line segment [0, 2π) (includingthe point 0 but excluding the point 2π ). Again, one may add an arbitrary constant δEto this continuum of eigenvalues. What we have then, is a model of an object moving ona lattice in one direction. At the beat of a clock, a state moves one step at a time to theright. Rather than a cogwheel, this is a rack. This is an important system, and we returnto it in Section 5.1.

The other option for a continuum limit is to keep the period T of the cogwheelconstant, while the time quantum δt tends to zero. This is also a cogwheel, now withinfinitely many, microscopic teeth, but still circular. Since now N δt = T is fixed, theontological7 states of the system can be described as an angle:

2πn/N → ϕ ,d

dtϕ(t) =

2π

T. (2.43)

The energy eigenvalues become

Ek = 2π k/T + δE , k = 0, 1, · · · , ∞ . (2.44)

If δE is chosen to be π/T , we have the spectrum Ek = (2π/T )(k + 12) . This is the

spectrum of a harmonic oscillator. In fact, any periodic system with period T , anda continuous time variable can be characterised by defining an angle ϕ obeying theevolution equation (2.43), and we can apply a mapping onto a harmonic oscillator withthe same period, as we now shall describe.

Mappings of one model onto another one will be frequently considered in this treatise.It will be of importance to understand what we mean by this. If one does not put anyconstraint on the nature of the mapping, one would be able to map any model onto anyother model; physically this might then be rather meaningless. Generally, what we are

7The words ‘ontological’ and ‘deterministic’ will be frequently used to indicate the same thing: amodel without any non deterministic features, describing things that are really there.

29

looking for are mappings that can be prescribed in a time-independent manner. Thismeans that, once we know how to solve the evolution law for one model, then afterapplying the mapping, we have also solved the evolution equations of the other model.The physical data of one model will reveal to us how the physical data of the other oneevolve.

This requirement might not completely suffice in case a model is exactly integrable. Inthat case, every integrable model can be mapped onto any other one, just by consideringthe exact solutions. In practice, however, time independence of the prescription can beeasily verified, and if we now also require that the mapping of the physical data of onemodel onto those of the other is one-to-one, we can be confident that we have a relationof the kind we are looking for. If now a deterministic model is mapped onto a quantummodel, we may demand that the classical states of the deterministic model map ontoan orthonormal basis of the quantum model. Superpositions, which look natural in thequantum system, might look somewhat contrived and meaningless in the deterministicsystem, but they are certainly acceptable as describing probabilistic distributions in thelatter. This report is about these mappings.

We have already seen how the periodic deterministic system can produce a discretespectrum of energy eigenstates. The continuous system described in this subsection gener-ates energy eigenstates that are equally spaced, and range from a lowest state to E →∞ .To map this onto the harmonic oscillator, all we have to do is map these energy eigen-states onto those of the oscillator, and since these are also equally spaced, both systemswill evolve in the same way. Of course this is nothing to be surprised about: both systemsare integrable and periodic with the same period T .

For the rest of this subsection, we will put T = 2π . The hamiltonian of this harmonicoscillator we now choose to be8

Hop = 12(p2 + x2 − 1) ; Hop |n〉H = n |n〉H , n = 0, 1, · · · ,∞ . (2.45)

The subscript H reminds us that we are looking at the eigen states of the hamiltonianHop . For later convenience, we subtracted the zero point energy9.

First, let us identify the operators x and p of the harmonic oscillator, with operatorsin the space of the ontological states |ϕ〉ont of our periodic system. This is straightforward.

In the energy basis of the oscillator, we have the annihilation operator a and thecreation operator a† with the following effect on the eigenstates of Hop :

a |n〉H =√n |n− 1〉H , a† |n〉H =

√n+ 1 |n+ 1 〉H . (2.46)

We also have

a = 1√2

(x+ ip) , a† = 1√2

(x− ip) , (2.47)

8In these expressions, x, p, a , and a† are all operators, but we omitted the subscript “op” to keepthe expressions readable.

9Interestingly, this zero point energy would have the effect of flipping the sign of the amplitudes afterexactly one period. Of course, this phase factor is not directly observable, but it may play some role infuture considerations. In what we do now, it is better to avoid these complications.

30

so, x = 1√2

(a + a†) , p = −i√2

(a − a†) , and therefore we can readily identify the matrix

elements (let us drop the subscript H ),

〈n− 1 |x |n 〉 =√n/2 = 〈n |x |n− 1〉 ,

〈n− 1 | p |n 〉 = −i√n/2 = −〈n | p |n− 1〉 , (2.48)

while all other matrix elements vanish.

2.6.1. The operator ϕop in the harmonic oscillator

The operator ϕop , corresponding to the ontological variable ϕ(t) in Eq. (2.43), can nowbe expressed in terms of the harmonic oscillator operators x and p . Taking 0 < ϕ < 2π ,we can write, analogously to Eq. (2.30),

ϕ = π +∞∑k=1

i

k(eikϕ − e−ikϕ) . (2.49)

This helps us to express the operator ϕop in terms of the more basic operators obtainedby raising the unitary operator eiϕop to the kth power.

It is easy to see that the operator e−iϕop only has non-vanishing elements when sand-wiched between two adjacent energy states, and, by choosing the appropriate normalisa-tion, to be read off from Eqs. (2.46), one obtains

e−iϕop =1√

Hop + 1a+ |N − 1〉〈0| ; eiϕop = a†

1√Hop + 1

+ |0〉〈N − 1| . (2.50)

Here, Hop is the hamiltonian (2.45), a and a† are the annihilation and creation operators,and |N−1〉 is the highest energy level that is included in our definition of the ontologicalstates. Note, that these last terms cannot be left out, since the operators e±iϕ wouldotherwise not be unitary. The order of the terms is chosen such that dividing by zero isavoided; Hop + 1 is bounded below by the value 1.

Let us however repeat the warning at the end of Section 2.5. The last terms in the twoequations (2.50) should not be ignored since these effectively represent the negative energystates that can emerge when the system is perturbed. The states were raised to occurat energies N − 1 in stead of −1 but that does not remove their possible contributions,when perturbations of the original equations of motion are considered. The eigen states|ϕ〉 of the operator ϕop are related to Glauber’s coherent states[11], but our operatorϕop is hermitean and its eigen states are orthonormal; Glauber’s states are eigen statesof the creation or annihilation operators, which were introduced by him using a differentphilosophy. Orthonormality is a prerequisite for the ontological states that are used inthis work.

2.6.2. The operators x and p in the ϕ basis

The next two sections are accounts of some technical calculations one can do to study thetransformation from the ontological ϕ -basis to the harmonic oscillator basis and back.They can be skipped unless the reader is particularly interested in these issues.

31

In the continuum limit of the cogwheel model, the energy eigenstates are normalisedstates in the ϕ variable, so Eqs. (2.22) become

ont〈ϕ |n 〉 = 1√2πe inϕ , 〈n |ϕ 〉ont = 1√

2πe− inϕ . (2.51)

We derive (dropping also the subscript ont )

〈ϕ1 |x |ϕ2 〉 =∞∑n=1

〈ϕ1 |n− 1 〉〈n− 1 |x |n 〉〈n |ϕ2〉 +

∞∑n=1

〈ϕ1 |n 〉〈n |x |n− 1 〉〈n− 1 |ϕ2〉 =

=1

2π

∞∑n=1

√n

2(e−iϕ1+in(ϕ1−ϕ2) + ein(ϕ1−ϕ2)+iϕ2) . (2.52)

We would like to elaborate this expression further, but then we have to use a math-ematical function that is not encountered very often in physics, the polylogarithm, orJonquiere’s function, of the order −1

2. The polylogarithm, Lis(z) is defined by

Lis(z) =∞∑k=1

zk

ks= z +

z2

2s+z3

3s+ · · · . (2.53)

We only need the case s = −12

, and call it L(z) :

L(z) = Li−1/2(z) =∞∑n=1

zk√k = z + z2

√2 + z3

√3 + · · · . (2.54)

This function has some important properties, to be found by contour integration andanalytic continuation:

L(z) =2√π

∫ ∞0

dtz e−t

2

(1− z e−t2)2; (2.55)

at z → 1 : L(z) →√π

2

1

(1− z)3/2, (2.56)

at z → −∞ : L(z) → −1√π log(−z)

. (2.57)

We can now work out Eq. (2.52) further:

〈ϕ1 |x |ϕ2 〉 =1

2π√

2(e−iϕ1 + eiϕ2)L(ei(ϕ1−ϕ2)) . (2.58)

For the momentum operator, we can do the same arithmetic, to obtain

〈ϕ1 | p |ϕ2 〉 =1

2π√

2(−i e−iϕ1 + i eiϕ2)L(ei(ϕ1−ϕ2)) . (2.59)

32

We see that these matrix elements are singular at ϕ1 − ϕ2 → 0 . They behave as

〈ϕ1 |x |ϕ2 〉 →1

4√π

cosϕ1 (−1± i) |ϕ1 − ϕ2|−3/2 ;

〈ϕ1 | p |ϕ2 〉 →−1

4√π

sinϕ1 (−1± i) |ϕ1 − ϕ2|−3/2 ,if ϕ1 − ϕ2

>< 0 . (2.60)

It is clear that, with such singularities present, these operators will have to be handledwith care.

2.6.3. The transformation matrix relating the harmonic oscillator with the periodicϕ system

Figure 4: Plot of the real part of the transformation matrix 〈ϕ |x 〉 . Horizontally,x runs from −10 to 10 , vertically, ϕ runs from −π to π . Eqs. (2.68), (2.70) and(2.71) were used. In the busiest parts of the picture, the rapid oscillations are nolonger visible.

The complete transformation from the ontological variable ϕ to the elementary op-erators x and p for the harmonic oscillator is described by the matrix elements 〈ϕ |x 〉and/or 〈ϕ | p 〉 . If the energy eigenstates are | k 〉 , then Eqs. (2.22) or (2.51) link themto the ϕ variable, and in the harmonic oscillator, they are given by the known solutions

33

of the oscillator in x or p space. With the proper normalisation,

〈ϕ |n 〉 = 1√2πeinϕ , 〈x |n 〉 =

Hn(x) e−12x2

2n/2π1/4√n!

, (2.61)

where Hn(x) are the Hermite polynomials. Thus, we get

〈ϕ |x 〉 =∞∑n=0

Hn(x) e−12x2

2(n+1)/2π3/4√n!e inϕ . (2.62)

Here, the sum converges, because the highest oscillator states peak at the highest positionstates, and since the oscillator moves fast near the origin, the wave functions tend tobecome small near the origin at high n . This convergence is slow however, and, as wewill see, this causes some practical difficulties when explicitly evaluating the sum.

A complication is caused by the square root of n! in the denominator; it impedes afurther reduction to simpler mathematical expressions. To get some impression about thegeneral form of this transformation matrix, one can try to evaluate the sum. It convergesslowly, and at least 100 terms are needed, at extremely high precision, to see where it leadsto. We can work out the sum in terms of more convergent expressions, but this is evenmore technical. This exercise in elementary mathematical function theory and contourintegration is elaborated in the next subsection. The real part of 〈ϕ |x 〉 is plotted inFigure 4.

The transformation matrix from ϕ to p is essentially the same. We see from Eqs.(2.58) — (2.60) that 〈ϕ | p 〉 is obtained from 〈ϕ |x 〉 by shifting ϕ→ ϕ+ 1

2π .

2.6.4. Appendix to Section 2.6: Explicitly evaluating the transformation matrix〈ϕ |x 〉

We wish to evaluate the expression (2.62) for 〈ϕ |x 〉 that was found in Subsection 2.6.3:

〈ϕ |x 〉 =∞∑n=0

Hn(x) e−12x2

2(n+1)/2π3/4√n!e inϕ . (2.63)

Because of the square root, we know of no easy way to evaluate the sum in terms ofelementary functions. There are now two directions for further study: we can try to devisealgorithms to obtain approximate expressions so as to get some idea about the nature ofthis matrix, or we can try to obtain as much information as is possible concerning theanalytic structure and properties of this matrix. In this Appendix we do a bit of each.We note that the number n counts all energy eigenstates, and these are all normalised.The very highest energy states will be highly oscillatory and therefore cease to contribute,which means that we expect the sum to converge properly. This convergence turns outto be quite slow, however.

To speed up convergence, we replace the square root of n! by an expression thatcoincides with the original for large n , whereas the difference stays quite small even for

34

n→ 0 . Use the doubling formula for n! ,√π

2nn! = Γ(1

2n+ 1

2) Γ(1

2n+ 1) . (2.64)

For large n , this is very nearly equal to Γ(12n + 3

4)2 , but also for all small values, the

deviation is only a few percent. The replacement should hardly modify the main featuresof the expression we wish to calculate. Now for this report, such guesses may be oflesser importance; it so happens, however, that the correction needed to obtain the exactexpression can also be calculated and since it converges much faster, this makes thisapproach quite useful.

Thus, we divide the amplitude into two parts, a dominant part, and the correctionterm,

〈ϕ |x 〉 = 〈ϕ |x 〉dom + 〈ϕ |x 〉corr . (2.65)

As will soon become clear, we wish to keep a factor n! in the denominator, so ourreplacement is:

〈ϕ |x 〉dom =∞∑n=0

e−12x2 Γ(1

2n+ 3

4)

π√

2

Hn(x) e inϕ

n!. (2.66)

To enable us to use the generating function for the Hermite polynomials Hn(x) ,

∞∑n=0

Hn(x) sn/n! = e− s2+2sx , (2.67)

we now use the integral expression for the Gamma function, to obtain

〈ϕ |x 〉dom =∞∑n=0

e−12x2

π√

2

∫ ∞0

dy

y1/4

Hn(x)

n!y

12n e−y e inϕ

=e−

12x2

π√

2

∫ ∞0

dy

y1/4exp(−y − ye2iϕ + 2x eiϕ

√y) . (2.68)

Substituting y =z2

1 + e2iϕ−ε , where ε is an infinitesimal, positive number to remind us

of a smooth cut-off at infinity of the sum in Eq. (2.67), we arrive at a smooth integralexpression, and by a few simple rearrangements,

〈ϕ |x 〉dom =

√2 e

12ix2 tanϕ

π(1 + e2iϕ−ε)3/4

∫ ∞0

√zdz exp

(− (z − x eiϕ√

1 + e2iϕ−ε)2).

(2.69)

At large x values, this expression splits into two different cases: if x cosϕ 0 , theintegral is dominated by its extremum point, at z = xeiϕ√

1+e2iϕ−ε, and we get

〈ϕ |x 〉dom →√

2x ei2x2 tanϕ+ 1

2iϕ

√π (1 + e2iϕ−ε)

, (2.70)

35

with some extra conditions concerning the sign choice for the square root.

When x cosϕ 0 , the integration path does not come close to the extremum, andonly small values of z contribute to the integral:

〈ϕ |x 〉dom →e−

12x2

4√π(−x eiϕ)3/2

. (2.71)

In practice, Eq. (2.68) turns out to be the most useful expression for the dominant partof the matrix.

But now we wish to know how, in principle, to compute the correction term. First,write it as

〈ϕ |x 〉corr =∞∑n=0

Hn(x) e−12x2+inϕ Γ(1

2n+ 3

4)

π√

2n!

( √n! π1/4

212n Γ(1

2n+ 3

4)− 1). (2.72)

Since we wish to use the generating function of the Hermite polynomials, Eq. (2.67), weneed to rewrite the terms inside the brackets as a series of nth powers. To this end, weextend it to the complex plane. The only singularities are the branch points caused bythe square root of n! . They lie on the negative part of the real axis. There, we use

Γ(n+ 1) =π

sin(−πn) Γ(−n); Γ(1

2n+ 3

4) =

π

sin π(14− 1

2n)Γ(1

4− 1

2n)

. (2.73)

At positive or vanishing n , there are no singularities; at negative n , the only singularitiesare caused by the zeros of the first sine function in (2.73). They are branch cuts, extendingfrom n = −1 to −2 , from n = −3 to −4 , etc., where the sine is negative. The residuesare easy to identify. And thus, by contour integration, we find that the brackets in Eq.(2.72) can de written as∫ ∞

t=1

%(t)dt

t+ n; %(t) =

212t Γ(1

2t+ 1

4)

π5/4√

Γ(t)

sin π(12t+ 1

4)√

| sin(πt)|. (2.74)

The integration goes over the line segments t ∈ [1, 2], [3, 4], · · · , where sin(πt) is negative,see Fig. 5. Due to the sine function in the numerator, %(t) has a rapidly alternatingsign, so that the integral

∫%(t)dt converges. Thus, the first part of the correction term

decreases as 1/n at large n , so that the constant −1 is automatically cancelled.

A very good approximation for Eq. (2.74) is

%(t) ≈C sin π(1

2t+ 1

4)√

| sin(πt)|, (2.75)

with C varying from ≈ 0.42 at small values of t , to√

2/π = 0.45 for large values of t .

Then, to work out the equation (2.72), we write

1

t+ n=

∫ ∞0

zn e−z(t+n) , (2.76)

36

985410

1

−1

t

Figure 5: The function %(t) calculated in Eq. (2.74)

so that the correction term can be written as

〈ϕ |x 〉corr =∞∑n=0

Hn(x) e−12x2+inϕ

π√

2 n!

∫ ∞0

dy

∫ ∞0

dz

∫dt %(t) y

12n− 1

4 e−y−z(t+n) =

=e−

12x2

π√

2

∫ ∞0

dy y−14

∫ ∞0

dz

∫dt %(t) e−y−zt−s

2+2sx ,

s =√y eiϕ−z . (2.77)

Combined with Eq. (2.68), this gives an accurate and convergent expression for the matrix〈ϕ |x 〉 . The matrix itself is illustrated as a plot in Fig. 4.

3. Other physical models

In the previous chapter, we have already seen some simple models that can be written interms of a completely deterministic physical rule: Zeeman atoms whose spectral statesare evenly spaced due to a weak external magnetic field, and the harmonic oscillator. Itmight seem as if we only have trivial, entirely integrable models of this sort. In princi-ple, however, our philosophy seems to be much more generally applicable. The multiplecogwheel model described in Section 2.3.1, giving rise to the energy levels illustrated inFig. 3, shows how large systems can generate quite complex behaviour. Any automatonwith a sufficiently complex memory will allow for many different periodic modes, so onemay expect these complex eigenvalue structures in its effective hamiltonian.

In this connection, remember also the doctrine that we outlined in Section 1.3: anytransformation of the basis in Hilbert space may be considered, so our description of thetemplates in terms of the ontological states can be quite complex; only the eigen valuesof the hamiltonian will be unaffected, and it is exactly these that can form quite generalspectra, as in Figure 3 c .

Indeed, in this chapter we show how such more generic systems can be studied system-atically. The real question will then be whether these models include systems as complexas the Standard Model, or at least models where Bell type experiments can be considered.Will we ever see ‘entangled photons’? This question will be discussed later (chapters 8and onwards).

37

In this chapter, we describe a number of models where our ideas can be further il-lustrated. Again in line with our general philosophy, we consider cases where one has aclassical, deterministic automaton on the one hand, and an apparently quantum mechani-cal system on the other. Then, the mathematical mapping is considered that shows thesetwo systems to be equivalent in the sense that the solutions of one can be used to describethe solutions of the other.

3.1. Time reversible cellular automata

A technically complicated, but very illustrative case will be briefly exhibited first. Westate right at the start that this particular example is not free from problems, but it doesillustrate the power of “quantum mathematics”. The technicalities will later be explained,in Chapter 11.

All our models can be characterised as ‘automata’. A cellular automaton is an au-tomaton where the data are imagined to form a discrete, d -dimensional lattice, in ann = d+ 1 dimensional space-time. The elements of the lattice are called ‘cells’, and eachcell can contain a limited amount of information. The data Q(~x, t) in each cell (~x, t)could be represented by an integer, or a set of integers, possibly but not necessarily limitedby a maximal value N . An evolution law prescribes the values of the cells at time t + 1if the values at time t (or t and t − 1 ) are given. Typically, the evolution law for thedata in a cell at the space-time position

(~x, t) , ~x = (x1, x2, · · · , xd) , xi, t ∈ Z , (3.1)

will only depend on the data in neighbouring cells at (~x ′, t − 1) and (~x, t − 2) , where‖~x ′− ~x‖ is limited by some bound. The cellular automaton is time-reversible if the datain the past cells can be recovered from the data at later times, and when the rule for thisis also a cellular automaton.

Time reversibility can easily be guaranteed if the evolution law is assumed to be ofthe form

Q(~x, t+ 1) = Q(~x, t− 1) + F (~x, Q(t)) , (3.2)

where Q(~x, t) represent the data at a given point (~x, t) of the space-time lattice, andF (~x, Q(t)) is some given function of the data of all cells neighbouring the point ~x attime t . The + here stands for addition, addition modulo some integer N , or some othersimple, invertible operation in the space of variables Q . Of course, we then have timereversibility:

Q(~x, t− 1) = Q(~x, t+ 1)− F (~x, Q(t)) , (3.3)

where − is the inverse of the operation + . Rules such as Eq. (3.2) are called a Margolusrule[26].

In principle, we can apply the procedures spelled out in Section 2.3.1, but we wouldlike to impose an other important requirement: it would be convenient if the hamiltonian

38

Hop could be described as the direct sum of hamiltonian density operators Hop(~x) thatare (more or less) locally defined:

Hop?=∑~x

Hop (~x) , [Hop (~x ′) , Hop (~x) ]→ 0 if ‖~x ′ − ~x‖ 1 . (3.4)

In that case, our quantum system could be directly related to some quantum fieldtheory in the continuum limit. We describe quantum field theories in Chapter 10. At firstsight, it may seem to be obvious that the hamiltonian should take the form of Eq. (3.4),but we have to remember that the hamiltonian definitions (2.8), (2.18), (2.26) and theexpressions illustrated in Fig. 3, are only well-defined modulo 2π/δt , so that they are notadditive at all.

These expressions do show that the hamiltonian Hop can be chosen in many ways,since one can add any operator that commutes with Hop . Therefore we do expect thata hamiltonian of the form (3.4) can be defined. An approach is explained in much moredetail in Chapter 11, but some ambiguity remains, and convergence of the procedure,even if we limit ourselves to low energy states, is far from obvious.

What we do see is this: a hamiltonian of the form (3.4) is the sum of terms that eachare finite and bounded below as well as above. Such a hamiltonian must have a groundstate, that is, an eigen state |0〉 with lowest eigen value, which could be normalised to zero.This eigenstate might be identified with the ‘vacuum’. This vacuum is stationary, even ifthe automaton itself may have no stationary solution. The next-to-lowest eigenstate maybe a one-particle state. In a Heisenberg picture, the fields Q(~x, t) will behave as operatorsQop(~x, t) when we pass on to a basis of template states, and as such they may create one-particle states out of the vacuum. Thus, we arrive at something that resembles a genuinequantum field theory. The states are quantum states in complete accordance with aCopenhagen interpretation. The fields Qop (~x, t) should obey the Wightman axioms.Quantum field theories will further be discussed in Chapter 10.

There are three ways, however, in which this theory differs from conventional quantumfield theories. One is, of course, that space and time are discrete. Well, maybe there is aninteresting ‘continuum limit’, in which the particle mass(es) is(are) considerably smallerthan the inverse of the time quantum, but this will in general not be the case.

Secondly, no attempt has been made to arrive at Lorentz invariance. Not only willthe one-particle states fail to exhibit Lorentz invariance, or even Gallilei invariance; thestates where particles may be moving with respect to the vacuum state will be completelydifferent from the static one-particle states. Also, rotation symmetry will be reduced tosome discrete lattice rotation group if anything at all. So, the familiar symmetries ofrelativistic quantum field theories will be totally absent.

Thirdly, it is not clear that the cellular automaton can be associated to a singlequantum model or perhaps many inequivalent ones. The addition or removal of otherconserved operators to Hop is akin to the addition of chemical potential terms. In theabsence of Lorentz invariance, it will be difficult to distinguish the different types of‘vacuum’ states one thus obtains.

39

For all these reasons, cellular automaton models are very different from the quantisedfield theories for elementary particles. The main issue discussed in this report, however, isnot whether it is easy to mimic the Standard Model in a cellular automaton, but whetherone can obtain quantum mechanics, and something resembling quantum field theory atleast in principle. The origin of the continuous symmetries of the Standard Model willstay beyond what we can handle in this report, but we can discuss the question to whatextent cellular automata can be used to approximate and understand the quantum natureof this world. See also our discussion of symmetry features in Section 6.

As will be explained later, it may well be that invariance under general coordinatetransformations will be a crucial ingredient in the explanation of continuous symmetries,so it may well be that the ultimate explanation of quantum mechanics will also requirethe complete solution of the quantum gravity problem. We cannot pretend to have solvedthat.

Most of the other models in this chapter will be explicitly integrable. The cellularautomata that we started off with in this first Section illustrate that our general philosophyalso applies to non-integrable systems. It is generally believed that cellular automatacontrolled by the Margolus rule can be computationally universal [25]. This means thatany such automaton can be arranged in special subsets of states that obey the rule of anyother computationally universal cellular automaton. One would be tempted to argue then,that any computationally universal cellular automaton can be used to mimic systems ascomplicated as the Standard Model of the subatomic particles. However, in that case, wephysicists would ask for one single special model that is more efficient than any other,so that any choice of initial state in this automaton describes a physically realisableconfiguration.

Technical details concerning the mapping of a cellular automaton onto quantum mod-els, and the question how this can be squared with locality, are postponed to Chapter 11.

3.2. Fermions

In order to find more precise links between the real quantum world on the one handand deterministic automaton models on the other, much more mathematical machineryis needed. For starters, fermions can be handled in an elegant fashion.

The model described in Fig. 2 (page 24), has 31 states, and the evolution law for onetime step is an element P of the permutation group for 31 elements: P ∈ P31 . Let itsstates be indicated as |1〉, · · · , |31〉 . We write the single time step evolution law as:

|i〉t → |i〉t+δt = |P (i)〉t = P |i〉t , i = 1, · · · , 31 , (3.5)

where the latter matrix P has matrix elements 〈j|P |i〉 that consists of 0s and 1s, withone 1 only in each row and in each column. As explained in Section 2.3.1, we assume thata hamiltonian matrix Hop

ij is found such that (when normalising the time step δt to one)

〈i|P |j〉 = (e−iHop

)ij , (3.6)

40

where possibly a zero point energy δE may be added that represents a conserved quantity:δE only depends on the cycle to which the index i belongs, but not on the item insidethe cycle.

Figure 6: The “second quantised” version of the multiple-cogwheel model of Fig. 2.Black dots represent fermions.

We now associate to this model a different one, whose variables are boolean ones,taking the values 0 or 1 (or equivalently, +1 or −1 ) at every one of these 31 sites. Thismeans that we now have 231 = 2,147,483,648 states, see Fig. 6. The evolution law isdefined such that these boolean numbers travel just as the sites in the original cogwheelmodel were dictated to move. Physically this means that, if in the original model, exactlyone particle was moving as dictated, we now have N particles moving, where N canvary between 0 and 31. In particle physics, this is known as “second quantisation”. Sinceno two particles are allowed to sit at the same site, we have fermions, obeying Pauli’sexclusion principle.

To describe these deterministic fermions in a quantum mechanical notation, we firstintroduce operator fields φop

i , and their hermitean conjugates, φop †i . Denoting our states

now as |n1, n2, · · · , n31〉 , where all n ’s are 0 or 1, we postulate

φopi |n1, · · · , ni, · · · , n31〉 = ni |n1, · · · , ni − 1, · · · , n31〉 ,

φop †i |n1, · · · , ni, · · · , n31〉 = (1− ni) |n1, · · · , ni + 1, · · · , n31〉 , (3.7)

These fields obey, at one given site i (omitting the superscript ‘op’ for brevity):

(φi)2 = 0, φ†iφi + φi φ

†i = I , (3.8)

where I is the identity operator; at different sites, the fields commute: φi φj = φj φi ;

φ†iφj = φjφ†i , if i 6= j .

To turn these into completely anti-commuting (fermionic) fields, we apply the so-calledJordan-Wigner transformation[3]:

ψi = (−1)n1+···+ni−1φi , (3.9)

41

where ni = φ†iφi = ψ†iψi are the occupation numbers at the sites i , i.e., we insert aminus sign if an odd number of sites j with j < i are occupied. As a consequence ofthis well-known procedure, one now has

ψi ψj + ψj ψi = 0 , ψ†iψj + ψjψ†i = δij , ∀(i, j) . (3.10)

The virtue of this transformation is that the anti-commutation relations (3.10) stay un-changed after any linear, unitary transformation of the ψi as vectors in our vector space(in the above example having 31 dimensions), provided that ψ†i transform as contra-vectors. Usually, the minus signs in Eq. (3.9) do no harm, but some care is asked for.

Now consider the permutation matrix P and write the hamiltonian in Eq. (3.6) as alower case hij ; it is a 31 × 31 component matrix. Writing Uji(t) = (e−iht)ji , we have,at integer time steps, P t

op = Uop(t) . We now claim that the permutation that moves thefermions around, is generated by the hamiltonian Hop

F defined as

HopF =

∑ij

ψ†j hji ψi . (3.11)

This we prove as follows:Let

ψi(t) = eiHopF t ψi e

−iHopF t ,

d

dte−iH

opF t = −iHop

F e−iHopF t = −i e−iH

opF tHop

F ; (3.12)

Then

d

dtψk(t) = i eiH

opF t∑ij

[ψ†j hji ψi, ψk]e−iHop

F t = (3.13)

i eiHopF t∑ij

hji

(− ψ†j , ψkψi + ψ†jψi, ψk

)e−iH

opF t = (3.14)

− i eiHopF t∑i

hkiψi e−iHop

F t = − i∑i

hkiψi(t) , (3.15)

where the anti-commutator is defined as A, B ≡ AB +BA .

This is the same equation that describes the evolution of the states |k〉 of the originalcogwheel model. So we see that, at integer time steps t , the fields ψi(t) are permutedaccording to the permutation operator P t . Note now, that the vacuum state |0〉 doesnot evolve at all. The N particle state, obtained by applying N copies of the fieldoperators ψ†i , therefore evolves with the same permutator. The Jordan Wigner minussign, (3.9) gives the transformed state a minus sign if after t permutations the orderof the N particles has become an odd permutation of their original relative positions.Although we have to be aware of the existence of this minus sign, it plays no significantrole in most cases. Physically, this sign is not observable.

The conclusion of this Section is that, if the hamiltonian matrix hij describes a singleor composite cogwheel model, leading to classical permutations of the states |i〉 , i =1, · · · ,M at integer times, then the model with hamiltonian (3.11) is related to a system

42

where occupied states evolve according to the same permutations, the difference beingthat now the total number of states is 2M instead of M .

The importance of the procedure displayed here is that we can read off how anti-commuting fermionic field operators ψi , or ψi(x) , can emerge from deterministic systems.The minus signs in their (anti-)commutators is due to the Jordan-Wigner transformation(3.9), without which we would not have any commutator expressions at all, so that thederivation (3.15) would have failed.

One might object that in most physical systems the hamiltonian matrix hij wouldnot lead to classical permutations at integer time steps, but our model is just a first step.A next step could be that hij is made to depend on the values of some local operatorfields ϕ(x) . This is what we have in the physical world, and this may result if thepermutation rules for the evolution of these fermionic particles are assumed to depend onother variables in the system.

In fact, there does exist a simplified fermionic model where hij does appear to generatepure permutations. This will be exhibited in the next Section.

3.3. ‘Neutrinos’ in three space dimensions

In some cases, it is worth-while to start at the other end. Given a typical quantum system,can one devise a deterministic classical automaton that would generate all its quantumstates? We now show a new case of interest.

One way to determine whether a quantum system may be mathematically equivalentto a deterministic model is to search for a complete set of beables. As defined in Subsec-tion 2.1.1, beables are operators that may describe classical observables, and as such theymust commute with one another, always, at all times. Thus, for conventional quantumparticles such as the electron in Bohr’s hydrogen model, neither the operators x nor pare beables because [ x(t) , x(t′) ] 6= 0 and [ p(t), p(t′) ] 6= 0 as soon as t 6= t′ . Typicalmodels where we do have such beables are ones where the hamiltonian is linear in themomenta, such as in Section 2.5, Eq. (2.38), rather than quadratic in p . But are theythe only ones?

Maybe the beables only form a space-time grid, whereas the data on points in betweenthe points on the grid do not commute. This would actually serve our purpose well, sinceit could be that the physical data characterising our universe really do form such a grid,while we have not yet been able to observe that, just because the grid is too fine for today’stools, and interpolations to include points in between the grid points could merely havebeen consequences of our ignorance.

Beables form a complete set if, in the basis where they are all diagonal, the collectionof eigenvalues completely identify the elements of this basis.

No such systems of beables do occur in nature, as far as we know today; that is, ifwe take all known forces into account, all operators that we can construct today ceaseto commute at some point. We can, and should, try to search better, but, alternatively,we can produce simplified models describing only parts of what we see, which do allow

43

transformations to a basis of beables. In Section 2.6, we already discussed the harmonicoscillator as our first important example. Here, we discuss another such model: massless‘neutrinos’, in 3 space-like and one time-like dimension.

A single quantised, non interacting Dirac fermion obeys the hamiltonian

Hop = βm+ αipi , (3.16)

where β, αi are Dirac 4× 4 matrices obeying

αiαj + αjαi = 2δij ; β2 = 1 ; αi β + β αi = 0 . (3.17)

Only in the case m = 0 can we construct a complete set of beables, in a straightforwardmanner10. In that case, we can omit the matrix β , and replace αi by the three Paulimatrices, the 2 × 2 matrices σi . The particle can then be looked upon as a masslessMajorana or chiral “neutrino”, having only two components in its spinor wave function.The neutrino is entirely ‘sterile’, as we ignore any of its interactions. This is why we callthis the ‘neutrino’ model, with ‘neutrino’ between quotation marks.

There are actually two choices here: the relative signs of the Pauli matrices could bechosen such that the particles have positive (left handed) helicity and the antiparticles areright handed, or they could be the other way around. We take the choice that particleshave the right handed helicities, if our coordinate frame (x, y, z) is oriented as the fingers1,2,3 of the right hand. The Pauli matrices σi obey

σ1 σ2 = iσ3 , σ2 σ3 = iσ1 , σ3 σ1 = iσ2 ; σ21 = σ2

2 = σ23 = 1 . (3.18)

The beables are:

Oopi = q, s, r , where qi ≡ ±pi/|p| , s ≡ q · ~σ , r ≡ 1

2(q · ~x+ ~x · q) . (3.19)

To be precise, q is a unit vector defining the direction of the momentum, modulo its sign.What this means is that we write the momentum ~p as

~p = pr q , (3.20)

where pr can be a positive or negative real number. This is important, because we needits canonical commutation relation with the variable r , being [ r, pr ] = i , without furtherrestrictions on r or pr . If pr would be limited to the positive numbers |p| , this wouldimply analyticity constraints for wave functions ψ(r) .

The caret ˆ on the operator q is there to remind us that it is a vector with length one,|q| = 1 . To define its sign, one could use a condition such as qz > 0 . Alternatively, wemay decide to keep the symmetry Pint (for ‘internal parity’),

10Massive ‘neutrinos’ could be looked upon as massless ones in a space with one or more extra dimen-sions, and that does also have a beable basis. Projecting this set back to 4 space-time dimensions howeverleads to a rather contrived construction.

44

q ↔ −q , pr ↔ −pr , r ↔ −r , s↔ −s , (3.21)

after which we would keep only the wave functions that are even under this reflection.The variable s can only take the values s = ±1 , as one can check by taking the squareof q · ~σ . In the sequel, the symbol p will be reserved for p = +~p/|p| , so that q = ±p .

The last operator in Eq. (3.19), the operator r , was symmetrised so as to guaranteethat it is hermitean. It can be simplified by using the following observations. In the ~pbasis, we have

~x = i∂

∂~p;

∂

∂~ppr = q ; [xi, pr ] = iqi ; [xi, qj ] =

i

pr(δij − qiqj) ; (3.22)

xiqi − qixi =2i

pr→ 1

2(q · ~x+ ~x · q) = q · ~x+

i

pr. (3.23)

This can best be checked first by checking the case pr = |p| > 0, q = p , and noting thatall equations are preserved under the reflection symmetry (3.21).

It is easy to check that the operators (3.19) indeed form a completely commuting set.The only non-trivial commutator to be looked at carefully is [r, q] = [ q · ~x , q ] . Consideragain the ~p basis, where ~x = i∂/∂~p : the operator ~p · ∂/∂~p is the dilatation operator.But, since q is scale invariant, it commutes with the dilatation operator:

[ ~p · ∂∂~p

, q ] = 0 . (3.24)

Therefore,

[ q · ~x, q ] = i [ p−1r ~p · ∂

∂~p, q ] = 0 , (3.25)

since also [ pr, q ] = 0 , but of course we could also have used equation (3.22), # 4.

The unit vector q lives on a sphere, characterised by two angles θ and ϕ . If wedecide to define q such that qz > 0 then the domains in which these angles must lie are:

0 ≤ θ ≤ π/2 , 0 ≤ ϕ < 2π . (3.26)

The other variables take the values

s = ±1 , −∞ < r <∞ . (3.27)

An important question concerns the completeness of these beables and their relationto the more usual operates ~x, ~p and ~σ , which of course do not commute so that thesethemselves are no beables. This we discuss in the next subsection, which can be skipped atfirst reading. For now, we mention the more fundamental observation that these beablescan describe ontological observables at all times, since the hamiltonian (3.16), which herereduces to

H = ~σ · ~p , (3.28)

45

generates the equations of motion

d

dt~x = −i[~x, H] = ~σ ,

d

dt~p = 0 ,

d

dtσi = 2εijk pj σk ; (3.29)

d

dtp = 0 ;

d

dt(p · ~σ) = 2εijk (pi/|p|) pjσk = 0 ,

d

dt(p · ~x) = p · ~σ , (3.30)

where p = ~p/|p| = ±q , and thus we have:

d

dtθ = 0 ,

d

dtϕ = 0 ,

d

dts = 0 ,

d

dtr = s = ± 1 . (3.31)

The physical interpretation is simple: the variable r is the position of a ‘particle’projected along a predetermined direction q , given by the two angles θ and ϕ , and thesign of s determines whether it moves with the speed of light towards larger or towardssmaller r values.

Note, that a rotation over 180 along an axis orthogonal to q may turn s into −s ,which is characteristic for half-odd spin representations of the rotation group, so that wecan still consider the neutrino as a spin 1

2particle.11

What we have here is a representation of the wave function for a single ‘neutrino’ inan unusual basis. As will be clear from the calculations presented in the subsection below,in this basis the ‘neutrino’ is entirely non localised in the two transverse directions, butits direction of motion is entirely fixed by the unit vector q and the boolean variable s .In terms of this basis, the ‘neutrino’ is a deterministic object. Rather than saying thatwe have a particle here, we have a flat sheet, a plane. The unit vector q describes theorientation of the plane, and the variable s tells us in which of the two possible directionsthe plane moves, always with the speed of light. Neutrinos are deterministic planes, orflat sheets. The basis in which the operators q, r , and s are diagonal will be called theontological basis.

Finally, we could use the boolean variable s to define the sign of q , so that it becomesa more familiar unit vector, but this can better be done after we studied the operatorsthat flip the sign of the variable s , because of a slight complication, which is discussedwhen we work out the algebra, in Subsections 3.3.1 and 3.3.2.

Clearly, operators that flip the sign of s exist. For that, we take any vector q ′ that isorthogonal to q . Then, the operator q ′ ·~σ obeys (q ′ ·~σ) s = −s (q ′ ·~σ) , as one can easilycheck. So, this operator flips the sign. The problem is that, at each point on the sphereof q values, there are two such vectors q ′ orthogonal to q . Which one should we take?Whatever our choice, it depends on the angles θ and ϕ . This implies that we necessarilyintroduce some rather unpleasant angular dependence. This is inevitable; it is caused bythe fact that the original neutrino had spin 1

2, and we cannot mimic this behaviour in

terms of the q dependence because all wave functions have integral spin. One has to keepthis in mind whenever the Pauli matrices are processed in our descriptions.

11But rotations in the plane, or equivalently, around the axis q , give rise to complications, which canbe overcome, see later in this Section.

46

Thus, in order to complete our operator algebra in the basis determined by the eigen-values q, s, and r , we introduce two new operators whose squares ore one. Define twovectors orthogonal to q , one in the θ -direction and one in the ϕ -direction:

q =

q1

q2

q3

, θ =1√

q21 + q2

2

q3 q1

q3 q2

q23 − 1

, ϕ =1√

q21 + q2

2

−q2

q1

0

. (3.32)

All three are normalised to one, as indicated by the caret. Their components obey

qi = εijk θjϕk , θi = εijk ϕjqk , ϕi = εijk qjθk . (3.33)

Then we define two sign-flip operators: write s = s3 , then

s1 = θ · ~σ , s2 = ϕ · ~σ , s3 = s = q · ~σ . (3.34)

They obey:

s2i = I , s1 s2 = is3 , s2 s3 = is1 , s3 s1 = is2 . (3.35)

Considering now the beable operators q, r, and s3 , the translation operator pr for thevariable r , the spin flip operators (“changeables”) s1 and s2 , and the rotation operatorsfor the unit vector q , how do we transform back to the conventional neutrino operators~x , ~p and ~σ ?

Obtaining the momentum operators is straightforward:

pi = pr qi , (3.36)

and also the Pauli matrices σi can be expressed in terms of the si , simply by invertingEqs. (3.34). Using Eqs (3.32) and the fact that q2

1 + q22 + q2

3 = 1 , one easily verifies that

σi = θis1 + ϕis2 + qis3 . (3.37)

However, to obtain the operators xi is quite a bit more tricky; they must commutewith the σi . For this, we first need the rotation operators ~Lont . This is not the standardorbital or total angular momentum. Our transformation from standard variables to beablevariables will not be quite rotationally invariant, just because we will be using either theoperator s1 or the operator s2 to go from a left-moving neutrino to a right moving one.Note, that in the standard picture, chiral neutrinos have spin 1

2. So flipping from one

mode to the opposite one involves one unit ~ of angular momentum in the plane. Theontological basis does not refer to neutrino spin, and this is why our algebra gives somespurious angular momentum violation. As long as neutrinos do not interact, this effectstays practically unnoticeable, but care is needed when either interactions or mass areintroduced.

The only rotation operators we can use in the beable frame, are the operators thatrotate the planes with respect to the origin of our coordinates. These we call ~Lont :

Lonti = −iεijkqj

∂

∂qk. (3.38)

47

By definition, they commute with the si , but care must be taken at the equator, wherewe have a boundary condition, which can be best understood by imposing the symmetrycondition (3.21).

Note that they do not coincide with any of the conventional angular momentum op-erators because those do not commute with the si , as the latter depend on θ and ϕ .One finds the following relation between the angular momentum ~L of the neutrinos and~L ont :

Lonti ≡ Li + 1

2

(θis1 + ϕis2 −

q3 θi√1− q2

3

s3

); (3.39)

the derivation of this equation is postponed to Subsection 3.3.1.

Since ~J = ~L+ 12~σ , one can also write, using Eqs. (3.37) and (3.32),

Lonti = Ji −

1

1− q23

q1

q2

0

s3 . (3.40)

We then derive, in Subsection 3.3.1, Eq. (3.63), the following expression for the op-erators xi in the neutrino wave function, in terms of the beables q, r and s3 , and thechangeables12 Lont

k , pr , s1 and s2 :

xi = qi (r −i

pr) + εijk qj L

ontk /pr +

1

2pr

(−ϕi s1 + θi s2 +

q3√1− q2

3

ϕi s3

)(3.41)

(note that θi and ϕi are beables since they are functions of q ).

The complete transformation from the beable basis to one of the conventional basesfor the neutrino can be derived from

〈~p, α | q, pr, s〉 = pr δ3(~p− q pr)χsα(q) , (3.42)

where α is the spin index of the wave functions in the basis where σ3 is diagonal, andχsα is a standard spinor solution for the equation (q · ~σαβ)χsβ(q) = s χsα(q) .

In Subsection 3.3.2, we show how this equation can be used to derive the elements ofthe unitary transformation matrix mapping the beable basis to the standard coordinateframe of the neutrino wave function basis13 (See Eq. 3.84):

〈 ~x, α | q, r, s 〉 =i

2πδ ′(r − q · ~x )χsα(q) , (3.43)

where δ ′(z) ≡ ddzδ(z) . This derivative originates from the factor pr in Eq. (3.42), which

is necessary for a proper normalisation of the states.

12See Eq. (3.50) and the remarks made there concerning the definition of the operator 1/pr in theworld of the beables, as well as in the end of Section 3.3.2.

13In this expression, there is no need to symmetrise q ·~x , because both q and ~x consist of C-numbersonly.

48

3.3.1. Algebra of the beable ‘neutrino’ operators

This subsection is fairly technical and can be skipped at first reading. It handles thealgebra needed for the transformations from the (q, s, r) basis to the (~x, σ3) or (~p, σ3)basis and back. This algebra is fairly complex, again, because, in the beable representa-tion, no direct reference is made to neutrino spin. Chiral neutrinos are normally equippedwith spin +1

2or −1

2with spin axis in the direction of motion. The flat planes that

are moving along here, are invariant under rotations about an orthogonal axis, and theassociated spin-angular momentum does not leave a trace in the non-interacting, beablepicture.

This forced us to introduce some axis inside each plane that defines the phases of thequantum states, and these coordinates explicitly break rotation invariance.

We consider the states specified by the variables s and r , and the polar coordinatesθ and ϕ of the beable q , in the domains given by Eqs. (3.26), (3.27). Thus, we havethe states |θ, ϕ, s, r〉 . How can these be expressed in the more familiar states |~x, σz〉and/or |~p, σz〉 , where σz = ±1 describes the neutrino spin in the z -direction, and viceversa?

Our ontological states are specified in the ontological basis spanned by the operatorsq, s (= s3), and r . We add the operators (changeables) s1 and s2 by specifying theiralgebra (3.35), and the operator

pr = −i∂/∂r ; [r, pr] = i . (3.44)

The original momentum operators are then easily retrieved. As in Eq, (3.20), define

~p = pr q . (3.45)

The next operators that we can reproduce from the beable operators q, r , and s1,2,3

are the Pauli operators σ1,2,3 :

σi = θis1 + ϕis2 + qis3 . (3.46)

Note, that these now depend non-trivially on the angular parameters θ and ϕ , since thevectors θ and ϕ , defined in Eq. (3.32), depend non-trivially on q , which is the radialvector specified by the angles θ and ϕ . One easily checks that the simple multiplicationrules from Eqs. (3.35) and the right-handed orthonormailty (3.33) assure that these Paulimatrices obey the correct multiplication rules also. Given the trivial commutation rulesfor the beables, [qi, θj] = [qi, ϕj] = 0 , and [pr, qi] = 0 , one finds that [pi, σj] = 0 , sohere, we have no new complications.

Things are far more complicated and delicate for the ~x operators. To reconstructan operator ~x = i∂/∂~p , obeying [xi, pj] = iδij and [xi, σj] = 0 , we first introduce theorbital angular momentum operator

Li = εijkxipk = −iεijkqj∂

∂qk(3.47)

49

(where σi are kept fixed), obeying the usual commutation rules

[Li, Lj] = iεijkLk , [Li, qj] = iεijkqk] , [Li, pj] = iεijkpk , etc., (3.48)

while [Li, σj] = 0 . In terms of these operators, we can now recover the original spaceoperators xi, i = 1, 2, 3 , of the neutrinos:

xi = qi (r −i

pr) + εijk qj Lk/pr . (3.49)

The operator 1/pr , the inverse of the operator pr = −i∂/∂r , should be −i times theintegration operator. This leaves the question of the integration constant. It is straight-forward to define that in momentum space, but eventually, r is our beable operator.For wave functions in r space, ψ(r, · · ·) = 〈r, · · · |ψ〉 , where the ellipses stand for otherbeables (of course commuting with r ), the most careful definition is:

1

prψ(r) ≡

∫ ∞−∞

12i sgn(r − r′)ψ(r′)dr′ , (3.50)

which can easily be seen to return ψ(r) when pr acts on it. “sgn(x )” stands for the signof x . We do note that the integral must converge at r → ±∞ . This is a restrictionon the class of allowed wave functions: in momentum space, ψ must vanish at pr → 0 .Restrictions of this sort will be encountered more frequently in this report.

The anti-hermitean term −i/pr in Eq. (3.49) arises automatically in a careful calcu-lation, and it just compensates the non hermiticity of the last term, where qj and Lkshould be symmetrised to get a hermitean expression. Lk commutes with pr . The xidefined here ends up being hermitean.

This perhaps did not look too hard, but we are not ready yet. The operators Licommute with σj , but not with the beable variables si . Therefore, an observer of thebeable states, in the beable basis, will find it difficult to identify our operators Li . It willbe easy for such an observer to identify operators Lont

i , which generate rotations of the qivariables while commuting with si . He might also want to rotate the Pauli-like variablessi , employing a rotation operator such as 1

2si , but that will not do, first, because they

no longer obviously relate to spin, but foremost, because the si in the conventional basishave a much less trivial dependence on the angles θ and ϕ , see Eqs. (3.32) and (3.34).

Actually, the reconstruction of the ~x operators from the beables will show a non-trivial dependence on the variables si and the angles θ and ϕ . This is because ~x andthe si do not commute. From the definitions (3.32) and the expressions (3.32) for thethe vectors θ and ϕ , one derives, from judicious calculations:

[xi, θj] =i

pr

( q3√1− q2

3

ϕi ϕj − θi qj), (3.51)

[xi, ϕj] =−iϕipr

(qj +

θj q3√1− q2

3

), (3.52)

[xi, qj] =i

pr(δij − qi qj) . (3.53)

50

The expression q3/√

1− q23 = cot(θ) emerging here is singular at the poles, clearly due

to the vortices there in the definitions of the angular directions θ and ϕ .

From these expressions, we now deduce the commutators of xi and s1,2,3 :

[xi, s1] =i

pr

( ϕi q3√1− q2

3

s2 − θi s3

), (3.54)

[xi, s2] =−iprϕi

(s3 +

q3√1− q2

3

s1

), (3.55)

[xi, s3] =i

pr(σi − qi s3) =

i

pr(θi s1 + ϕi s2) . (3.56)

In the last expression Eq. (3.46) for ~σ was used. Now, observe that these equations canbe written more compactly:

[xi, sj] = 12

[ 1

pr

(− ϕis1 + θis2 +

q3 ϕi√1− q2

3

s3

), sj

]. (3.57)

To proceed correctly, we now need also to know how the angular momentum operatorsLi commute with s1,2,3 . Write Li = εijkxj qk pr , where only the functions xi do notcommute with the sj . It is then easy to use Eqs. (3.54)—(3.57) to find the desiredcommutators:

[Li, sj] = 12

[− θis1 − ϕis2 +

q3 θi√1− q2

3

s3, sj

], (3.58)

where we used the simple orthonormality relations (3.33) for the unit vectors θ, ϕ , andq . Now, this means that we can find new operators Lont

i that commute with all the sj :

Lonti ≡ Li + 1

2

(θis1 + ϕis2 −

q3 θi√1− q2

3

s3

), [Lont

i , sj] = 0 (3.59)

(Eq. 3.39). It is then of interest to check the commutator of two of the new “angularmomentum” operators. One thing we know: according to the Jacobi identity, the com-mutator of two operators Lont

i must also commute with all sj . Now, expression (3.59)seems to be the only one that is of the form expected, and commutes with all s operators.It can therefore be anticipated that the commutator of two Lont operators should againyield an Lont operator, because other expressions could not possibly commute with alls . The explicit calculation of the commutator is a bit awkward. For instance, one mustnot forget that Li also commutes non-trivially with cot(θ) :

[Li,q3√

1− q23

] =iϕi

1− q23

. (3.60)

But then, indeed, one finds [Lonti , Lont

j

]= iεijkL

ontk . (3.61)

51

The commutation rules with qi and with r and pr were not affected by the additionalterms:

[Lonti , qj] = iεijk qk , [Lont

i , r] = [Lonti , pr] = 0 . (3.62)

This confirms that we now indeed have the generator for rotations of the beables qi , whileit does not affect the other beables si, r and pr .

Thus, to find the correct expression for the operators ~x in terms of the beable variables,we replace Li in Eq. (3.49) by Lont

i , leading to

xi = qi (r −i

pr) + εijk qj L

ontk /pr +

1

2pr

(−ϕi s1 + θi s2 +

q3√1− q2

3

ϕi s3

). (3.63)

This remarkable expression shows that, in terms of the beable variables, the ~x coordi-nates undergo finite, angle-dependent displacements proportional to our sign flip operatorss1, s2 , and s3 . These displacements are in the plane. However, the operator 1/pr doessomething else. From Eq. (3.50) we infer that, in the r variable,

〈r1|1

pr|r2〉 = 1

2i sgn(r1 − r2) . (3.64)

Returning now to a remark made earlier in this chapter, one might decide to use thesign operator s3 (or some combination of the three s variables) to distinguish oppositesigns of the q operators. The angles θ and ϕ then occupy the domains that are moreusual for an S2 sphere: 0 < θ < π , and 0 < ϕ ≤ 2π . In that case, the operators s1,2,3

refer to the signs of q3, r and pr . Not much would be gained by such a notation.

The hamiltonian in the conventional basis is

H = ~σ · ~p . (3.65)

It is linear in the momenta pi , but it also depends on the non commuting Pauli matricesσi . This is why the conventional basis cannot be used directly to see that this is adeterministic model. Now, in our ontological basis, this becomes

H = s pr . (3.66)

Thus, it multiplies one momentum variable with the commuting operator s . The hamiltonequation reads

dr

dt= s , (3.67)

while all other beables stay constant. This is how our ‘neutrino’ model became determin-istic. In the basis of states | q, r, s 〉 our model clearly describes planar sheets at distancer from the origin, oriented in the direction of the unit vector q , moving with the velocityof light in a transverse direction, given by the sign of s .

52

Once we defined, in the basis of the two eigenvalues of s , the two other operators s1

and s2 , with (see Eqs. (3.35))

s1 =

(0 11 0

), s2 =

(0 −ii 0

), s3 = s =

(1 00 −1

), (3.68)

in the basis of states | r 〉 the operator pr = −i∂/∂r , and, in the basis | q 〉 the operatorsLonti by

[Lonti , Lont

j ] = iεijkLontk , [Lont

i , qj] = iεijk qk , [Lonti , r] = 0 , [Lont

i , sj] = 0 , (3.69)

we can write, in the ‘ontological’ basis, the conventional ‘neutrino’ operators ~σ (Eq. (3.46)),~x (Eq. (3.63)), and ~p (Eq. (3.45)). By construction, these will obey the correct commu-tation relations.

3.3.2. Orthonormality and transformations of the ‘neutrino’ beable states

The quantities that we now wish to determine are the inner products

〈~x, σz|θ, ϕ, s, r〉 , 〈~p, σz|θ, ϕ, s, r〉 . (3.70)

The states | θ, ϕ, s, r〉 will henceforth be written as | q, s, r〉 . The use of momentumvariables q ≡ ±~p/|p| , qz > 0, together with a real parameter r inside a Dirac bracketwill always denote a beable state in this subsection.

Special attention is required for the proper normalisation of the various sets of eigen-states. We assume the following normalisations:

〈~x, α | ~x ′, β〉 = δ3(~x− ~x ′) δαβ , 〈~p, α | ~p ′, β〉 = δ3(~p− ~p ′) δαβ , (3.71)

〈~x, α | ~p, β〉 = (2π)−3/2 ei ~p·~x δαβ ; (3.72)

〈q, r, s | q ′, r′, s′ 〉 = δ2(q, q ′) δ(r − r′) δs s′ , δ2(q, q′) ≡ δ(θ − θ′) δ(ϕ− ϕ′)sin θ

, (3.73)

and α and β are eigenvalues of the Pauli matrix σ3 ; furthermore∫δ3~x

∑α

| ~x, α〉〈~x, α | = I =

∫d2q

∫ ∞−∞

dr∑s=±

| q, r, s〉〈q, r, s | ;

∫d2q ≡

∫ π/2

0

sin θ dθ

∫ 2π

0

dϕ . (3.74)

The various matrix elements are now straightforward to compute. First we define thespinors χ±α (q) by solving

(q · ~σαβ)χsβ = s χsα ;

(q3 − s q1 − iq2

q1 + iq2 −q3 − s

)(χs1χs2

)= 0 , (3.75)

53

which gives, after normalising the spinors,

χ+1 (q) =

√12(1 + q3)

χ+2 (q) =

q1 + i q2√2(1 + q3)

;

χ−1 (q) = −√

12(1− q3)

χ−2 (q) =q1 + i q2√2(1− q3)

, (3.76)

where not only the equation s3 χ±α = ±χ±α was imposed, but also

s α1β χ

±α = χ∓β , s α

2β χ±α = ± i χ∓β , (3.77)

which implies a constraint on the relative phases of χ+α and χ−α . The sign in the second of

these equations is understood if we realise that the index s here, and later in Eq. (3.81),is an upper index.

Next, we need to know how the various Dirac deltas are normalised:

d3~p = p2r d2q dpr ; δ3(q pr − q ′ pr′) =

1

p2r

δ2(q, q ′) δ(pr − pr′) , (3.78)

We demand completeness to mean∫d2q

∫ ∞−∞

dpr∑s=±

〈 ~p, α | q, pr, s 〉〈 q, pr, s | ~p ′, α′〉 = δαα′δ3(~p− ~p ′) ; (3.79)

∫d3~p

2∑α=1

〈 q, pr, s | ~p, α 〉〈 ~p, α | q ′, pr′, s′ 〉 = δ2(q, q ′)δ(pr − pr′) δss′ , (3.80)

which can easily be seen to imply14

〈~p, α | q, pr, s〉 = pr δ3(~p− q pr)χsα(q) , (3.81)

since the norm p2r has to be divided over the two matrix terms in Eqs (3.79) and (3.80).

This brings us to derive, using 〈 r | pr 〉 = (2π)−1/2 e i pr r ,

〈 ~p, α | q, r, s 〉 =1√2π

1

prδ2(±~p|p|

, q)e−i(q · ~p ) r χsα(q) , (3.82)

where the sign is the sign of p3 .

The Dirac delta in here can also be denoted as

δ2(±~p|p|

, q)

= (q · ~p )2 δ2(~p ∧ q) , (3.83)

where the first term is a normalisation to ensure the expression to become scale invariant,and the second just forces ~p and q to be parallel or antiparallel. In the case q = (0, 0, 1) ,this simply describes p2

3 δ(p1) δ(p2) .

14Note that the phases in these matrix elements could be defined at will, so we could have chosen |p|in stead of pr . Our present choice is for future convenience.

54

Finally then, we can derive the matrix elements 〈 ~x, α | q, r, s 〉 . Just temporarily, weput q in the 3-direction: q = (0, 0, 1) ,

〈 ~x, α | q, r, s 〉 =1√2π

(2π)−3/2

∫d3~p

(q · ~p )2

prδ2(~p ∧ q)e−i(q · ~p ) r + i ~p · ~x χsα(q) =

=1

(2π)2

∫d3~p p3 δ(p1) δ(p2) ei p3(x3 − r) χsα(q) =

=1

2π

id

drδ(r − q · ~x )χsα(q) =

i

2πδ ′(r − q · ~x )χsα(q) . (3.84)

With these equations, our transformation laws are now complete. We have all matrixelements to show how to go from one basis to another. Note, that the states with vanishingpr , the momentum of the sheets, generate singularities. Thus, we see that the states |ψ 〉with 〈pr = 0 |ψ 〉 6= 0 , or equivalently, 〈 ~p = 0 |ψ 〉 6= 0 , must be excluded. We call suchstates ‘edge states’, since they have wave functions that are constant in space (in r andalso in ~x ), which means that they stretch to the ‘edge’ of the universe. There is an issuehere concerning the boundary conditions at infinity, which we will need to avoid. We seethat the operator 1/pr , Eq. (3.50), is ill defined for these states.

3.3.3. Second quantisation of the ‘neutrinos’

Being a relativistic Dirac fermion, the object described in this chapter so-far suffers fromthe problem that its hamiltonian, (3.28) and (3.66), is not bounded from below. Thereare positive and negative energy states. The cure will be the same as the one used byDirac: second quantisation. For every given value of the unit vector q , we consider anunlimited number of ‘neutrinos’, which can be in positive or negative energy states. Tobe more specific, one might, temporarily, put the variables r on a discrete lattice:

r = rn = n δr , (3.85)

but often we ignore this, or in other words, we let δr tend to zero.

We now assume that these particles, having spin 12

, have to be described by anti-commuting fermionic operators. We have operator fields ψα(~x ) and ψ†α(~x ) obeyinganticommutation rules,

ψα(~x ), ψ†β(~x ′ ) = δ3(~x− ~x ′)δαβ . (3.86)

Using the transformation rules of Subsection 3.3.2, we can transform these fields intofields ψ(q, r, s) and ψ†(q, r, s) obeying

ψ(q, r, s), ψ†(q ′, r′, s′ ) = δ2(q, q ′ ) δ(r − r′)δs s′ → δ2(q, q ′) δnn′ δs s′ . (3.87)

At any given value of q (which could also be chosen discrete if so desired), we havea straight line of r values, limited to the lattice points (3.85). On a stretch of N sitesof this lattice, we can imagine any number of fermions, ranging from 0 to N . Each of

55

these fermions obeys the same evolution law (3.67), and therefore also the entire systemis deterministic.

There is no need to worry about the introduction of anti-commuting fermionic oper-ators (3.86), (3.87). The minus signs can be handled through a Jordan-Wigner transfor-mation, implying that the creation or annihilation of a fermion that has an odd number offermions at one side of it, will be accompanied by an artificial minus sign. This minus signhas no physical origin but is exclusively introduced in order to facilitate the mathematicswith anti-commuting fields. Because, at any given value of q , the fermions propagateon a single line, and they all move with the same speed in one direction, the Jordan-Wigner transformation is without complications. Of course, we still have not introducedinteractions among the fermions, which indeed would not be easy as yet.

This ‘second quantised’ version of the neutrino model has one big advantage: we candescribe it using a hamiltonian that is bounded from below. The argument is identicalto Dirac’s own ingenious procedure. The hamiltonian of the second quantised system is(compare the first quantised hamiltonian (3.28)):

H =

∫d3~x

∑α

ψ∗α(~x)h βα ψβ(~x) , h β

α = −i~σ βα ·

∂

∂~x. (3.88)

Performing the transformation to the beable basis described in Subsection 3.3.2, we find

H =

∫d2q

∫dr∑s

ψ∗(q, r, s) (−i s) ∂

∂rψ(q, r, s) . (3.89)

Let us denote the field in the standard notation as ψ standα (~x) or ψ stand

α (~p) , and thefield in the ‘beable’ basis as ψont

s (q, r) . Its Fourier transform is not a beable field, butto distinguish it from the standard notation we will sometimes indicate it nevertheless asψonts (q, pr) .

In momentum space, we have (see Eq. 3.42):

ψ standα (~p ) =

1

pr

∑s

χsα(q)ψonts (q, pr) ; (3.90)

ψonts (q, pr) = pr

∑α

χsα(q)∗ ψ standα (~p) , ~p ≡ q pr , (3.91)

where ‘stand’ stands for the standard representation, and ‘ont’ for the ontological one,although we did the Fourier transform replacing the variable r by its momentum variablepr . The normalisation is such that∑

α

∫d3~p |ψ stand

α (~p )|2 =∑s

∫q3>0

d2q

∫ π/δr

−π/δrdpr|ψont

s (q, pr)|2 , (3.92)

see Eqs. (3.78)—(3.81).

In our case, ψ has only two spin modes, it is a Weyl field, but in all other respectsit can be handled just as the Dirac field. Following Dirac, in momentum space, each

56

momentum ~p has two energy eigen modes (eigen vectors of the operator h βα in the

hamiltonian (3.88)), which we write, properly normalised, as

u stand±α (~p ) =

1√2|p|(|p| ± p3)

(±|p|+ p3

p1 + ip2

); E = ±|p| . (3.93)

Here, the spinor lists the values for the index α = 1, 2 . In the basis of the beables:

uont±s (q, pr) =

(10

)if ± pr > 0 ,

(01

)if ± pr < 0 ; E = ±|pr| . (3.94)

Here, the spinor lists the values for the index s = + and − .

In both cases, we write

ψ(~p) = u+ a1(~p ) + u− a†2(−~p ) ; a1, a2 = a1, a†2 = 0 , (3.95)

a1(~p ), a†1(~p ′) = a2(~p ), a†2(~p ′) = δ3(~p− ~p ′) or δ(pr − pr′) δ2(q, q ′) ; (3.96)

Hop = |p| (a†1a1 + a†2a2 − 1) , (3.97)

where a1 is the annihilation operator for a particle with momentum ~p , and a†2 is thecreation operator for an antiparticle with momentum −~p . We drop the vacuum energy−1 . In case we have a lattice in r space, the momentum is limited to the values |~p | =|pr| < π/δr .

3.4. The ‘neutrino’ vacuum correlations

The vacuum state |∅〉 is now the state of lowest energy. This means that, at eachmomentum value ~p or equivalently, at each (q, pr) , we have

ai|∅〉 = 0 . (3.98)

The beable states are the states where, at each value of the set (q, r, s) the number of‘particles’ is specified to be either 1 or 0. This means, of course, that the vacuum state(3.98) is not a beable state; it is a superposition of all beable states.

One may for instance compute the correlation functions of right- and left moving‘particles’ (sheets, actually) in a given direction. In the beable (ontological) basis, onefinds that left-movers are not correlated to right movers, but two left-movers are correlatedas follows:

P (r1, r2)− P (r1)P (r2) = 〈∅|ψ∗s(r1)ψs(r1) ψ∗s(r2)ψs(r2)|∅〉conn =∣∣∣ 1

2π

∫ π/δr

0

dp eip(r2−r1)∣∣∣2 =

1

4π2 |r1 − r2|2, (3.99)

where ‘conn’ stands for the connected diagram contribution only, that is, the particle andantiparticle created at r2 are both annihilated at r1 . The same applies to two rightmovers. In the case of a lattice, where δr is not yet tuned to zero, this calculation is stillexact if r1 − r2 is an integer multiple of δr .

57

An important point about the second quantised hamiltonian (3.88), (3.89): on thelattice, we wish to keep the hamiltonian (3.97) in momentum space. In position space,Eqs. (3.88) or (3.89) cannot be valid since one cannot differentiate in the space variabler . But we can have the induced evolution operator over finite integer time intervalsT = nt δr . This evolution operator then displaces left movers one step to the left andright movers one step to the right. The hamiltonian (3.97) does exactly that, while itcan be used also for infinitesimal times; it is however not quite local when re-expressedin terms of the fields on the lattice coordinates, since now momentum is limited to staywithin the Brillouin zone |pr| < π/δr . We return to this subject in Section 5.

Correlations of data at two points that are separated in space but not in time, ornot sufficiently far in the time-like direction to allow light signals to connect these twopoints, are called space-like correlations. The space-like correlations found in Eq. (3.99)are important. They probably play an important role in the mysterious behaviour of thesebeable models when Bell’s inequalities are considered, in Chapter 8.7 and beyond.

Note that we are dealing here with spacelike correlations of the ontological degrees offreedom. The correlations are a consequence of the fact that we are looking at a specialstate that is preserved in time, a state we call the vacuum. All physical states normallyencountered deviate only very slightly from this vacuum state, so we will always havesuch correlations.

The mapping we found between massless, non interacting neutrinos and an ontological,deterministic model cotains an important lesson about correlations. In the chapters aboutBell inequalities and the Cellular Automaton Interpretation (Chapters 8 and 12), it isargued that the ontological theories proposed in this work must feature strong, spacelikecorrelations throughout the universe. This would be the only way to explain how theBell, or CHSH inequalities can be so strongly violated in these models. Now since our‘neutrinos’ are non interacting, one cannot really do EPR-like experiments with them, soalready for that reason, there is no direct contradiction. However, we also see that strongspace-like correlations are present in this model.

Indeed, one’s first impression might be that the ontological ‘neutrino sheet’ model ofthe previous section is entirely non local. The sheets stretch out infinitely far in twodirections, and if a sheet moves here, we immediately have some information about whatit does elsewhere. But on closer inspection one should concede that the equations ofmotion are entirely local. These equations tell us that if we have a sheet going througha space-time point x , carrying a sign function σ , and oriented in the direction q , then,at the point x , the sheet will move with the speed of light in the direction dictated by qand σ . No information is needed from points elsewhere in the universe. This is locality.

The thing that is non local is the ubiquitous correlations in this model. If we havea sheet at ~x, t , oriented in a direction q , we instantly know that the same sheet willoccur at other points ~y, t , if q · (~y− ~x ) = 0 , and it has the same values for q and σ . Itwill be explained in Chapter 10, see Section 10.5 that space-like correlations are normalin physics, both in classical systems such as crystals or star clusters and in quantummechanical ones such as quantised fields. In the neutrino sheets, the correlations areeven stronger, so that assuming their absence is a big mistake when one tries to draw

58

conclusions from Bell’s theorem.

4. PQ theory

Most quantum theories describe operators whose eigenvalues form continua of real num-bers. Examples are one or more particles in some potential well, harmonic oscillators,but also bosonic quantum fields and strings. If we want to relate these to deterministicsystems we could consider ontological observables that take values in the real numbers aswell, such as what we did in Section 2.5 and the harmonic oscillator in Section 2.6. Thereis however an other option.

One important application of the transformations described in this report could beour attempts to produce fundamental theories about nature at the Planck scale. Here,we have the holographic principle at work, and the Bekenstein bound. What these tell usis that Hilbert space assigned to any small domain of space should be finite-dimensional.Real numbers are described by an unlimited sequence of digits and therefore requireinfinite dimensional Hilbert spaces. That’s too large. One may suspect that uncertaintyrelations, or non-commutativity, add some blur to these numbers. In this chapter, weoutline a mathematical procedure for a systematic approach.

For PQ theory, as our approach will be called, we employed a new notation in earlierwork [73][74], where not ~ but h is normalised to one. Wave functions then take theform e 2πipx = ε ipx , where ε = e2π = 535.5 . This notation was very useful to avoid factors1/√

2π for the normalisation of wave functions on a circle. Yet we decided not to usesuch notation in this report, so as to avoid clashes with other discussions in the standardnotation in various other chapters. Therefore, we return to the normalisation ~ = 1 .Factors 2π will now occur more frequently, and hopefully they won’t deter the reader.

In this chapter, and part of the following ones, dynamical variables can be real num-bers, indicated with lower case letters: p, q, r, x, · · · , they can be integers indicated bycapitals: N, P, Q, X, · · · , or they are angles (numbers defined on a circle), indicatedby Greek letters α, η, κ, %, · · · , usually obeying −π < α ≤ π , or sometimes definedmerely modulo 2π .

A real number r , for example the number r = 137.035999074 · · · , is composed ofan integer, here R = 137 , and an angle, %/2π = 0.035999074 · · · . In examples such asa quantum particle on a line, Hilbert space is spanned by a basis defined on the line: | r 〉 . In PQ theory, we regard such a Hilbert space as the product of Hilbert spacespanned by the integers |R 〉 and Hilbert space spanned by the angles, | % 〉 . So, we have

1√2π| r 〉 = |R, % 〉 = |R 〉 | % 〉 . (4.1)

Note that continuity of a wave function |ψ〉 implies twisted boundary conditions:

〈R + 1, % |ψ〉 = 〈R, %+ 2π |ψ〉 . (4.2)

The fractional part, or angle, is defined unambiguously, but the definition of the integralpart depends on how the angle is projected on the segment [0, 2π] , or: how exactly dowe round a real number to its integer value?

59

So far our introduction; it suggests that we can split up a coordinate q into an integerpart Q and a fractional part ξ/2π and its momentum into an integer part 2πP and afractional part κ . Now we claim that this can be done in such a way that both [P,Q] = 0and [ξ, κ] = 0 .

Now, let us set up our algebra as systematically as possible.

4.1. The algebra of finite displacements

Let there be given an operator qop with non-degenerate eigenstates |q〉 having eigenvaluesq spanning the entire real line. The associated momentum operator is pop with eigenstates|p〉 having eigenvalues p , also spanning the real line. The usual quantum mechanicalnotation, now with ~ = 1 , is

qop|q〉 = q|q〉 ; pop|p〉 = p|p〉 ; [qop, pop] = i ;

〈q|q ′〉 = δ(q − q ′) ; 〈p|p ′〉 = δ(p− p ′) ; 〈q|p〉 = 1√2πeipq (4.3)

(often, we will omit the subscript “op” denoting that we refer to an operator, if this shouldbe clear from the context)

Consider now the displacement operators e−ipop a in position space, and eiqop b inmomentum space, where a and b are real numbers:

e−ipop a|q〉 = |q + a〉 ; eiqop b|p〉 = |p+ b〉 . (4.4)

We have

[qop , pop ] = i , eiqop b pop = (pop − b) eiqop b ;

eiqop b e−ipop a = e−ipop a eiab eiqop b

= e−ipop a eiqop b , if ab = 2π × integer . (4.5)

Let us consider the displacement operator in position space for a = 1 . It is unitary,and therefore can be written uniquely as e−iκ , where κ is a hermitean operator witheigenvalues κ obeying −π < κ ≤ π . As we see in Eq. (4.4), κ also represents themomentum modulo 2π . Similarly, eiξ , with −π < ξ ≤ π , is defined to be an operatorthat causes a shift in the momentum by a step b = 2π . This means that ξ/2π is theposition operator q modulo one. We write

p = 2πK + κ , q = X + ξ/2π , (4.6)

where both K and X are integers and κ and ξ are angles. We suggest that the readerignore the factors 2π at first reading; these will only be needed when doing precisecalculations.

4.1.1. From the one-dimensional infinite line to the two-dimensional torus

As should be clear from Eqs. (4.5), we can regard the angle κ as the generator of a shiftin the integer X , and the angle ξ as generating a shift in K :

e−iκ |X 〉 = |X + 1 〉 , eiξ |K 〉 = |K + 1 〉 . (4.7)

60

Since κ and ξ are uniquely defined as generating these elementary shifts, we deducefrom Eqs. (4.5) and (4.7) that

[ξ, κ] = 0 . (4.8)

Thus, consider the torus spanned by the eigenvalues of the operators κ and ξ . Wenow claim that the Hilbert space generated by the eigenstates |κ, ξ〉 is equivalent to theHilbert space spanned by the eigenstates |q〉 of the operator qop , or equivalently, theeigenstates |p〉 of the operator pop (with the exception of exactly one state, see later).

It is easiest now to consider the states defined on this torus, but we must proceedwith care. If we start with a wave function |ψ〉 that is continuous in q , we have twistedperiodic boundary conditions, as indicated in Eq. (4.2). Here, in the ξ coordinate,

〈X + 1, ξ |ψ〉 = 〈X, ξ + 2π |ψ〉 , or

〈κ, ξ + 2π |ψ〉 = 〈κ, ξ |eiκ|ψ〉 , (4.9)

whereas, since this wave function assumes X to be integer, we have strict periodicity inκ :

〈κ+ 2π, ξ |ψ〉 = 〈κ, ξ |ψ〉 . (4.10)

If we would consider the same state in momentum space, the periodic boundary con-ditions would be the other way around, and this is why, in the expression used here, thetransformations from position space to momentum space and back are non-trivial. For oursubsequent calculations, it is much better to transform first to a strictly periodic torus.To this end, we introduce a phase function φ(κ, ξ) with the following properties:

φ(κ, ξ + 2π) = φ(κ, ξ) + κ; φ(κ+ 2π, ξ) = φ(κ, ξ) ; (4.11)

φ(κ, ξ) = −φ(−κ, ξ) = −φ(κ,−ξ) ; φ(κ, ξ) + φ(ξ, κ) = κξ/2π . (4.12)

An explicit expression for such a function is derived in Appendix 4A (Section 4.2) andsummarised in Appendix 4B (Section 4.3). Here, we just note that this function suffersfrom a singularity at the point (ξ = ±π, κ = ±π) . This singularity is an inevitableconsequence of the demands (4.11) and (4.12). It is a topological defect that can bemoved around on the torus, but not disposed of.

Transforming |ψ〉 now with the unitary transformation

U(κ, ξ) = e−iφ(κ, ξ) = eiφ(ξ, κ)−ξκ/2π , (4.13)

turns the boundary conditions (4.9) and (4.10) both into strictly periodic boundaries.

For the old wave function, we had X = −i∂/∂κ , so, qop = −i∂/∂κ + ξ/2π . Theoperator pop would simply be −2πi∂/∂ξ , assuming that the boundary condition (4.9)ensures that this reduces to the usual differential operator. Our new, transformed wavefunction now requires a modification of these operators to accommodate for the unusual

61

phase factor φ(κ, ξ) . Now our two operators become

qop = −i ∂∂κ

+( ∂∂κφ(ξ, κ)

)= −i ∂

∂κ+

ξ

2π−( ∂∂κφ(κ, ξ)

); (4.14)

pop = −2πi∂

∂ξ− 2π

( ∂∂ξφ(κ, ξ)

)= −2πi

∂

∂ξ− κ+ 2π

( ∂∂ξφ(ξ, κ)

). (4.15)

This is how the introduction of a phase factor φ(κ, ξ) can restore the symmetry be-tween the operators qop and pop . Note that, although φ is not periodic, the derivative∂φ(κ, ξ)/∂ξ is periodic, and therefore, both qop and pop are strictly periodic in κ andξ (beware the reflections ξ ↔ κ in Eqs. (4.14) and (4.15)).

We check that they obey the correct commutation rule:

[qop, pop] = i . (4.16)

It is very important that these operators are periodic. It implies that we have no thetajumps in their definitions. If we had not introduced the phase function φ(κ, ξ) , we wouldhave such theta jumps and in such descriptions the matrix elements in Q, P space wouldbe much less convergent.

The operators −i∂/∂κ and −i∂/∂ξ now do not exactly correspond to the operatorsK and X anymore, because of the last terms in Eqs. (4.14) and (4.15). They are integershowever, which obviously commute, and these we shall call Q and P . To obtain theoperators qop and pop in the basis of the states |Q, P 〉 , we simply expand the wavefunctions in κ, ξ space in terms of the Fourier modes,

〈κ, ξ |Q, P 〉 =1

2πeiP ξ+iQκ . (4.17)

We now need the Fourier coefficients of the phase function φ(κ, ξ) . They are given inAppendix 4A, Section 4.2, where we also derive the explicit expressions for the operatorsqop and pop in the Q, P basis:

qop = Qop + aop ; 〈Q1, P1|Qop |Q2, P2〉 = Q1 δQ1Q2 δP1 P2 ,

〈Q1, P1| aop |Q2, P2〉 =(−1)P+Q+1 iP

2π(P 2 +Q2), (4.18)

where Q stands short for Q2 −Q1 , and P = P2 − P1 .

For the p operator, it is derived analogously,

pop = 2πPop + bop , 〈Q1, P1|Pop|Q2, P2〉 = P1 δQ1Q2δP1P2 ; (4.19)

〈Q1, P1| bop |Q2, P2〉 =(−1)P+Q iQ

P 2 +Q2. (4.20)

And now for some surprise. Let us compute now the commutator, [ qop, pop ] , in thebasis of the integers Q and P . We have

[Qop, Pop] = 0 ; [ aop, bop ] = 0 ; [qop, pop] = [Qop, bop ] + 2π[ aop, Pop] ;

〈Q1, P1| [ qop, pop ] |Q2, P2〉 = −i(−1)P+Q)(1− δQ1Q2δP1P2) . (4.21)

62

Here, the delta function is inserted because the commutator vanishes if Q1 = Q2 andP1 = P2 . So, the commutator is not equal to i times the identity, but it can be writtenas

[ qop, pop ] = i ( I− |ψe 〉〈ψe | ) , where 〈Q,P |ψe 〉 = (−1)P+Q . (4.22)

Apparently, there is one state (with infinite norm), for which the standard commutatorrule is violated. We encounter more such states in this report, to be referred to asedge states, that have to be factored out of our Hilbert space. From a physical pointof view it will usually be easy to ignore the edge states, but when we do mathematicalcalculations it is important to understand their nature. The edge state here coincides withthe state δ(κ−π) δ(ξ−π) , so its mathematical origin is easy to spot: it is located at thesingularity of our auxiliary phase function φ(κ, ξ) ; apparently, we must limit ourselvesto wave functions that vanish at that spot in (κ, ξ ) space.

4.1.2. The states |Q,P 〉 in the q basis

As in other chapters, we now wish to identify the transformation matrix enabling usto transform from one basis to an other. Thus, we wish to find the matrix elementsconnecting states |Q,P 〉 to states |q〉 or to states |p〉 . If we can find the function〈q|0, 0〉 , which gives the wave function in q space of the state with Q = P = 0 , findingthe rest will be easy. Appendix 4C, Section 4.4, shows the derivation of this wave function.In (κ, ξ) space, the state |q〉 is

〈κ, ξ |q〉 = ei φ(κ,2πq) δ(ξ − ξq) , (4.23)

if q is written as q = X + ξq/2π , and X is an integer. Appendix 4C shows that thenthe wave function for P = Q = 0 is

〈q|0, 0〉 = 12π

∫ 2π

0

dκ e−iφ(κ,2πq) . (4.24)

The general wave function is obtained by shifting P and Q by integer amounts:

〈q|Q,P 〉 = 12πe2πiPq

∫ 2π

0

dκ e−iφ(κ, 2π(q−Q)) . (4.25)

The wave function (4.24), which is equal to its own Fourier transform, is special becauseit is close to a block wave, having just very small tails outside the domain |q| < 1

2, see

Fig. 7

The wave 〈q|Q,P 〉 is similar to the so-called wavelets [37], which are sometimes usedto describe pulsed waves, but this one has two extra features. Not only is it equal to itsown Fourier transform, it is also orthogonal to itself when shifted by an integer. Thismakes the set of waves |Q,P 〉 in Eq. (4.25) an orthonormal basis.

This PQ formalism is intended to be used to transform systems based on integernumbers only, to systems based on real numbers. The integers may be assumed to un-dergo switches described by a permutation operator Pop . After identifying some useful

63

1

43210-1 q

0.4

0.2

0-1 (X+.5)(q−X−.5)

(X+.5).ψ

Asymptotic wave form:

Figure 7: The wave function of the state |Q,P 〉 when P = Q = 0 . Below, theasymptotic form of the little peaks far from the centre, scaled up.

expression for a hamiltonian Hop , with Pop = e−iHop δt , one can now transform this to aquantum system with that hamiltonian in its new basis.

For a single PQ pair, constructing a deterministic model whose evolution operatorresembles a realistic quantum hamiltonian is difficult. A precise, canonical, discrete hamil-tonian formalism is possible in the PQ scheme, but it requires some more work that wepostpone to Section 7. Interesting hamiltonians are obtained in the multidimensionalcase: Pi, Qi . Such a system is considered in Chapter 5.

4.2. Appendix 4A Transformations in the PQ theory

The Appendices 4 A, B and C are needed only when explicit calculations are wanted,connecting the basis sets of the real numbers q , their Fourier transforms p , the integers(Q, P ) and the torus (κ, ξ) .

We wish to construct a solution to the boundary conditions (4.11) and (4.12) for aphase function φ(κ, ξ) ,

φ(κ, ξ + 2π) = φ(κ, ξ) + κ; φ(κ+ 2π, ξ) = φ(κ, ξ) ; (4.26)

φ(κ, ξ) = −φ(−κ, ξ) = −φ(κ,−ξ) ; φ(κ, ξ) + φ(ξ, κ) = κξ/2π . (4.27)

64

At first sight, these conditions appear to be contradictionary. If we follow a closed contour(κ, ξ) = (0, 0) → (2π, 0) → (2π, 2π) → (0, 2π) → (0, 0) , we pick up a term 2π . Thisimplies that a function that is single valued everywhere does not exist, hence, we musthave a singularity. We can write down an amplitude ψ(κ, ξ) = r eiφ of which this is thephase, but this function must have a zero or a pole. Let’s assume it has a zero, and r issimply periodic. Then one can find a solution. If r and φ are real functions:

r(κ, ξ) eiφ(κ, ξ) =∞∑

N=−∞

e−π(N− ξ

2π)2+iNκ , (4.28)

which is obviously periodic in κ , while the substitution

ξ → ξ + 2π , N → N + 1 , (4.29)

gives the first part of Eq. (4.26).

This sum is a special case of the elliptic function ϑ3 , and it can also be written as aproduct[18]:

r(κ, ξ) eiφ(κ,ξ) = e−ξ2

4π

∞∏N=1

(1− e−2πN)∞∏N=0

(1 + e ξ+iκ−2πN−π)(1 + e−ξ−iκ−2πN−π) , (4.30)

from which we can easily read off the zeros: they are at (κ, ξ) = (2πN1 + π, 2πN2 + π) .We deliberately chose these to be at the corners (±π, ±π) of the torus, but it does notreally matter where they are; they are simply unavoidable.

The matrix elements 〈Q1, P1| qop |Q2, P2〉 are obtained by first calculating them onthe torus. We have15

qop = Qop + a(κ, ξ) ; Qop = −i ∂∂κ

, a(κ, ξ) =( ∂∂κφ(ξ, κ)

). (4.31)

To calculate a(κ, ξ) we can best take the product formula (4.30):

a(κ, ξ) =∞∑N=0

aN(κ, ξ) ,

aN(κ, ξ) =∂

∂κ

(arg(1 + eκ+iξ−2π(N+

12

)) + arg(1 + e−κ−iξ−2π(N+12

))). (4.32)

Evaluation gives:

aN(κ, ξ) =12

sin ξ

cos ξ + cosh(κ− 2πN − π)+

12

sin ξ

cos ξ + cosh(κ+ 2πN + π), (4.33)

which now allows us to rewrite it as a single sum for N running from −∞ to ∞ insteadof 0 to ∞ .

15With apologies for interchanging the κ and ξ variables at some places, which was unavoidable,please beware.

65

Let us first transform from the ξ basis to the P basis, leaving κ unchanged. Thisturns a(κ, ξ) into an operator aop . With

〈P1|ξ〉〈ξ|P2〉 = 12πeiP ξ , P = P2 − P1 , (4.34)

we get the matrix elements of the operator aop in the (κ, P ) frame:

〈P1|aop(κ)|P2〉 =∞∑

N=−∞

aN(κ, P ) , P ≡ P2 − P1 , (4.35)

and writing z = e iξ , we find

aN(κ, P ) =

∮dz

2πizzP

−12i(z − 1/z)

z + 1/z + eR + e−R, R = κ+ 2πN + π ,

aN(κ, P ) = 12sgn(P )(−1)P−1ie−|P (κ+2πN+π) | , (4.36)

where sgn(P ) is defined to be ±1 if P >< 0 and 0 if P = 0 . The absolute valuetaken in the exponent indeed means that we always have a negative exponent there; itoriginated when the contour integral over the unit circle forced us to choose a pole insidethe unit circle.

Next, we find the (Q,P ) matrix elements by integrating this over κ with a factor〈Q1|κ〉〈κ|Q2〉 = 1

2πeiQκ , with Q = Q2 − Q1 . The sum over N and the integral over κ

from 0 to 2π combine into an integral over all real values of κ , to obtain the remarkablysimple expression

〈Q1, P1| aop |Q2, P2〉 =(−1)P+Q+1 iP

2π(P 2 +Q2). (4.37)

In Eq. (4.14), this gives for the q operator:

qop = Qop + aop ; 〈Q1, P1| qop |Q2, P2〉 = Q1 δQ1Q2 δP1 P2 + 〈Q1, P1| aop |Q2, P2〉 . (4.38)

For the p operator, the role of Eq. (4.31) is played by

pop = 2πP + b(κ, ξ) ; P = −i ∂∂ξ

, b(κ, ξ) = −2π( ∂∂ξφ(κ, ξ)

), (4.39)

and one obtains analogously, writing P ≡ P2 − P1 ,

pop = 2πPop + bop , 〈Q1, P1|Pop|Q2, P2〉 = P1 δQ1Q2δP1P2 ; (4.40)

〈Q1, P1| bop |Q2, P2〉 =(−1)P+Q iQ

P 2 +Q2. (4.41)

66

4.3. Appendix 4B. Resume of the quasi-periodic phase function φ(κ, ξ)

Let Q and P be integers while κ and ξ obey −π < κ < π , −π < ξ < π . The operatorsobey

[Qop, Pop] = [κop, Pop] = [Qop, ξop] = [κop, ξop] = 0 . (4.42)

Inner products:

〈κ|Q〉 = 1√2πeiQκ , 〈ξ|P 〉 = 1√

2πeiP ξ . (4.43)

The phase angle function φ(κ, ξ) and φ(κ, ξ) are defined obeying (4.26) and (4.27), and

φ(κ, ξ) = φ(ξ, κ) , φ(κ, ξ) + φ(κ, ξ) = κξ/2π . (4.44)

When computing matrix elements, we should not take the operator φ itself, since it ispseudo periodic instead of periodic, so that the edge state δ(ξ ± π) gives singularities.Instead, the operator ∂φ(κ, ξ)/∂ξ ≡ a(κ, ξ) is periodic, so we start from that. Notethat, on the (κ, ξ) -torus, our calculations often force us to interchange the order of thevariables κ and ξ . Thus, in a slightly modified notation,

a(κ, ξ) =∞∑

N=−∞

aN(κ, ξ) ; aN(κ, ξ) =12

sinκ

cosκ+ cosh(ξ + 2π(N + 12)). (4.45)

With Q2 −Q1 ≡ Q , we write in the (Q, ξ) basis

〈Q1, ξ1|aopN |Q2, ξ2〉 = δ(ξ1 − ξ2)aN(Q, ξ1)

aN(Q, ξ) = 12sgn(Q)(−1)Q−1ie−|Q(ξ+2πN+π)| ,

a(Q, ξ) = 12i(−1)Q−1 cosh(Qξ)

sinh(Qπ). (4.46)

Conversely, we can derive in the (κ, P ) basis, with P2 − P1 ≡ P :

〈κ1, P1|aopN |κ2, P2〉 = δ(κ1 − κ2)aN(κ1, P )

aN(κ, P ) =1

4πsinκ

∫ π

−πdξ

eiP ξ

cosκ+ cosh(ξ + 2πN + π);

a(κ, P ) =∑N

aN(κ, P ) = (−1)Psinh(Pκ)

2 sinh(Pπ)(4.47)

(the latter expression is found by contour integration).

Finally, in the (Q,P ) basis, we have, either by Fourier transforming (4.46) or (4.47):

〈Q1, P1|aop|Q2, P2〉 =−i2π

(−1)Q+P Q

Q2 + P 2. (4.48)

The operators qop and pop are now defined on the torus as

qop(κ, ξ) = Qop + a(ξ, κ) , pop(κ, ξ) = 2π(Pop − a(κ, ξ)) , (4.49)

with Qop = −i∂/∂κ , Pop = −i∂/∂ξ . This reproduces Eqs. (4.37) and (4.41).

67

4.4. Appendix 4C. The wave function of the state |0, 0〉

We calculate the state |q〉 in the (κ, ξ) torus. Its wave equation is (see Eq. (4.14))

qop|q〉 = −i e−iφ(ξ,κ) ∂

∂κ

(eiφ(ξ,κ) |q〉

)= q|q〉 . (4.50)

The solution is

〈κ, ξ|q〉 = C(ξ) e−iφ(ξ,κ)+iqκ . (4.51)

Since the solution must be periodic in κ and ξ , while we have the periodicity condi-tions (4.26) for φ , we deduce that this only has a solution if ξ/2π is the fractional partof q , or, q = X + ξ/2π , where X is integer. In that case, we can write

〈κ, ξ|q〉 = C(ξ) eiXκ+iφ(κ,ξ) = C(ξ) eiφ(κ,2πq) . (4.52)

The complete matrix element is then, writing q = X + ξq/2π ,

〈κ, ξ|q〉 = C eiφ(κ,2πq) δ(ξ − ξq) (4.53)

(note that the phase φ(κ, ξ) is not periodic in its second entry ξ ; the entries are reversedcompared to Eqs (4.26)). The normalisation follows from requiring∫ 2π

0

dκ

∫ 2π

0

dξ 〈q1|κ, ξ〉〈κ, ξ|q2〉 = δ(q1 − q2) ;∫ ∞−∞

dq 〈κ1, ξ1|q〉〈q|κ2, ξ2〉 = δ(κ1 − κ2)δ(ξ1 − ξ2) , (4.54)

from which

C = 1 . (4.55)

Note that we chose the phase to be +1 . As often in this report, phases can be chosenfreely.

Since

〈κ, ξ|Q,P 〉 = 12πeiP ξ+iQκ , (4.56)

we have

〈q|Q,P 〉 = 12π

∫ 2π

0

dκ eiP ξ+iQκ−iφ(κ,2πq) = 12πe2πiPq

∫ 2π

0

dκ e−iφ(κ,2π(q−Q)) . (4.57)

68

5. Models in two space-time dimensions without interactions

5.1. Two dimensional model of massless bosons

An important motivation for our step from real numbers to integers is that we requiredeterministic theories to be infinitely precise. Any system based on a classical action,requires real numbers for its basic variables. But this also introduces limited precision. If,as one might be inclined to suspect, the ultimate physical degrees of freedom merely formbits and bytes, then these can only be discrete, and the prototypes of discrete systems arethe integers. Perhaps later one might want to replace these by integers with a maximalsize, such as integers modulo a prime number, the elements of Z/pZ .

The question is how to phrase a systematic approach. For instance, how do we mimica quantum field theory? If such a theory is based on perturbative expansions, can wemimic such expansions in terms of integers? Needless to observe that standard pertur-bation expansions seem to be impossible for discrete systems, but various special kindsof expansions can still be imagined, such as 1/N expansions, where N could be somecharacteristic of an underlying algebra.

We shall not be able to do this in this report, but we make a start at formulatingsystematic approaches. In this chapter, we consider a quantised field, whose field variables,of course, are operators with continua of eigenvalues in the real numbers. If we want toopen the door to perturbative field theories, we first need to understand free particles.One example was treated in Section 3.3. These were fermions. Now, we try to introducefree bosons.

Such theories obey linear field equations, such as

∂2t φ(~x, t) =

d∑i=1

∂2i φ(~x, t)−m2 φ(~x, t) . (5.1)

In the case of fermions, we succeeded, to some extent, to formulate the massless casein three space dimensions (Section 3.3, Subsection 3.3.3), but applying PQ theory tobosonic fields in higher dimensions has not been successful. The problem is that equationssuch as Eq. (5.1) are difficult to apply to integers, even if we may fill the gaps betweenthe integers with generators of displacements.

In our search for systems where this can be done, we chose, as a compromise, masslessfields in one space-like dimension only. The reason why this special case can be handledwith PQ theory is, that the field equation, Eq. (5.1), can be reduced to first orderequations by distinguishing left-movers and right-movers. Let us first briefly summarisethe continuum quantum field theory for this case.

69

5.1.1. Second-quantised massless bosons in two dimensions

We consider a single, scalar, non interacting, massless field q(x, t) . Both x and t areone-dimensional. The lagrangian and hamiltonian are:

L = 12(∂tq

2 − ∂xq2) ; Hop =

∫dx(1

2p2 + 1

2∂xq

2) , (5.2)

where we use the symbol p(x) to denote the canonical momentum field associated to thescalar field q(x) , which, in the absence of interactions, obeys p(x) = ∂tq(x) . The fieldsq(x) and p(x) are operator fields. The equal-time commutation rules are, as usual:

[q(x), q(y)] = [p(x), p(y)] = 0 ; [q(x), p(y)] = iδ(x− y) . (5.3)

Let us regard the time variable in q(x, t) and p(x, t) to be in Heisenberg notation.We then have the field equations:

∂2t q = ∂2

xq , (5.4)

and the solution of the field equations can be written as follows:

aL(x, t) = p(x, t) + ∂xq(x, t) = aL(x+ t) ; (5.5)

aR(x, t) = p(x, t)− ∂xq(x, t) = aR(x− t) . (5.6)

The equations force the operators aL to move to the left and aR to move to the right.In terms of these variables, the hamiltonian (at a given time t ) is

Hop =

∫dx 1

4

((aL(x))2 + (aR(x))2

). (5.7)

The commutation rules for aL,R are:

[aL, aR] = 0 , [aL(x1), aL(x2)] = 2i δ ′(x1 − x2) ,

[aR(x1), aR(x2)] = −2i δ ′(x1 − x2) , (5.8)

where δ ′(z) = ∂∂zδ(z) .

Now let us Fourier transform in the space direction, by moving to momentum spacevariables k , and subtract the vacuum energy. We have in momentum space (note thatwe restrict ourselves to positive values of k ):

aL,R(k) = 1√2π

∫dx e−ikx aL,R(x) , a†(k) = a(−k) ; (5.9)

Hop =

∫ ∞0

dk 12

((aL†(−k)aL(−k) + aR†(k)aR(k)

), (5.10)

k, k′ > 0 : [aL(−k1), aL(−k2)] = 0 , [aL(−k1), aL†(−k2)] = 2k1δ(k1 − k2) . (5.11)

[aR(k1), aR(k2)] = 0 , [aR(k1), aR†(k2)] = 2k1δ(k1 − k2) . (5.12)

70

In this notation, aL,R(k) are the annihilation and creation operators apart from a factor√2k , so the hamiltonian (5.7) can be written as

Hop =

∫ ∞0

dk (kNL(−k) + kNR(k)) . (5.13)

where NL,R(∓k)dk are the occupation numbers counting the left and right moving par-ticles. The energies of these particles are equal to the absolute values of their momentum.All of this is completely standard and can be found in all the text books.

Inserting a lattice cut-off for the UV divergences in quantum field theories is alsostandard practice. Restricting ourselves to integer values of the x coordinate, and usingthe lattice length in x space as our unit of length, we replace the commutation rules(5.3) by

[q(x), q(y)] = [p+(x), p+(y)] = 0 ; [q(x), p+(y)] = i δx,y (5.14)

(the reason for the superscript + will be explained later, Eqs. (5.30) — (5.32)). The exactform of the hamiltonian on the lattice depends on how we wish to deal with the latticeartefacts. The choices made below might seem somewhat artificial or special, but it canbe verified that most alternative choices one can think of can be transformed to these bysimple lattice field transformations, so not much generality is lost. It is important howeverthat we wish to keep the expression (5.13) for the hamiltonian: also on the lattice, wewish to keep the same dispersion law as in the continuum, so that all excitations mustmove left or right exactly with the same speed of light (which of course will be normalisedto c = 1 ).

The lattice expression for the left- and right movers will be

aL(x+ t) = p+(x, t) + q(x, t)− q(x− 1, t) ; (5.15)

aR(x− t) = p+(x, t) + q(x, t)− q(x+ 1, t) . (5.16)

They obey the commutation rules

[aL, aR] = 0 ; [aL(x), aL(y)] = ± i if y = x± 1 ; else 0 ; (5.17)

[aR(x), aR(y)] = ∓ i if y = x± 1 ; else 0 . (5.18)

They can be seen to be the lattice form of the commutators (5.8). In momentum space,writing

aL,R(x) ≡ 1√2π

∫ π

−πdκ aL,R(κ) eiκx , aL,R †(κ) = aL,R(−κ) ; (5.19)

the commutation rules (5.11) and (5.12) are now

[aL(−κ1), aL†(−κ2)] = 2 sinκ1 δ(κ1 − κ2) ; (5.20)

[aR(κ1), aR†(κ2)] = 2 sinκ1 δ(κ1 − κ2) , (5.21)

71

so our operators aL,R(∓κ) and aL,R †(∓κ) are the usual annihilation and creation oper-ators, multiplied by the factor √

2| sinκ| . (5.22)

If we want the hamiltonian to take the form (5.13), then, in terms of the creation andannihilation operators (5.20) and (5.21), the hamiltonian must be

Hop =

∫ π

0

dκκ

2 sinκ

(aL†(−κ) aL(−κ) + a†R(κ) aR(κ)

). (5.23)

Since, in momentum space, Eqs. (5.15) and (5.16) take the form

aL(κ) = p+(κ) + (1− e−iκ)q(κ) , aR(κ) = p+(κ) + (1− eiκ)q(κ) , (5.24)

after some shuffling, we find the hamiltonian (ignoring the vacuum term)

Hop = 12

∫ π

0

dκ( κ

tan 12κ|p+(κ)|2 + 4k tan 1

2κ |q(κ) + 1

2p+(κ)|2

), (5.25)

where |p+(κ)|2 stands for p+(κ) p+(−κ) . Since the field redefinition q(x)+ 12p+(x)→ q(x)

does not affect the commutation rules, and

limκ→0

κ

2 tan(12κ)

= 1 , 4 sin2(12κ)|q(κ)|2 → |(∂xq)(κ)|2 , (5.26)

we see that the continuum limit (5.2), (5.10) is obtained when the lattice length scale issent to zero.

Because of the factor (5.22), the expression (5.23) for our hamiltonian shows that theoperators a and a† , annihilate and create energies of the amount |κ| , as usual, and theHamilton equations for aL,R are

d

dtaL(−κ, t) = −i[ aL(−κ, t), Hop ] =

−i κ2 sinκ

2 sinκ aL(−κ) = −iκ aL(−κ, t) ;

d

dtaR(κ, t) = −iκ aR(κ, t) . (5.27)

Consequently,

aL(−κ, t) e−iκx = aL(−κ, 0)e−iκx−iκt ; aR(κ, t)eiκx = aR(κ, 0)eiκx−iκt . (5.28)

We now notice that the operators aL(x, t) = aL(x + t) and aR(x, t) = aR(x − t) moveexactly one position after one unit time step. Therefore, even on the lattice,

aL(x, 1) = aL(x+ 1, 0) , aR(x, 1) = aR(x− 1, 0) , etc. (5.29)

and now we can use this to eliminate p+(x, t) and q(x, t) from these equations. Writing

p+(x, t) ≡ p(x, t+ 12) , (5.30)

72

one arrives at the equations

q(x, t+ 1) = q(x, t) + p(x, t+ 12) ; (5.31)

p(x, t+ 12) = p(x, t− 1

2) + q(x− 1, t)− 2q(x, t) + q(x+ 1, t) . (5.32)

We now see why we had to shift the field q(x, t) by half the field momentum in Eq. (5.25):it puts the field at the same position t+ 1

2as the momentum variable p+(x, t) .

Thus, we end up with a quantum field theory where not only space but also time is ona lattice. The momentum values p(x, t + 1

2) can be viewed as variables on the time-like

links of the lattice.

At small values of κ , the hamiltonian (5.23), (5.25) closely approaches that of thecontinuum theory, and so it obeys locality conditions there. For this reason, the modelwould be interesting indeed, if this is what can be matched with a cellular automaton.However, there is a problem with it. At values of κ approaching κ → ±π , the kernelsdiverge. Suppose we would like to write the expression (5.23) in position space as

Hop = 12

∑x, s

M|s|

(aL(x) aL(x+ s) + aR(x) aR(x+ s)

), (5.33)

then Ms would be obtained by Fourier transforming the coefficient κ/2 sin(κ) on theinterval [−π, π] for k . The factor 1

2comes from symmetrising the expression (5.33) for

positive and negative s . One obtains

Ms = 12π

∫ π−λ

−π+λ

κdκ

2 sinκe−isκ = 1

2

log 2

λ−∑s/2−1

k=01

k+1/2if s = even

log(2λ) +∑(s−1)/2

k=11k

if s = odd(5.34)

where λ is a tiny cut-off parameter. The divergent part of Hop is

14( log

1

λ)∑x,y

(−1)x−y(aL(x)aL(y) + aR(x)aR(y)

)=

14( log

1

λ)((∑

x

(−1)xaL(x))2

+(∑

x

(−1)xaR(x))2)

. (5.35)

Also the kernel 4κ tan 12κ in Eq. (5.25) diverges as κ→ ±π . Keeping the divergence

would make the hamiltonian non-local, as Eq. (5.35) shows. We can’t just argue that thelargest κ values require infinite energies to excite them because they do not; accordingto Eq. (5.13), the energies of excitations at momentum κ are merely proportional to κitself. We therefore propose to make a smooth cut-off, replacing the divergent kernelssuch as 4κ tan 1

2κ by expressions such as

(4κ tan 12κ)(1− e−Λ2(π−κ)2) , (5.36)

where Λ can be taken to be arbitrarily large but not infinite.

73

We can also say that we keep only those excitations that are orthogonal to plane waveswhere aL(x) or aR(x) are of the form C(−1)x . Also these states we refer to as edgestates.

What we now have is a lattice theory where the hamiltonian takes the form (5.13),where NL(−κ) and NR(κ) (for positive κ ) count excitations in the left- and the rightmovers, both of which move with the same speed of light at all modes. It is this system thatwe can now transform to a cellular automaton. Note, that even though the lattice modelmay look rather contrived, it has a smooth continuum limit, which would correspond toa very dense automaton, and in theories of physics, one usually only can compare thecontinuum limit with actual observations, such as particles in field theories. Our point isthat, up this point, our theory can be seen as a conventional quantum system.

5.1.2. The cellular automaton with integers in 2 dimensions

Our cellular automaton is a model defined on a square lattice with one space dimensionx and one time coordinate t , where both x and t are restricted to be integers. Thevariables are two sets of integers, one set being integer numbers Q(x, t) defined on thelattice sites, and the other being defined on the links connecting the point (x, t) with(x, t+ 1) . These will be called P (x, t+ 1

2) , but they may sometimes be indicated as

P+(x, t) ≡ P−(x, t+ 1) ≡ P (x, t+ 12) . (5.37)

The automaton obeys the following time evolution laws:

Q(x, t+ 1) = Q(x, t) + P (x, t+ 12) ; (5.38)

P (x, t+ 12) = P (x, t− 1

2) + Q(x− 1, t)− 2Q(x, t) +Q(x+ 1, t) , (5.39)

just analogously to Eqs. (5.31) and (5.32). It is also a discrete version of a classical fieldtheory where Q(x, t) are the field variables and P (x, t) = ∂

∂tQ(x, t) are the classical field

momenta.

Alternatively, one can write Eq. (5.39) as

Q(x, t+ 1) = Q(x− 1, t) +Q(x+ 1, t)−Q(x, t− 1) , (5.40)

which, incidentally, shows that the even lattice sites evolve independently from the oddones. Later, this will become important.

As the reader must understand by now, we introduce Hilbert space for this system, just

as a tool. The basis elements of this Hilbert space are the states∣∣∣ Q(x, 0), P+(x, 0)

⟩.

If we consider a superposition of such states, we will simply define the squares of the am-plitudes to represent the probabilities. The total probability is the length-squared of thevector, which will usually be taken to be one. At this stage, superpositions mean nothingmore than this, and it is obvious that any chosen superposition, whose total length is one,may represent a reasonable set of probabilities. The basis elements all evolve in terms of apermutation operator that permutes the basis elements in accordance with the evolution

74

equations (5.38) and (5.39). As a matrix in Hilbert space, this permutation operator onlycontains ones and zeros, and it is trivial to ascertain that statistical distributions, writtenas “quantum” superpositions, evolve with the same evolution matrix.

As operators in this Hilbert space, we shall introduce shift generators that are angles,defined exactly as in Eq. (4.7), but now at each point x1 at time t = 0 , we have anoperator κ(x1) that generates an integer shift in the variable Q(x1) and an operatorξ+(x1) generating a shift in the integer P+(x1) :

e−iκ(x1) | Q,P+ 〉 = | Q ′(x), P+(x) 〉 ; Q ′(x) = Q(x) + δxx1 ; (5.41)

e iξ+(x1) | Q,P+ 〉 = | Q(x), P+′(x) 〉 ; P+′(x) = P+(x) + δxx1 ; (5.42)

The time variable t is an integer, so what our evolution equations (5.38) and (5.39)generate is an operator Uop (t) obeying Uop (t1 + t2) = Uop (t1)Uop (t2) , but only forinteger time steps. In Section 2.4, Eq. (2.30), a hamiltonian Hop was found that obeysUop (t) = e−iHop t , by Fourier analysis. The problem with that hamiltonian is that

1. It is not unique: one may add 2π times any integer to any of its eigenvalues; and

2. It is not extensive: if two parts of a system are space-like separated, we would likethe hamiltonian to be the sum of the two separate hamiltonians, but then it willquickly take values more than π , whereas, by construction, the hamiltonian (2.30)will obey |H| ≤ π .

Thus, by adding appropriate multiples of real numbers to its eigenvalues, we would liketo transform our hamiltonian into an extensive one. The question is how to do this.

Indeed, this is one of the central questions that forced us to do the investigationsdescribed in this report; the hamiltonian of the quantum field theory considered here isan extensive one, and also naturally bounded from below.

At first sight, the similarity between the automaton described by the equations (5.38)and (5.39), and the quantum field theory of Section (5.1.1) may seem to be superficial atbest. Quantum physicists will insist that the quantum theory is fundamentally different.

However, we claim that there is an exact mapping between the basis elements of thequantised field theory of Subsection 5.1.1 and the states of the cellular automaton (again,with an exception for possible edge states). We shall show this by concentrating on theleft-movers and the right-movers separately.

Our procedure will force us first to compare the left-movers and the right-movers inboth theories. The automaton equations (5.38) and (5.39) ensure that, if we start withintegers at t = 0 and t = 1

2, all entries at later times will be integers as well. So this

is a discrete automaton. We now introduce the combinations AL(x, t) and AR(x, t) asfollows:

AL(x, t) = P+(x, t) +Q(x, t)−Q(x− 1, t) ; (5.43)

AR(x, t) = P+(x, t) +Q(x, t)−Q(x+ 1, t) , (5.44)

75

and we derive

AL(x, t+ 1) = P+(x, t) +Q(x− 1, t+ 1)− 2Q(x, t+ 1) +Q(x− 1, t+ 1)

+Q(x, t+ 1)−Q(x− 1, t+ 1) =

P+(x, t) +Q(x− 1, t+ 1)−Q(x, t+ 1) =

P+(x, t) +Q(x− 1, t) + P+(x− 1, t)−Q(x, t)− P+(x, t) =

P+(x− 1, t) +Q(x− 1, t)−Q(x, t) = AL(x− 1, t) . (5.45)

So, we have

AL(x− 1, t+ 1) = AL(x, t) = AL(x+ t) ; AR(x, t) = AR(x− t) , (5.46)

which shows that AL is a left-mover and AR is a right mover. All this is completelyanalogous to Eqs. (5.15) and (5.16).

5.1.3. The mapping between the boson theory and the automaton

The states of the quantised field theory on the lattice were generated by the left- andright moving operators aL(x + t) and aR(x − t) , where x and t are integers, but aL

and aR have continua of eigenvalues, and they obey the commutation rules (5.17) and(5.18):

[aL, aR] = 0 ; [aL(x), aL(y)] = ± i if y = x± 1 ; else 0 ; (5.47)

[aR(x), aR(y)] = ∓ i if y = x± 1 ; else 0 . (5.48)

In contrast, the automaton has integer variables AL(x + t) and AR(x − t) , Eqs, (5.43)and (5.44). They live on the same space-time lattice, but they are integers, and theycommute.

Now, PQ theory suggests what we have to do. The shift generators κ(x1) and ξ(x1)(Eqs. 5.41 and 5.42) can be combined to define shift operators ηL(x1) and ηR(x1) forthe integers AL(x1, t) and AR(x1, t) . Define

e iηL(x1)|AL, AR〉 = |AL′, AR〉 , AL

′(x) = AL(x) + δx,x1 , (5.49)

e iηR(x1)|AL, AR〉 = |AL, AR′〉 , AR

′(x) = AR(x) + δx,x1 . (5.50)

These then have to obey the following equations:

ξ(x) = ηL(x) + ηR(x) ;

−κ(x) = ηL(x) + ηR(x)− ηL(x+ 1)− ηR(x− 1) . (5.51)

The first of these tells us that, according to Eqs. (5.43) and (5.44), raising P+(x) byone unit, while keeping all others fixed, implies raising this combination of AL and AR .The second tells us what the effect is of raising only Q(x) by one unit while keeping theothers fixed. Of course, the additions and subtractions in Eqs. (5.51) are modulo 2π .

76

Inverting Eqs. (5.51) leads to

ηL(x+ 1)− ηL(x− 1) = ξ(x) + κ(x)− ξ(x− 1) ,

ηR(x− 1)− ηR(x+ 1) = ξ(x) + κ(x)− ξ(x+ 1) . (5.52)

These are difference equations whose solutions involve infinite sums with a boundaryassumption. This has no further consequences; we take the theory to be defined bythe operators ηL,R(x) , not the ξ(x) and κ(x) . As we have encountered many timesbefore, there are some non-local modes of measure zero, ηL(x + 2n) = constant , andηR(x+ 2n) = constant .

What we have learned from the PQ theory, is that, in a sector of Hilbert space thatis orthogonal to the edge state, an integer variable A , and its shift operator η , obey thecommutation rules

Ae iη = e iη(A+ 1) ; [η, A] = i . (5.53)

This gives us the possibility to generate operators that obey the commutation rules (5.47)and (5.48) of the quantum field theory:

aL(x)?=√

2π AL(x)− 1√2π

ηL(x− 1) ; (5.54)

aR(x)?=√

2π AR(x)− 1√2π

ηR(x+ 1) . (5.55)

The factors√

2π are essential here. They ensure that the spectrum is not larger orsmaller than the real line, that is, without gaps or overlaps (degeneracies).

The procedure can be improved. In the expressions (5.54) and (5.55), we have an edgestate whenever ηL,R take on the values ±π . This is an unwanted situation: these edgestates make all wave functions discontinuous on the points aL,R(x) =

√2π (N(x) + 1

2) .

Fortunately, we can cancel most of these edge states by repeating more precisely theprocedure explained in our treatment of PQ theory: these states were due to vorticesin two dimensional planes of the tori spanned by the η variables. Let us transform, bymeans of standard Fourier transforming the A lattices to the η circles, so that we get amulti-dimensional space of circles – one circle at every point x .

As in the simple PQ theory (see Eqs. (4.14) and (4.15)), we can introduce a phasefunction ϕ(ηL) and a ϕ(ηR) , so that Eqs. (5.54) and (5.55) can be replaced with

aL(x) = − i√

2π∂

∂ηL(x)+√

2π( ∂

∂ηL(x)ϕ(ηL)

)− 1√

2πηL(x− 1) , (5.56)

aR(x) = − i√

2π∂

∂ηR(x)+√

2π( ∂

∂ηR(x)ϕ(ηR)

)− 1√

2πηR(x+ 1) , (5.57)

where ϕ(η(x)) is a phase function with the properties

ϕ(ηL(x) + 2πδx,x1 = ϕ(ηL(x)) + ηL(x1 + 1) ; (5.58)

ϕ(ηR(x) + 2πδx,x1 = ϕ(ηR(x)) + ηR(x1 − 1) . (5.59)

77

Now, as one can easily check, the operators aL,R(x) are exactly periodic for all η variables,just as we had in Section 4.

A phase function with exactly these properties can be written down. We start withthe elementary function φ(κ, ξ) derived in Appendix 4A, Section 4.2, Eq. (4.28), havingthe properties

φ(κ, ξ + 2π) = φ(κ, ξ) + κ ; φ(κ+ 2π, ξ) = φ(κ, ξ) ; (5.60)

φ(κ, ξ) = −φ(−κ, ξ) = −φ(κ,−ξ) ; φ(κ, ξ) + φ(ξ, κ) = κξ/2π . (5.61)

The function obeying Eqs. (5.58) and (5.59) is now not difficult to construct:

ϕ(ηL) =∑x

φ(ηL(x+ 1), ηL(x)

);

ϕ(ηR) =∑x

φ(ηR(x− 1), ηR(x)

), (5.62)

and as was derived in Appendix 4A, a phase function with these properties can be givenas the phase of an elliptic function,

r(κ, ξ)eiφ(κ,ξ) ≡∞∑

N=−∞

e−π(N− ξ2π

)2−iNκ , (5.63)

where r and φ are both real functions of κ and ξ .

We still have edge states, but now these only sit at the corners where two consecutiveη variables take the values ±π . This is where the phase function φ , and therefore alsoϕ , become singular. We suspect that we can simply ignore them.

We then reach an important conclusion. The states of the cellular automaton canbe used as a basis for the description of the quantum field theory. These models areequivalent. This is an astounding result. For generations we have been told by our physicsteachers, and we explained to our students, that quantum theories are fundamentallydifferent from classical theories. No-one should dare to compare a simple computer modelsuch as a cellular automaton based on the integers, with a fully quantised field theory.Yet here we find a quantum field system and an automaton that are based on states thatneatly correspond to each other, they evolve identically. If we describe some probabilisticdistribution of possible automaton states using Hilbert space as a mathematical device,we can use any wave function, certainly also waves in which the particles are entangled,and yet these states evolve exactly the same way.

Physically, using 19th century logic, this should have been easy to understand: whenquantising a classical field theory, we get energy packets that are quantised and behaveas particles, but exactly the same are generated in a cellular automaton based on theintegers; these behave as particles as well. Why shouldn’t there be a mapping?

Of course one can, and should, be skeptic. Our field theory was not only constructedwithout interactions and without masses, but also the wave function was devised in sucha way that it cannot spread, so it should not come as a surprise that no problems are

78

encountered with interference effects, so yes, all we have is a primitive model, not veryrepresentative for the real world. Or is this just a beginning?

Note: being exactly integrable, the model discussed in this Section has infinitely manyconservation laws. For instance, one may remark that the equation of motion (5.40) doesnot mix even sites with odd sites of the lattice; similar equations select out sub latticeswith x+ t = 4k and x and t even, from other sub lattices.

5.1.4. An alternative ontological basis: the compactified model

In the above chapters and sections of this chapter, we have seen various examples ofdeterministic models that can be mapped onto quantum models and back. The readermay have noticed that, in many cases, these mappings are not unique. Modifying thechoices of the constant energy shifts Ei in the composite cogwheel model, Section 2.3.1,we saw that many apparently different quantum theories can be mapped onto the sameset of cogwheels, although there, the Ei could have been regarded as various chemicalpotentials, having no effect on the evolution law. In our PQ theory, one is free to addfractional constants to Q and P , thus modifying the mapping somewhat. Here, theeffect of this would be that the ontological states obtained from one mapping do not quitecoincide to those of the other, they are superpositions, and this is an example of theoccurrence of sets of ontological states that are not equivalent, but all equally legal.

The emergence of inequivalent choices of an ontological basis is particularly evident ifthe system in question has a symmetry groups that are larger than those of the ontologicalsystem. If the ontological system is based on a lattice, it can only have some of the discretelattice groups as its symmetries, whereas the quantum system, based on real coordinates,can have continuous symmetry groups such as the rotation, translation and Lorentz group.Performing a symmetry transformation that is not a symmetry of the ontological modelthen leads to a new set of ontological states (or “wave functions”) that are superpositionsof the other states. Only one of these sets will be the “real” ontological states; forour theory, and in particular the cellular automaton interpretation, Chapters 12 – 14,this has no further consequences, except that it will be almost impossible to single outthe “true” ontological basis as opposed to the apparent ones, obtained after symmetrytransformations.

In this subsection, we point out that even more can happen. Two (or perhaps more)systems of ontological basis elements may exist that are entirely different from one another.This is the case for the model of the previous subsection, which handles the mathematicsof non-interacting massless bosons in 1 + 1 dimensions. We argued that an ontologicalbasis is spanned by all states where the field q(x, t) is replaced by integers Q(x, t) . Alattice in x, t was introduced, but this was a temporary lattice, we could send the meshsize to zero in the end.

In Subsection 5.1.2, we introduced the integers AL(x) and AR(x) , which are theinteger-valued left movers and right movers; they span an ontological basis. Equivalently,one could have taken the integers Q(x, t) and P+(x, t) at a given time t ), but this isjust a reformulation of the same ontological system.

79

But why not take the continuous degrees of freedom ηL(x) and ηR(x) ? At each x ,these variables take values between −π and π . Since they are also left- and right movers,their evolution law is exactly as deterministic as that of the integers AL and AR :

∂

∂tηL(x, t) =

∂

∂xηL(x, t) ,

∂

∂tηR(x, t) = − ∂

∂xηR(x, t) , (5.64)

while all η ’s commute.

Actually, for the η fields, it is much easier now to regard the continuum limit for thespace–time lattice. The quantum operators aL,R are still given by Eqs, (5.56) and (5.57).There is a singularity when two consecutive η fields take the values ±π , but if they don’tat t = t0 , they never reach those points at other times, of course.

There exists a somewhat superior way to rephrase the mapping by making use of thefact that the η fields are continuous, so that we can do away with, or hide, the lattice.This is shown in more detail in Subsection 5.3.5.

What we conclude from this subsection is that both our quantum model of bosons andthe model of left and right moving integers are mathematically equivalent to a classicaltheory of scalar fields where the values are only defined monulo 2π . From the ontologicalpoint of view our new model is entirely different from both previous models.

Because the variables of the classical model only take values on the circle, we call theclassical model a compactified classical field theory. At other places in this report, theauthor warned that classical, continuous theories may not be the best ontological systemsto assume for describing Nature, because they tend to be chaotic: as time continues,more and more decimal places of the real numbers describing the initial state will becomerelevant, and this appears to involve unbounded sets of digital data. To our presentcontinuous field theory in 1 + 1 dimensions, this objection does not apply, because thereis no chaos; the theory is entirely integrable. Of course, in more complete models ofthe real world we do not expect integrability, so there this objection against continuummodels does apply.

5.1.5. The quantum ground state

The mappings we found are delicate ones, and not always easy to implement. For instance,one would like to ask which solution of the cellular automaton, or the compactified fieldtheory, corresponds to the quantum ground state of the quantised field theory. First, weanswer the question: if you have the ground state, how do we add a single particle to it?

Now this should be easy. We have the creation and annihilation operators for leftmovers and right movers, which are the Fourier transforms of the operators aL,R(x) ofEqs. (5.56) and (5.57). When the Fourier parameter, the lattice momentum κ , is in theinterval −π < κ < 0 , then aL(κ) is an annihilation operator and aR(κ) is a creationoperator. When 0 < κ < π , this is the other way around. Since nothing can be annihi-lated from the vacuum state, the annihilation operators vanish, so aL,R(x) acting on thevacuum can only give a superposition of one–particle states.

To see how a single left-moving particle is added to a classical cellular automaton

80

state, we consider the expression (5.56) for aL(x) , acting on the left-mover’s coordinatex + t → x when t = 0 . The operator ∂/∂ηL(x) multiplies the amplitude for the statewith iAL(x) ; the other operators in Eq. (5.56) are just functions of ηL at the point xand the point x− 1 . Fourier transforming these functions gives us the operators e±Niη

L

multiplied with the Fourier coefficients found, acting on our original state. According toEq. (5.49), this means that, at the two locations x and x − 1 , we add or subtract Nunits to the number AL there, and then we multiply the new state with the appropriateFourier coefficient. Since the functions in question are bounded, we expect the Fourierexpansion to converge reasonably well, so we can regard the above as being a reasonableanswer to the question how to add a particle. Of course, our explicit construction addeda particle at the point x . Fourier transforming it, gives us a particle with momentum−κ and energy κ .

In the compactified field model, the action of the operators (5.56) and (5.57) is straight-forward; we find the states with one or more particles added, provided that the wavefunction is differentiable. The ontological wave functions are not differentiable – they aredelta peaks, so particles can only be added for the non-ontological waves, which are to beregarded as probabilistic distributions of ontological states.

Finding the quantum ground state itself is harder. It is that particular superposition ofontological states from which no particles can be removed. Selecting out the annihilationparts of the operators aL,R(x) means that we have to apply the projection operator P−on the function aL(x) and P+ on aR(x) , where the projection operators P± are givenby

P±a(x) =∑x′

P±(x− x′)a(x′) , P−(y) = δy,0 − P+(y) ,

P+(y) =1

2π

∫ π

0

dκ eiyκ =i

πyif y odd , 1

2δy,0 if y even. (5.65)

The state for which the operators P+aL(x) and P−aR(x) vanish at all x is thequantum ground state. It is a superposition of all cellular automaton states.

Note that the theory has a Goldstone mode[10], which means that an excitation inwhich all fields q(x, t) , or all automaton variables Q(x, t) , get the same constant Q0

added to them, does not affect the total energy. This is an artefact of this particularmodel16. Note also that the projection operators P±(x) are not well-defined on x -independent fields; for these fields, the vacuum is ill-defined.

5.2. Bosonic theories in higher dimensions?

At first sight, it seems that the model described in previous sections may perhaps begeneralised to higher dimensions. In this section, we begin with setting up a scheme thatshould serve as an approach towards handling bosons in a multiply dimensional space as a

16Paradoxically, models in two space-time dimensions are known not to allow for Goldstone modes;this theorem[17], however, only applies when there are interactions. Ours is a free particle model.

81

cellular automaton. Right-away, we emphasise that a mapping in the same spirit as whatwas achieved in previous sections and chapters will not be achieved. Nevertheless, we willexhibit here the initial mathematical steps, which start out as seemingly beautiful. Then,we will exhibit, with as much clarity as we can, what, in this case, the obstacles are, andwhy we call this a failure in the end. As it seems today, what we have here is a looseend, but it could also be the beginning of a theory where, as yet, we were forced to stophalf-way.

5.2.1. Instability

We would have been happy with either a discretised automaton or a compactified classicalfield theory, and for the time being, we keep both options open.

Take the number of space-like dimensions to be d , and suppose that we replaceEqs. (5.38) and (5.39) by

Q(~x, t+ 1) = Q(~x, t) + P (~x, t+ 12) ; (5.66)

P (~x, t+ 12) = P (~x, t− 1

2) +

d∑i=1

(Q(~x− ei, t)− 2Q(~x, t) +Q(~x+ ei, t)

), (5.67)

where ei are unit vectors in the i th direction in space.

Next, consider a given time t . We will need to localise operators in time, and cando this only by choosing the time at which an operator acts such that, at that particulartime, the effect of the operator is as concise as is possible. This was why, in Eqs. (5.66)and (5.67), we chose to indicate time as t± 1

2(where t is integer). Let κop (~x, t+ 1

2) be

the generator of a shift of Q(~x, t) and the same shift in Q(~x, t+ 1) (so that P (~x, t+ 12)

does not shift), while ξop (~x, t) generates identical, negative shifts of P (~x, t + 12) and of

P (~x, t− 12) , without shifting Q(~x, t) at the same t , and with the signs both as dictated

in Eqs (5.41) and (5.42). Surprisingly, one finds that these operators obey the sameequations (5.66) and (5.67): the operation

κop (~x, t+ 12) has the same effect as

κop (~x, t+ 32)−

2d∑i=1

(ξop (~x+ ei, t+ 1)− ξop (~x, t+ 1)

),

and ξop (~x, t) has the same effect as

ξop (~x, t+ 1)− κop (~x, t+ 12) (5.68)

(where the sum is the same as in Eq. (5.67) but in a more compact notation) and therefore,

ξop (~x, t+ 1) = ξop (~x, t) + κop (~x, t+ 12) ; (5.69)

κop (~x, t+ 12) = κop (~x, t− 1

2) +

+d∑i=1

(ξop (~x− ei, t)− 2ξop (~x, t) + ξop (~x+ ei, t)

), (5.70)

82

and, noting that Q and P are integers, while κop and ξop are confined to the inter-val [0, 2π) , we conclude that again the same equations are obeyed by the real numberoperators

qop (~x, t)?= Q(~x, t) + 1

2πξop (~x, t) , and

pop (~x, t+ 12)

?= 2πP (~x, t+ 1

2) + κop (~x, t+ 1

2) . (5.71)

These operators, however, do not obey the correct commutation rules. There even

appears to be a factor 2 wrong, if we would insert the equations [Q(~x), κop (~x′)]?= iδ~x,~x′ ,

[ξop (~x), P (~x′)?= iδ~x,~x′ . Of course, the reason for this failure is that we have the edge

states, and we have not yet restored the correct boundary conditions in ξ, κ space byinserting the phase factors ϕ , as in Eqs. (5.56), (5.57), or φ in (4.14), (4.15). Thisis where our difficulties begin. These phase factors also aught to obey the correct fieldequations, and this seems to be impossible to realise.

In fact, there is an other difficulty with the equations of motion, (5.66), (5.67): they areunstable. It is true that, in the continuum limit, these equations generate the usual fieldequations for smooth functions q(~x, t) and p(~x, t) , but we now have lattice equations.Fourier transforming the equations in the space variables ~x and time t , one finds

−2i(sin 12ω)q(~k, ω) = p(~k, ω) ,

−2i(sin 12ω)p(~k, ω) =

d∑i=1

2(cos ki − 1) q(~k, ω) . (5.72)

This gives the dispersion relation

4 sin2 12ω = 2(1− cosω) =

d∑i=1

2(1− cos ki) . (5.73)

In one space-like dimension, this just means that ω = ±k , which would be fine, but ifd > 1 , and ki take on values close to ±π , the r.h.s. of this equation exceeds the limitvalue 4, the cosine becomes an hyperbolic cosine, and thus we find modes that oscillateout of control exponentially with time.

To mitigate this problem, we would have to, somehow, constrain the momenta ki to-wards small values only, but, both in a cellular automation where all variables are integers,and in the compactified field model, where we will need to respect the intervals (−π, π) ,this is hard to accomplish. Note that we used Fourier transforms on functions such as Qand P in Eqs. (5.66) and (5.67) that take integer values. In itself, that procedure is fine,but it shows the existence of exponentially exploding solutions. These solutions can alsobe attributed to the non-existence of an energy function that is conserved and boundedfrom below. Such an energy function does exist in one dimension:

E = 12

∑x

P+(x, t)2 +

12

∑x

(Q(x, t) + P+(x, t)

)(2Q(x, t)−Q(x− 1, t)−Q(x+ 1, t)

), (5.74)

83

or in momentum space, after rewriting the second term as the difference of two squares,

E = 12

∫ ∞−∞

dk(

(cos 12k)2 |P+(k)|2 + 4(sin 1

2k)2 |Q(k) + 1

2P+(k)|2

). (5.75)

Up to a factor sin k / k , this is the hamiltonian (5.25) (Since the equations of motionat different k values are independent, conservation of one of these hamiltonians impliesconservation of the other).

In higher dimensions, models of this sort cannot have a non-negative, conserved energyfunction, and so these will be unstable.

5.2.2. Abstract formalism for the multidimensional harmonic oscillator

Our PQ procedure for coupled harmonic oscillators can be formalised more succinctlyand elegantly. Let us write a time-reversible harmonic model with integer degrees offreedom as follows. In stead of Eqs. (5.66) and (5.67) we write

Qi(t+ 1) = Qi(t) +∑j

TijPj(t+ 12) ; (5.76)

Pi(t+ 12) = Pi(t− 1

2)−

∑j

VijQj(t) . (5.77)

Here, t is an integer-valued time variable. It is very important that both matrices T andV are real and symmetric:

Tij = Tji ; Vij = Vji . (5.78)

Tij would often, but not always, be taken to be the Kronecker delta δij , and Vij wouldbe the second derivative of a potential function, here being constant coefficients. Since wewant Qi and Pi both to remain integer-valued, the coefficients Tij and Vij will also betaken to be integer-valued. In principle, any integer-valued matrix would do; in practice,we will find severe restrictions.

Henceforth, we shall omit the summation symbol∑

j , as its presence can be taken tobe implied by summation convention: every repeated index is summed over.

When we define the translation generators for Qi and Pi , we find that, in a Heisenbergpicture, it is best to use an operator κop

i (t+ 12) to add one unit to Qi(t) while all other

integers Qj(t) with j 6= i and all Pj(t + 12) are kept fixed. Note that, according to the

evolution equation (5.76), this also adds one unit to Qi(t + 1) while all other Qj(t + 1)are kept fixed as well, so that we have symmetry around the time value t+ 1

2. Similarly,

we define an operator ξopi (t) that shifts the value of both Pi(t− 1

2) and Pi(t+ 1

2) , while

all other Q and P operators at t− 12

and at t+ 12

are kept unaffected; all this was alsoexplained in the text between Eqs. (5.67) and (5.68).

So, we define the action by operators κopi and ξop

i by

84

e−iκopi (t+ 1

2)|Qj(t), Pj(t+ 1

2)〉 = |Q′j(t), Pj(t+ 1

2)〉 or

e−iκopi (t+ 1

2)|Qj(t+ 1), Pj(t+ 1

2)〉 = |Q′j(t+ 1), Pj(t+ 1

2)〉 with (5.79)

Q′j(t) = Qj(t) + δji , Q′j(t+ 1) = qj(t+ 1) + δji ;

eiξopi (t)|Qj(t), Pj(t+ 1

2)〉 = |Q′j(t), Pj(t+ 1

2)〉 or

eiξopi (t)|Qj(t), Pj(t− 1

2)〉 = |Qj(t), P

′j(t− 1

2)〉 with (5.80)

P ′j(t± 12) = Pj(t± 1

2) + δji .

The operators ξopi (t) and κop

i (t+ 12) then obey exactly the same equations as Qi(t)

and Pi(t+ 12) , as given in Eqs (5.76) and (5.77):

ξopi (t+ 1) = ξop

i (t) + Tij κopj (t+ 1

2) ; (5.81)

κopi (t+ 1

2) = κop

i (t− 12)− Vij ξop

j (t) . (5.82)

The stability question can be investigated by writing down the conserved energy func-tion. After careful inspection, we find that this energy function can be defined at integertimes:

H1(t) = 12Tij Pi(t+ 1

2)Pj(t− 1

2) + 1

2Vij Qi(t)Qj(t) , (5.83)

and at half-odd integer times:

H2(t+ 12) = 1

2Tij Pi(t+ 1

2)Pj(t+ 1

2) + 1

2Vij Qi(t)Qj(t+ 1) . (5.84)

Note that H1 contains a pure square of the Q fields but a mixed product of the P fieldswhile H2 has that the other way around. It is not difficult to check that H1(t) = H2(t+ 1

2) :

H2 −H1 = −12Tij Pi(t+ 1

2)VjkQk(t) + 1

2Vij Qi(t)Tjk Pk(t+ 1

2) = 0 . (5.85)

Similarly, we find that the hamiltonian stays the same at all times. Thus, we have aconserved energy, and that could guarantee stability of the evolution equations.

However, we still need to check whether this energy function is indeed non-negative.This we do by rewriting it as the sum of two squares. In H1 , we write the momentumpart (kinetic energy) as

12Tij

(Pi(t+ 1

2) + 1

2VikQk(t)

)(Pj(t+ 1

2) + 1

2Vj`Q`(t)

)− 1

8TijVik Vj`Qk(t)Q`(t) , (5.86)

so that we get at integer times (in short-hand notation)

H1 = ~Q(12V − 1

8V T V ) ~Q+ 1

2(~P+ + 1

2~QV )T (~P+ + 1

2V ~Q) , (5.87)

and at half-odd integer times:

H2 = ~P (12T − 1

8T V T )~P + 1

2( ~Q− + 1

2~P T )V ( ~Q− + 1

2T ~P ) , (5.88)

85

where P+(t) stands for P (t+ 12) , and Q−(t+ 1

2) stands for Q(t) .

The expression (5.84) for H2 was the one used in Eq. (5.74) above. It was turned intoEq. (5.88) in the next expression, Eq. (5.75).

Stability now requires that the coefficients for these squares are all non-negative. Thishas to be checked for the first term in Eq. (5.87) and in (5.88). If V and/or T have oneor several vanishing eigen vectors, this does not seem to generate real problems, and wereplace these by infinitesimal numbers ε > 0 . Then , we find that, on the one hand

〈T 〉 > 0 , 〈V 〉 > 0 ; (5.89)

but also, by multiplying left and right by V −1 and T−1 :

〈 4V −1 − T 〉 ≥ 0 , 〈 4T−1 − V 〉 ≥ 0 . (5.90)

Unfortunately, there are not so many integer-valued matrices V and T with theseproperties. Limiting ourselves momentarily to the case Tij = δij , we find that the matrixVij can have at most a series of 2’s on its diagonal and sequences of ±1 ’s on both sidesof the diagonal. Or, the model displayed above in Eq. (5.66) and (5.67), on a lattice withperiodicity N , is the most general multi-oscillator model that can be kept stable by anonnegative energy function.

If we want more general, less trivial models, we have to search for a more advanceddiscrete hamiltonian formalism (see Sect. 7).

If it were not for this stability problem, we could have continued to construct real-valued operators qop

i and popi by combining Qi with ξop

i and Pi with κopi . The

operators eiQξopi and e−iPκ

opi give us the states |Qi, Pi〉 from the ‘zero-state’ |0, 0〉 .

This means that only one wave function remains to be calculated by some other means,after which all functions can be mapped by using the operators eiaipi and eibjqj . Butsince we cannot obtain stable models in more than 1 space-dimensions, this procedureis as yet of limited value. It so happens, however, that the one-dimensional model isyet going to play a very important role in this work: (super) string theory, see the nextSection.

5.3. (Super)strings

So-far, most of our models represented non-interacting massless particles in a limitednumber of space dimensions. Readers who are still convinced that quantum mechanicalsystems will never be explained in terms of classical underlying models, will not be shockedby what they have read until now. After all, one cannot do Gedanken experiments withparticles that do not interact, and anyway, massless particles in one spacial dimension donot exhibit any dispersion, so here especially, interference experiments would be difficultto imagine. This next chapter however might make him/her frown a bit: we argue thatthe bulk region of the (super)string equations can be mapped onto a deterministic, onto-logical theory. The reason for this can be traced to the fact that string theory, in a flatbackground, is essentially just a one-space, one-time massless quantum field theory.

86

As yet, however, our (super)strings do not interact, so the string solutions will act asnon-interacting particles; for theories with interactions, go to Chapters 7 and 11.

Superstring theory started off as an apparently rather esoteric and formal approachto the question of unifying the gravitational force with other forces. The starting pointwas a dynamical system of relativistic string-like objects, subject to the rules of quantummechanics. As the earliest versions of these models were beset by anomalies, they appearedto violate Lorentz invariance, and also featured excitation modes with negative mass-squared, referred to as “tachyons”. These tachyons would have seriously destabilisedthe vacuum and for that reason had to be disposed of. It turned out however, that bycarefully choosing the total number of transverse string modes, or, the dimensions of thespace-time occupied by these strings, and then by carefully choosing the value of theintercept a(0) , which fixes the mass spectrum of the excitations, and finally by imposingsymmetry constraints on the spectrum as well, one could make the tachyons disappearand repair Lorentz invariance.[29][38] It was then found that, while most excitation modesof the string would describe objects whose rest mass would be close to the Planck scale,a very specific set of excitation modes would be massless or nearly massless. It is thesemodes that are now identified as the set of fundamental particles of the Standard Modeltogether with possible extensions of the Standard Model at mass scales that are still smallcompared to the Planck mass.

A string is a structure that is described by a sheet wiped out in space-time, the string‘world sheet’. The sheet requires two coordinates that describe it, usually called σ andτ . The coordinates occupied in an n = d+ 1 dimensional space-time, temporarily takento be Minkowski space-time, are described by the symbols Xµ(σ, τ), µ = 0, 1, · · · d .

Precise mathematical descriptions of a classical relativistic string and its quantumcounterpart are given in several excellent text books[29][38] and they will not be repeatedhere, but we give a brief summary. We emphasise, and we shall repeat doing so, that ourdescription of a superstring will not deviate from the standard picture, when discussingthe fully quantised theory. We do restrict ourselves to standard perturbative string theory,which means that we begin with a simply connected piece of world sheet, while topologi-cally non-trivial configurations occur at higher orders of the string coupling constant gs .We restrict ourselves to the case gs = 0 .

Also, we do have to restrict ourselves to a flat Minkowski background. These maywell be important restrictions, but we do have speculations concerning the back reactionof strings on their background; the graviton mode, after all, is as dictated in the stan-dard theory, and these gravitons do represent infinitesimal curvatures in the background.Strings in black hole or (anti)-de Sitter backgrounds are as yet beyond what we can do.

5.3.1. String basics

An infinitesimal segment d` of the string at fixed time, multiplied by an infinitesimal timesegment dt , defines an infinitesimal surface element dΣ = d` dt . A Lorentz invariant

87

description of dΣ is

dΣµν =∂Xµ

∂σ

∂Xν

∂τ− ∂Xν

∂σ

∂Xµ

∂τ. (5.91)

Its absolute value dΣ is then given by

± dΣ2 = 12ΣµνdΣµν = (∂σX

µ)2(∂τXν)2 − (∂σX

µ∂τXµ)2 , (5.92)

where the sign distinguishes space-like surfaces (+) from time-like ones (−) . The stringworld sheet is supposed to be time-like.

The string evolution equations are obtained by finding the extremes of the NambuGoto action,

S = −T∫

dΣ = −T∫

dσdτ√

(∂σXµ∂τXµ)2 − (∂σXµ)2(∂τXν)2 , (5.93)

where T is the string tension constant; T = 1/(2πα′) .

The light cone gauge is defined to be the coordinate frame (σ, τ) on the string worldsheet where the curves σ = const. and the curves τ = const. both represent light raysconfined to the world sheet. More precisely:

(∂σXµ)2 = (∂τX

µ)2 = 0 . (5.94)

In this gauge, the Nambu-Goto action takes the simple form

S = T (∂σXµ)(∂τX

µ) (5.95)

(the sign being chosen such that if σ and τ are both pointing in the positive timedirection, and our metric is (−,+,+,+) ), the action is negative. Imposing both lightcone conditions (5.94) is important to ensure that also the infinitesimal variations of theaction (5.95) yield the same equations as the variations of (5.93). They give:

∂2σX

µ = ∂2τ X

µ = 0 , (5.96)

but we must remember that these solutions must always be subject to the non-linearconstraint equations (5.94) as well.

The solutions to these equations are left- and right movers:

Xµ(σ, τ) = XµL(σ) +Xµ

R(τ) ; (∂σXµL)2 = 0 , (∂τX

µR)2 = 0 (5.97)

(indeed, one might decide here to rename the coordinates σ = σ+ and τ = σ− ). Wenow will leave the boundary conditions of the string free, but concentrate on the bulkproperties.

The re-parametrisation invariance on the world sheet has not yet been removed com-pletely by the gauge conditions (5.94), since we can still transform

σ → σ(σ) ; τ → τ(τ) . (5.98)

88

The σ and τ coordinates are usually fixed by using one of the space-time variables; onecan choose

X± = (X0 ±Xd)/√

2 , (5.99)

to define

σ = σ+ = X+L , τ = σ− = X+

R . (5.100)

Substituting this in the gauge condition (5.94), one finds:

∂σX+L ∂σX

−L = ∂σX

−L = 1

2

d−1∑i=1

(∂σXiL)2 , (5.101)

∂τX+R∂τX

−R = ∂τX

−R = 1

2

d−1∑i=1

(∂τXiR)2 . (5.102)

So, the longitudinal variables X± , or, X0 and Xd are both fixed in terms of the d−1 =n− 2 transverse variables X i(σ, τ) .

The boundary conditions for an open string are then

XµL(σ + `) = Xµ

L(σ) + uµ , XµR(σ) = Xµ

L(σ) , (5.103)

while for a closed string,

XµL(σ + `) = Xµ

L(σ) + uµ XµR(τ + `) = Xµ

R(τ) + uµ , (5.104)

where ` and uµ are constants. uµ is the 4-velocity. One often takes ` to be fixed, like2π , but it may be instructive to see how things depend on this free world-sheet coordinateparameter ` . One finds that, for an open string, the action over one period is17

Sopen = 12T

∫ `

0

dσ

∫ `

0

dτ ∂σXµ ∂τX

µ = 12Tu2 . (5.105)

For a particle, the action is S = pµuµ , and from that, one derives that the open string’smomentum is

pµopen = 12T uµ . (5.106)

For a closed string, the action over one period is

Sclosed = T

∫ `

0

dσ

∫ `

0

dτ ∂σXµ ∂τX

µ = Tu2 , (5.107)

and we derive that the closed string’s momentum is

pµclosed = T uµ . (5.108)

17The factor 1/2 originates from the fact that, over one period, only half the given domain is covered.Do note, however, that the string’s orientation is reversed after one period.

89

Note that the length ` of the period of the two world sheet parameters does not enterin the final expressions. This is because we have invariance under re-parametrisation ofthese world sheet coordinates.

Now, in a flat background, the quantisation is obtained by first looking at the indepen-dent variables. These are the transverse components of the fields, being X i(σ, τ) , withi = 1, 2, · · · , d − 1 . This means that these components are promoted to being quantumoperators. Everything is exactly as in Chapter 5. X i

L are the left movers, XµR are the

right movers. One subsequently postulates that X+(σ, τ) is given by the gauge fixingequation (5.100), or

X+L (σ) = σ , X+

R (τ) = τ , (5.109)

while finally the coordinate X−(σ, τ) is given by the constraint equations (5.101) and(5.102).

The theory obtained this way is manifestly invariant under rotations among the trans-verse degrees of freedom, X i(σ, τ) in coordinate space, forming the space-like groupSO(d− 1) . To see that it is also invariant under other space-like rotations, involving thedth direction, and Lorentz boosts, is less straightforward. To see what happens, one has towork out the complete operator algebra of all fields Xµ , the generators of a Lorentz trans-formation, and finally their commutation algebra. After a lengthy but straightforwardcalculation, one obtains the result that the theory is indeed Lorentz invariant providedthat certain conditions are met:

- the sequences J = a(s) of string excited modes (“Regge trajectories”) must showan intercept a(0) that must be limited to the value a(0) = 1 (for open strings),and

- the number of transverse dimensions must be 24 (for a bosonic string) or 8 (for asuperstring), so that d = 25 or 9 , and the total number of space-time dimensionsD is 26 or 10.

So, one then ends up with a completely Lorentz invariant theory. It is this theory thatwe will study, and compare with a deterministic system. As stated at the beginning ofthis section, many more aspects of this quantised relativistic string theory can be foundin the literature.

The operators X+(σ, τ) and X−(σ, τ) are needed to prove Lorentz invariance, and,in principle, they play no role in the dynamical properties of the transverse variablesX i(σ, τ) . It is the quantum states of the theory of the transverse modes that we planto compare with classical states in a deterministic theory. At the end, however, we willneed X+ and X− as well. Of these, X+ can be regarded as the independent target timevariable for the theory, without any further dynamical properties, but then X−(σ, τ) maywell give us trouble. It is not an independent variable, so it does not affect our states, butthis variable does control where in space-time our string is. We return to this question inSubsection 5.3.5.

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5.3.2. Strings on a lattice

To relate this theory to a deterministic system[75], one more step is needed: the worldsheet must be put on a lattice[30], as we saw in Section 5.1.1. How big or how smallshould we choose the meshes to be? It will be wise to choose these meshes small enough.Later, we will see how small; most importantly, most of our results will turn out to betotally independent of the mesh size alattice . This is because the dispersion propertiesof the hamiltonian (5.25) have been deliberately chosen in such a way that the latticeartefacts disappear there: the left- and right movers always go with the local speed oflight. Moreover, since we have re-parametrisation invariance on the world sheet, in steadof sending alattice to zero, we could decide to send ` to infinity. This way, we can keepalattice = 1 throughout the rest of the procedure. Remember, that the quantity ` did notenter in our final expressions for the physical properties of the string, not even if they obeyboundary conditions ensuring that we talk of open or closed strings. Thus, the physicallimit will be the limit `→∞ , for open and for closed strings.

We now proceed as in Sections 5.1.1 and 5.1.2. Assuming that he coordinates x andt used there, are related to σ and τ by18

σ = 1√2(x+ t) , τ = 1√

2(t− x) ; x = 1√

2(σ − τ) , t = 1√

2(σ + τ) , (5.110)

we find that the Nambu Goto action (5.95) amounts to d−1 copies of the two-dimensionalaction obtained by integrating the lagrangian (5.2):

L =d−1∑i−1

∂σXi ∂τX

i , (5.111)

provided the string constant T is normalised to one. (Since all d+ 1 modes of the stringevolve independently as soon as the on shell constraints (5.100) — (5.102) are obeyed,and we are now only interested in the transverse modes, we may here safely omit the 2longitudinal modes).

If our units are chosen such that T = 1 , we can use the lattice rules (5.43) and (5.44),for the transverse modes, or

X iL(x, t) = pi(x, t) +X i(x, t)−X i(x− 1, t) ; (5.112)

X iR(x, t) = pi(x, t) +X i(x, t)−X i(x+ 1, t) . (5.113)

where P i(x, t) = ∂tXi(x, t) (cf Eqs. (5.15) and (5.16), or (5.43) and (5.44)). Since these

obey the commutation rules (5.17) and (5.18), or

[X iL, X

jR] = 0 ; [X i

L(x), XjL(y)] = ± i δij if y = x± 1 , else 0 , (5.114)

[X iR(x), Xj

R(y)] = ∓ i δij if y = x± 1 , else 0 , (5.115)

18With apologies for a somewhat inconsistent treatment of the sign of time variables for the right-movers; we preferred to have τ go in the +t direction while keeping the notation of Section 5.1 for left-and right movers on the world sheet.

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we can write these operators in terms of integer-valued operators AiL,R(x) and theirassociated shift generators ηiL,R , as in Eqs. (5.56) and (5.57). There, the η basis wasused, so that the integer operators AiL,R are to be written as −i∂/∂ηiL,R . To make animportant point, let us, momentarily, replace the coefficients there by α, β, and γ :

X iL(x) = − iα ∂

∂ηiL(x)+ β

( ∂

∂ηiL(x)ϕ(ηiL)

)− γ ηiL(x− 1) , (5.116)

and similarly for X iR , and ϕ(η(x)) is the phase function introduced in Eqs. (5.58) and

(5.59).

What fixes the coefficients α, β and γ in these expressions? First, we must have theright commutation relations (5.114) and (5.115). This fixes the product α γ = 1 . Next,we insist that the operators X i

L are periodic in all variables ηiL(x) . This was why thephase function ϕ(η) was introduced. It itself is pseudo periodic, see Eq. (5.58). Exactperiodicity of X i

L requires β = 2πγ . Finally, and this is very important, we demand thatthe spectrum of values of the operators X−L,R runs smoothly from −∞ to ∞ withoutoverlaps or gaps; this fixes the ratios of the coefficients α and γ : we have α = 2πγ .Thus, we retrieve the coefficients:

α = β =√

2π ; γ = 1/√

2π . (5.117)

The reason why we emphasise the fixed values of these coefficients is that we have toconclude that, in our units, the coordinate functions X i

L,R(x, t) of the cellular automaton

are√

2π times some integers. In our units, T = 1/(2πα′) = 1 ; α′ = 1/(2π) . In arbitrarylength units, one gets that the variables X i

L,R are integer multiples of a lattice mesh lengtha , with

a =√

2π/T = 2π√α′ . (5.118)

What we find is that our classical string lives on a square lattice with mesh size a .According to the theory explained in the last few sections, the fully quantised bosonicstring is entirely equivalent to this classical string; there is a dual mapping between thetwo. The condition (5.118) on the value of the lattice parameter a is essential for thismapping. If string theoreticians can be persuaded to limit themselves to string coordinatesthat live on this lattice, they will see that the complete set of quantum states of the bosonicstring still spans the entire Hilbert space they are used to, while now all basis elements ofthis Hilbert space propagate classically, according to the discrete analogues of the classicalstring equations.

Intuitively, in the above, we embraced the lattice theory as the natural ontologicalsystem corresponding to a non-interacting string theory in Minkowski space. However, inprinciple, we could just as well have chosen the compactified theory. This theory wouldassert that the transverse degrees of freedom of the string do not live on R d−1 , but onT d−1 , a (continuous) torus in d − 1 dimensions, again with periodicity conditions overlengths 2π

√α′ , and these degrees of freedom would move about classically.

In Subsection 5.3.5 we elaborate further on the nature of the deterministic stringversions.

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5.3.3. The lowest string excitations

String theory is a quantum field theory on the 1 + 1 dimensional world sheet of thestring. If this quantum field theory is in its ground state, the corresponding string modedescribes the lightest possible particle in this theory. As soon as we put excited statesin the world sheet theory, the string goes into excited states, which means that we aredescribing heavier particles. This way, one describes the mass- or energy spectrum of thestring.

In the original versions of the theory, the lightest particle turned out to have a negativemass squared; it would behave as a tachyon, which would be an unwanted feature of atheory. All of its excited modes have zero or positive values for m2 .

The more sophisticated, modern versions of the theory are rearranged in such a waythat the tachyon mode can be declared to be unphysical, but it still acts as a descriptionof the formal string vacuum. To get the string spectrum, one starts with this unphysicaltachyon state and then creates descriptions of the other states by considering the actionof creation operators.

To relate these string modes to the ontological states at the deterministic, classicalsides of our mapping equation, we again consider the ground state, as it was described inSubsection 5.1.5, to describe the tachyon solution. Thus, the same procedure as in thatsubsection will have to be applied. Similarly one can get the physical particles by havingthe various creation operators act on the tachyon (ground) state. This way, we get ourdescription of the photon (the first spin one state of the open string) and the graviton(the spin 2 excited state of the closed string).

5.3.4. The Superstring

To construct theories containing fermions, it was proposed to plant fermionic degrees offreedom on the string world sheet. Again, anomalies were encountered, unless the bosonicand fermionic degrees of freedom can be united to form super multiplets. Each bosoniccoordinate degree of freedom Xµ(σ, τ) would have to be associated with a fermionic degreeof freedom ψµ(σ, τ) . This should be done for the left-moving modes independently of theright-moving ones. A further twist can be given to the theory by only adding fermionicmodes to the left-movers, not to the right movers (or vice versa). This way, chiralitycan be introduced in string theory, not unlike the chirality that is clearly present in theStandard Model. Such a theory is called a heterotic string theory.

Since the world sheet is strictly two-dimensional, we have no problems with spin andhelicity within the world sheet, so, here, the quantisation of fermionic fields — at least atthe level of the world sheet — is simpler than in the case of the ‘neutrinos’ discussed insubsection 3.3.3.

Earlier, we used the coordinates σ and τ as light cone coordinates on the world sheet;now, temporarily, we want to use there a space-like coordinate and a time-like one, whichwe shall call x1 = x and x0 = t .

On the world sheet, spinors are 2-dimensional rather than 4-dimensional, and we take

93

them to be hermitean operators, called Majorana fields, which we write as

ψµ∗(x, t) = ψµ(x, t) , and ψµ†(x, t) = ψµ∗ T (x, t) (5.119)

(assuming a real Minkowskian target space; the superscript T means that if ψ = (ψ1

ψ2)

then ψT = (ψ1, ψ2) ).

There are only two Dirac matrices, call them %0 and %1 , or, after a Wick rotation,%1 and %2 . They obey

%0 = i%2 , %α, %β = %α%β + %β%α = 2ηαβ , (α, β = 1, 2) . (5.120)

A useful representation is

%1 = σx =

(0 11 0

), %2 = σy =

(0 −ii 0

); %0 =

(0 1−1 0

). (5.121)

The spinor fields conjugated to the fields ψµ(x, t) are ψµ(x, t) , here defined by

ψµ

= ψµT%2 , (5.122)

Skipping a few minor steps, concerning gauge fixing, which can be found in the text booksabout superstrings, we find the fermionic part of the lagrangian:

L(x, t) = −d∑

µ=0

ψµ(x, t) %α∂α ψ

µ(x, t) . (5.123)

Since the %2 is antisymmetric and fermion fields anti-commute, two different spinorsψ and χ obey

χψ = χT%2ψ = i(−χ1ψ2 + χ2ψ1) = i(ψ2χ1 − ψ1χ2) = ψT%2χ = ψχ . (5.124)

Also we have

χ%µψ = −ψ%µχ , (5.125)

an antisymmetry that explains why the lagrangian (5.123) is not a pure derivative. TheDirac equation on the string world sheet is found to be

2∑α=1

%α∂αψµ = 0 . (5.126)

In the world sheet light cone frame, one writes

(%+∂− + %−∂+)ψ = 0 , (5.127)

where, in the representation (5.121),

%± = 1√2(%0 ± %1) , and (5.128)

%+ = −%− =√

2

(0 01 0

), %− = −%+ =

√2

(0 −10 0

). (5.129)

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The solution of Eq. (5.126) is simply

ψµ(σ, τ) =

(ψµL(σ)

ψµR(τ)

). (5.130)

Thus, one finds that the fermionic left-movers and right-movers have no further spinorindices.

As is the case for the bosonic coordinate fields XµL,R(x) , also the fermionic field com-

ponents ψµL,R have two longitudinal modes, µ = ± , that are determined by constraintequations. These equations are dictated by supersymmetry. So for the fermions also, weonly keep the d − 1 transverse components as independent dynamical fields ( d is thenumber of space-like dimensions in target space).

The second-quantised theory for such fermionic fields has already briefly been discussedin our treatment of the second-quantised ‘neutrino’ system, in Section 3.3.3. Let us repeathere how it goes for these string world sheet fermions. Again, we assume a lattice on theworld sheet, while the Dirac equation on the lattice now reduces to a finite-step equation,so chosen as to yield exactly the same solutions (5.130):

ψi(x, t) = − 1√2%0(%+ψi(x− 1, t− 1) + %−ψi(x+ 1, t− 1)

), i = 1, · · · , d− 1 . (5.131)

The deterministic counterpart is a boolean variable set si(x, t) , which we assume to betaking the values ±1 . One may write their evolution equation as

si(x, t) = si(x− 1, t− 1) si(x+ 1, t− 1) si(x, t− 2) . (5.132)

of which the solution can be written as

si(x, t) = siL(x, t) siR(r, t) , (5.133)

where siL and siR obey

siL(x, t) = siL(x+ 1, t− 1) ; siR(x, t) = siR(x− 1, t− 1) , (5.134)

which is the boolean analogue of the Dirac equation (5.131).

One can see right away that all basis elements of the Hilbert space for the Diracequation can be mapped one-to-one onto the states of our boolean variables. If we wouldstart with these states, there is a straightforward way to construct the anti-commutingfield operators ψiL,R(x, t) of our fermionic system, the Jordan-Wigner transformation[3],also alluded to in Section 3.3.3. At every allowed value of the parameter set (x, i, α) ,where α stands for L or R , we have an operator aiα(x) acting on the boolean variablesiα(x) as follows:

a|+〉 = |−〉 ; a|−〉 = 0 . (5.135)

These operators, and their hermitean conjugates a† obey the mixed commutation - anti-commutation rules

aiα(x), aiα(x) = 0 , aiα(x), ai†α (x) = I , (5.136)

[aiα(x1), ajβ(x2)] = 0 , [aiα(x1), aj†β (x2)] = 0 if

x1 6= x2

and/or i 6= jand/or α 6= β .

(5.137)

95

Turning the commutators in Eq. (5.137) into anti-commutators is easy, if one can putthe entire list of variables x, i and α in some order. Call them y and consider theordering y1 < y2 . Then, the operators ψ(y) can be defined by:

ψ(y) ≡( ∏y1<y

s(y1))a(y) . (5.138)

This turns the rules (5.136) and (5.137) into the anti-commutation rules for fermionicfields:

ψ(y1), ψ(y2) = 0 ; ψ(y1), ψ†(y2) = δ(y1, y2) . (5.139)

In terms of the original variables, the latter rule is written as

ψiα(x1), ψj†β (x2) = δ(x1 − x2)δijδaβ . (5.140)

Let us denote the left-movers L by α = 1 or β = 1 , and the right movers R by α = 2or β = 2 . Then, we choose our ordering procedure for the variables y1 = (x1, i, α) andy2 = (x2, j, β to be defined by

if α < β then y1 < y2 ;if α = β and x1 < x2 then y1 < y2 ;

if α = β and x1 = x2 and i < j then y1 < y2 ,else y1 = y2 or y1 > y2 . (5.141)

This ordering is time-independent, since all left movers are arranged before all rightmovers. Consequently, the solution (5.130) of the ‘quantum’ Dirac equation holds withoutmodifications in the Hilbert space here introduced.

It is very important to define these orderings of the fermionic fields meticulously, asin Eqs. (5.141). The sign function between brackets in Eq. (5.138), which depends on theordering, is typical for a Jordan-Wigner transformation. We find it here to be harmless,but this is not always the case. Such sign functions can be an obstruction against morecomplicated procedures one might wish to perform, such as interactions between severalfermions, between right-movers and left-movers, or in attempts to go to higher dimensions.

At this point, we may safely conclude that our dual mapping between quantised stringsand classical lattice strings continues to hold in case of the superstring.

5.3.5. Deterministic strings and the longitudinal modes

The transverse modes of the (non interacting) quantum bosonic and superstrings (in flatMinkowski space-time) could be mapped onto a deterministic theory of strings movingalong a target space lattice. How do we add the longitudinal coordinates, and how dowe check Lorentz invariance? The correct way to proceed is first to look at the quantumtheory, where these questions are answered routinely in terms of quantum operators.

Now we did have to replace the continuum of the world sheet by a lattice, but weclaim that this has no physical effect because we can choose this lattice as fine as we

96

please whereas rescaling of the world sheet has no effect on the physics since this is just acoordinate transformation on the world sheet. We do have to take the limit `→∞ butthis seems not to be any difficulty.

Let us first eliminate the effects of this lattice as much as possible. Rewrite Eqs. (5.5)and (5.6) as:

p(x, t) = ∂xk(x, t) , aL,R(x, t) = ∂xbL,R(x, t) ; (5.142)

bL(x+ t) = k(x, t) + q(x, t) ; bR(x− t) = k(x, t)− q(x, t) ; (5.143)

the new fields now obey the equal time commutation rules

[q(x, t), k(x′, t)] = 12i sgn(x− x′ ) ; (5.144)

[bL(x), bL(y)] = − [bR(x), bR(y)] = −i sgn(x− y) , (5.145)

where sgn(x) = 1 if x > 0 , sgn(x) = −1 if x < 0 and sgn(0) = 0 .

Staying with the continuum for the moment, we cannot distinguish two “adjacent”sites, so there will be no improvement when we try to replace an edge state that is singularat η(x) = ±π by one that is singular when this value is reached at two adjacent sites;in the continuum, we expect our fields to be continuous. In any case, we now dropthe attempt that gave us the expressions (5.56) and (5.57), but just accept that thereis a single edge state at every point. This means that, now, we replace these mappingequations by

bL(x) =√

2πBLop (x) +

1√2πζL(x) , (5.146)

bR(x) =√

2πBRop (x) +

1√2πζR(x) , (5.147)

where the functions BL,R will actually play the role of the integer parts of the coordinatesof the string, and ζL,R(x) are defined by their action on the integer valued functionsBL,R(x) , as follows:

eiζL(x1)|BL,R(x)〉 = |B′L,R(x)〉 ,

B′L(x) = BL(x) + θ(x− x1 ) ,B′R(x) = BR(x) ;

(5.148)

eiζR(x1)|BL,R(x)〉 = |B′′L,R(x)〉 ,

B′′L(x) = BL(x) ,B′′R(x) = BR(x) + θ(x1 − x) ,

(5.149)

so that, disregarding the edge state,

[BL(x), ζL(y)] = −iθ(x− y) , [BR(x), ζR(y)] = −iθ(y − x) . (5.150)

This gives the commutation rules (5.145). If we consider again a lattice in x space, wherethe states are given in the ζ basis, then the operator BL(x) obeying commutation rule(5.150) can be written as

BL(x1) =∑y<x1

−i ∂

∂ζL(y). (5.151)

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Now the equations of motion of the transverse string states are clear. These justseparate into left-movers and right-movers, both for the discrete lattice sites X i(σ, τ)and for the periodic ηi(σ, τ) functions, where i = 1, · · · , d − 1 . Also, the longitudinalmodes split up into left moving ones and right moving ones. These, however, are fixedby the gauge constraints. In standard string theory, we can use the light cone gauge topostulate that the coordinate variable X+ is given in an arbitrary way by the world sheetcoordinates, and one typically chooses the constraint equations (5.100).

This means that

a+L(σ) = 1 , a+

R(τ) = 1 , (5.152)

but by simple coordinate transformations σ → σ1(σ) and τ → τ1(τ) , one can chooseany other positive function of the coordinate σ (left-mover) or τ (right mover). Now,Eqs. (5.94) here mean that

(aµL(σ))2 = (aµR(τ))2 = 0 , (5.153)

so that, as in Eqs. (5.101) and (5.102), we have the constraints

a+L(σ) = 1

2

d−1∑i=1

(aiL(σ))2 , a+R(τ) = 1

2

d−1∑i=1

(aiR(t))2 , (5.154)

where aiL,R(x) = ∂xbiR,L(x) (see the definitions 5.142).

In view of Eq. (5.151) for the operator BiL(x) , it is now tempting to write for the

longitudinal coordinate X+L : ∂σB

iL(σ) = −i ∂

∂ζiL(σ), so that

∂σX+L (σ)

?= 1

2

d−1∑i=1

(− 2π ∂2

∂ζ(σ)2+

1

2π(∂σζ(σ))2 − 2i ∂

∂ζ(σ), ∂σζ(σ)

), (5.155)

but the reader may have noticed that we now disregarded the edge states, which heremay cause problems: they occur whenever the functions ηi cross the values ±π , wherewe must postulate periodicity.

We see that we do encounter problems if we want to define the longitudinal coordi-nates in the compactified classical field theory. Similarly, this is also hard in the discreteautomaton model, where we only keep the Bi

L,R as our independent ontological variables.How do we take their partial derivatives in σ and τ ?

Here, we can bring forward that the gauge conditions (5.152) may have to be replacedby Dirac delta functions, so as to reflect our choice of a world sheet lattice.

These aspects of the string models we have been considering are not finished, andcertainly not well understood. This subsection was added to demonstrate briefly whathappens if we study the gauge constraints of the theory to get some understanding ofthe longitudinal modes, in terms of the ontological states. At first sight it seemed thatthe compactified deterministic theory would offer better chances to allow us to rigorously

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derive what these modes look like; it seems as if we can replace the world sheet lattice bya continuum, but the difficulties are not entirely resolved.

If we adopt the cellular automaton based on the integers BiL,R , use of a world sheet

lattice is almost inevitable. The continuum limit has to be taken with much care.

5.3.6. Some brief remarks on (super) string interactions

As long as our (super) strings do not interact, the effects of the constraints are minor.They tell us what the coordinates X−(σ, τ) are if we know all other coordinates on theworld sheet. In the previous Section, our point was that the evolution of these coordinateson the world sheet is deterministic. Our mappings from the deterministic string statesonto the quantum string states is one-to-one, apart from the edge states that we chooseto ignore. In the text books on string theory, superstring interactions are described byallowing topologically non-trivial world sheets. In practice, this means that strings mayexchange arms when they meet at one point, or their end points may join or tear apart.All this is then controlled by a string coupling constant gs ; an expansion in powers of gsyields string world sheet diagrams with successively higher topologies.

Curiously, one may very well imagine a string interaction that is deterministic, exactlyas the bulk theory obeys deterministic equations. Since, in previous sections, we did notrefer to topological boundary conditions, we regard the deterministic description obtainedthere as a property of the string’s ‘bulk’.

A natural-looking string interaction would be obtained if we postulate the following:

Whenever two strings meet at one point on the space-time lattice, they ex-change arms, as depicted in Fig. 8.

This “law of motion” is deterministic, and unambiguous, provided that both strings areoriented strings. The deterministic version of the interaction would not involve any freelyadjustable string constant gs .

If we did not have the problem how exactly to define the longitudinal componentsof the space-time coordinates, this would complete our description of the deterministicstring laws. Now, however, we do have the problem that the longitudinal coordinatesare ‘quantum’; they are obtained from constraints that are non-linear in the other fieldsXµ(σ, τ) , each of which contain integer parts and fractional parts that do not commute.

This problem, unfortunately, is significant. It appears to imply that, in terms of the‘deterministic’ variables, we cannot exactly specify where on the world sheet the exchangedepicted in Figure 8 takes place. This difficulty has not been resolved, so as yet we cannotproduce a ‘deterministic’ model of interacting ‘quantum’ strings.

We conclude from our exercise in string theory that strings appear to admit a descrip-tion in terms of ontological objects, but just not yet quite. The most severe difficulties liein the longitudinal modes. They are needed to understand how the theory can be madeLorentz invariant. It so happens that Local Lorentz invariance is a problem for everytheory that attempts to describe the laws of nature at the Planck scale, so it should not

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Figure 8: Deterministic string interaction. This interaction takes place whenevertwo pieces of string meet at one space-time point.

come as a surprise that we have these problems as well. We suspect that today’s incom-plete understanding of Lorentz invariance at the Planck scale needs to be replaced, butit may well be that this can only be done in full harmony with the Cellular AutomatonInterpretation. What this section suggests us is that this cannot be done solely within theframework of string theory, although strings may perhaps be helpful to lead us to furtherideas.

An example of a corner of string theory that has to be swept clean is the black holeissue. Here also, strings seem to capture the physical properties of black holes partlybut not completely; as long as this is the case one should not expect us to be able toformulate a concise ontological theory. This is why most parts of this report concentrateon the general philosophy of the CAI rather than attempting to construct a completemodel.

6. Symmetries

In classical and in quantum systems, we have the Noether theorem[1]:

Whenever there is a continuous symmetry in a system, there exists a conservedquantity associated with it.

Examples are conservation of momentum (translation symmetry), conservation of energy(symmetry with respect to time translations), and angular momentum (rotation symme-try). In classical systems, the Noether theorem is limited to continuous symmetries, inquantum systems, this theorem is even more universal: here also discrete symmetries havetheir associated conservation laws: parity P = ±1 of a system or particle (mirror sym-metry), non commuting discrete quantum numbers associated with more general discretepermutations, etc. Also, in a quantum system, one can reverse the theorem:

Any conserved quantity is associated to a symmetry,

for instance isospin symmetry follows from the conservation of the isospin vector ~I =(I1, I2, I3) , baryon number conservation leads to a symmetry with respect to U(1) rota-tions of baryonic wave functions, and so on.

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6.1. Classical and quantum symmetries

We now claim that this more generalised Noether theorem can also be applied to classicalsystems, simply by attaching a basis element of Hilbert space to every state the classicalsystem can be in. If, for instance, the evolution law Ut, t+t1 is independent of time t ,we have a conserved energy. This energy is the eigen value of (the logarithm of) Ut, t+t1for the smallest admissible value of t1 . Now since an energy eigen state will usually notbe an ontological state of the system, this energy conservation law only emerges in ourquantum procedure; it does not show up in standard classical considerations. For us, thisis very important: if t1 is as small as the Planck time, the energy eigen states, all theway to the Planck energy, are superpositions of ontological states. If, as we usually do,we limit ourselves to quantum systems with much lower energies, we are singling out asection of Hilbert space that is not represented by individual ontological states, and forthis reason we should not expect recognisable classical features in the quantum systemsthat we are usually looking at: atoms, molecules, elementary particles.

Often, our deterministic models are based on a lattice rather than a space-time con-tinuum. The classical space-time symmetries on a lattice are more restricted than thoseof a continuum. It is here that our mappings onto quantum systems may help. If weallow ontological states to have symmetry relations with superimposed states, much moregeneral symmetry groups may be encountered. This is further illustrated in this chapter.

Since we often work with models having only finite amounts of data in the form of bitsand bytes in given volume elements, we are naturally led to systems defined on a lattice.There are many ways in which points can be arranged in a lattice, as is well known fromthe study of the arrangement of atoms in crystalline minerals. The symmetry propertiesof the minerals are characterised by the set of crystallographic point groups, of whichthere are 32 in three dimensions.[35]

The simplest of these is the cubic symmetry group generated by a cubic lattice:

~x = (n1, n2, n3) , (6.1)

where n1, n2 and n3 are integers. What we call the cubic group here is the set of all 48orthogonal rotations including the reflections of these three integers into ± each other (6permutations and 23 signs). This group, called O(3,Z) , is obviously much smaller thanthe group O(3,R) of all orthonormal rotations. The cubic group is a finite subgroup ofthe infinite orthogonal group.

Yet in string theory, Section 5.3.2, something peculiar seems to happen: even thoughthe string theory is equivalent to a lattice model, it nevertheless appears not to lose itsfull orthogonal rotation symmetry. How can this be explained?

6.2. Continuous transformations on a lattice

Consider a classical model whose states are defined by data that can be arranged in a d di-mensional cubic lattice. Rotation symmetry is then usually limited by the group O(d,Z) .If now we introduce our Hilbert space, such that every state of the classical system is a

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basis element of that, then we can introduce superpositions, and much more symmetrygroups are possible. There are several ways now to introduce continuous translations androtations.

To this end, it is, again, very instructive to do the Fourier transformation:

〈~x|ψ〉 = (2π)−d/2∫|κi|<π

dd~κ〈~κ|ψ〉ei~κ·~x , (6.2)

Here, |ψ〉 describes a single particle living on the lattice, but we could also take it asthe operator field of a second-quantised system, as is usual in quantum field theories. Ofcourse, as is usual in physics notation, 〈~x| are the bras in x space (where ~x is Eq. (6.1),the lattice), whereas 〈~κ| are the bras in momentum space, where ~κ are continuous, andall its components κi obey |κi| < π .

The inverse of the Fourier transform (6.2) is:

〈~κ|ψ〉 = (2π)−d/2∑~x∈Zd〈~x|ψ〉 e−i~κ·~x . (6.3)

6.2.1. Continuous translations

Translations over any distance ~a can now be defined as the operation

〈~κ|ψ〉 → 〈~κ|ψ〉 e−i~κ·~a , (6.4)

although only if ~a has integer components, this represents a shift

〈~x|ψ〉 → 〈 ~x− ~a |ψ〉 , (6.5)

since ~x must sit in the lattice, otherwise this would represent a non existing state.

If ~a has fractional components, the transformation in x space can still be defined.Take for instance a fractional value for ax , or, ~a = (ax, 0, 0) . Then

〈κx|ψ〉 → 〈κx|ψ〉 e−iκxax , 〈x|ψ〉 →∑x′

〈x′|ψ〉∆ax(x− x′) ,

∆ax(x1) = (2π)−1

∫ π

−πdκx e

−iaxκx+iκx x1 =sin π(x1 − ax)π (x1 − ax)

, (6.6)

where we also used Eq. (6.3) for the inverse of Eq. (6.2).

One easily observes that Eq. (6.6) reduces to Eq. (6.5) if ax tends to an integer.Translations over a completely arbitrary vector ~a are obtained as the product of fractionaltranslations over (ax, 0, 0) , (0, ay, 0) and (0, 0, az) :

〈~x|ψ〉 →∑

x′, y′, z′

〈~x ′|ψ〉∆~a(~x− ~x ′) , ∆~a(~x1) = ∆ax(x1)∆ay(y1)∆az(z1) . (6.7)

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Notice that the kernel function ∆~a(~x1) maximises for the values of ~x1 closest to ~a ,so, even for translations over fractional values of the components of ~a , the translationoperation involves only the components of |ψ〉 closest to the target value ~x− ~a .

The generator for infinitesimal translations is the operator ~η . Translations over afinite distance ~a can then be described by the operator ei~a·~η . Writing ~η = (ηx, ηy, ηz) ,and taking ηx, ηy, and ηz each to act only in one dimension, we have

〈~κ| ~η |ψ〉 = −~κ 〈~κ|ψ〉 , (6.8)

〈x| eiηx ax|ψ〉 =∑x′

〈x′|ψ〉 sin π(x− x′ − ax)π(x− x′ − ax)

, (6.9)

and when ax is taken to be infinitesimal, while x and x′ are integers, one finds

〈x|(I + iηxax)|ψ〉 =∑x′

〈x′|ψ〉(δxx′ + (1− δxx′)

(−1)x−x′(−πax)

π(x− x′)

); (6.10)

〈x| ηx |x〉 = 0 , 〈x| ηx |x′〉 =i (−1)x−x

′

x− x′if x 6= x′ . (6.11)

The eigen states |ηx〉 of the operator ηx can be found: 〈x|ηx〉 = e−iηxx .

The expressions we found for this generator are the most natural ones but not theonly possible choices; we must always remember that one may add multiples of 2π to itseigen values. This modifies the matrix elements (6.11) while the effects of translationsover integer distances remain the same.

An important feature of our definition of fractional translations on a lattice is theircommutation rules. These translations are entirely commutative (as we can deduce fromthe definition Eq. (6.4)):

[~η, ~η ′] = 0 . (6.12)

6.3. Continuous rotations

What can be done with translations on a lattice, can also be done for rotations, in variousways. Let us first show how to obtain a perfectly unitary general rotation operator on alattice, in principle. Again, we start from the Fourier modes, eiκx , Eq. (6.2). How do wegenerate arbitrary rotations?

Taking again the cubic lattice as our prototype, we immediately see the difficulty: thespace of allowed values for ~κ is a square (in 2 dimensions) or a cube (in 3 dimensions).This square and this cube are only invariant under the discrete rotation group O(d,Z) .Therefore, rotations over other angles can at best be approximate, it seems. We illus-trate the situation for a two-dimensional rectangular lattice, but extrapolation to higherdimensions and/or other lattice configurations is straightforward.

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6.3.1. Covering the Brillouin zone with circular regions

The space of allowed momentum values is called the Brillouin zone, and it is the square inFigure 9 a . A first approximation for a rotation of the lattice by any angle ϕ is obtainedby drawing the largest possible circle in the Brillouin zone (or the largest possible spherein the 3 or higher dimensional case) and rotate the region inside that. The data on theremainder of the Brillouin zone, outside the circle, are ignored or replaced by zero.

This procedure would clearly violate unitarity, so we must do something better withthe remainder of the Brillouin zone. This is possible, see Fig 9 b . The rotation operatorcould then be defined as follows.

π

−π

π−π

BB

BB

A

B

A

C

b)a)

Figure 9: Rotations in the Brillouin zone of a rectangular lattice. a) We canlimit ourselves to the region inside the largest circle that fits in th Brillouin zone(A). The shaded region (B) is neglected and the amplitude there replaced by zero.This is good if strongly fluctuating modes in ~x space may be ignored, such as ina photograph with a rectangular grid of pixels. b) Unitarity is restored if we alsofill the remainder of the Brillouin zone also with circles, B, C, etc., the larger thebetter (as explained in the text), but never overlapping. In the picture, the shadedregions should also be filled with circles. The rotation operator now rotates everycircle by the same angle ϕ (arrows).

We fill the entire Brillouin zone with circular regions, such that they completely coverthe entire space without overlappings. As will be explained shortly, we prefer to keepthese circles as large as possible to get the best result. The action of the rotation operatorwill now be defined to correspond to a rotation over the same angle ϕ inside all of thesecircles (arrows in Figs. 9 a and b ). With “circles” we here mean circular regions, or, ifd > 2 , regions bounded by (d− 1) -spheres.

This is the best we can do in the Brillouin zone, being the space of the Fourier vectors~κ . The reason why we split the Brillouin zones into perfectly spherical regions, ratherthan other shapes, becomes clear if we inspect the action of this operator in ~x -space:how does this operator work in the original space of the lattice sites ~x ?

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Let us first consider the action of a single circle, while the data on the rest of theBrillouin zone are replaced by zero. First take a circle (sphere) whose centre is at theorigin, and its radius is r . Projecting out this circle means that, in ~x -space, a wavefunction ψ(~x) is smeared as follows:

ψ′(~x) = (2π)−d∑~x ′

∫|~κ|<r

dd~κ ei~κ·(~x−~x′) ψ(~x ′) =

=∑~x ′

( rπ

)dKd ( r

π|~x− ~x ′|)ψ(~x ′) . (6.13)

The kernel of this rotationally symmetric expression turns out to be a Bessel function:

Kd (y) =πd−12

2dΓ(d+12

)

∫ 1

−1

dk(1− k2)d−12 eiπky = (2y)−d/2Jd/2(πy) . (6.14)

It is a smooth function, dropping off at infinity as a power of y (see Figure 10):

Kd (y) −→2 sinπ(y + 1−d

4)

π(2y)d+12

as y →∞ . (6.15)

543210-1 y

b

a

Figure 10: The function Kd (y) for d = 2 ( a ) and for d = 5 ( b ).

We infer from Eq, (6.13) that, projecting out the inside of a circle with radius r inthe Brillouin zone, implies smearing the data on the lattice over a few lattice sites inall directions, using the kernel Kd(y) . The smaller the radius r , the further out thesmearing, which is why we should try to keep our circles (spheres) as large as possible.

Next, we notice that most of the circles in Fig. 9 b are off-centre. A displacement bya vector ~κ1 in the Brillouin zone corresponds to a multiplication in configuration spaceby the exponent ei~κ1·~x . Projecting out a circle with radius r and its origin on the spot~κ1 in the Brillouin zone, means dividing the wave function ψ(~x) by the exponent ei~κ1·~x ,smearing it with the kernel Kd(y) , then multiplying with the exponent again (thus, webring the circle to the origin, project out the centralised circle, then move it back to whereit was). This amounts to smearing the original wave function with the modified kernel

Kd (y,~κ1) = Kd (y)ei~κ1·(~x−~x′) , y = r

π|~x− ~x ′| . (6.16)

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If we add the effects of all circles with which we covered the Brillouin zone, the totaleffect should be that we recover the original wave function on the lattice.

And now we can rotate. Rotating a circle (r, κ1) in the Brillouin zone over any angleϕ has exactly the same effect as 1) finding the smeared wave function using the kernelKd (y) e−i~κ1·~x

′, rotating the resulting continuous function over the angle ϕ in ~x -space,

and then multiplying with the exponential ei~κ1·~x . If we add together the effects of allcircles, we get the rotation operator. If we want the effect of an orthogonal rotation Ωin ~x -space, then this results in

ψ′(~x) =∑~x ′

∑i

Kd (riπ|Ω~x− ~x ′|) ei~κi·(~x−~x ′) ψ(~x ′) , (6.17)

where the index i counts the circles covering the Brillouin zone.

The transformations described in this subsection form a perfectly acceptable rotationgroup, converging to the usual rotations in the continuum limit. This can easily be seenby noting that the continuum case is dominated by the small values of ~κ , which are all inthe primary circle. The other circles also rotate the wave functions to the desired location,but they only move along the rapidly oscillating parts, while the vectors ~κ1 stay orientedin the original direction.

Unitarity of this operator follows from the fact that the circles cover the Brillouin zoneexactly once. We also see that these rotations are good representations of the rotationgroup,

Ω3 = Ω1Ω2 . (6.18)

Of course, the operation (6.17), to be referred to as R(Ω) , is not quite an ordinary rota-tion. If T (~a) is the translation over a non-lattice vector ~a as described in Section 6.2.1,then

R(Ω)T (~a) 6= T (Ω~a)R(Ω) , (6.19)

and furthermore, if Ω is chosen to be one of the elements of the crystal group of thelattice, R(Ω) does still not coincide with Ω itself. This latter defect can be cured, butwe won’t go into these details.

This rotation operator has one disadvantage: it will be extremely difficult to constructsome deterministic evolution law that respects this transformation as a symmetry. Forthis reason, we now consider other continuous symmetry groups, which in fact will alsoinclude rotations.

6.3.2. Rotations using the P, Q formalism

If our theory is defined on a lattice, there is another great way to recover many of thesymmetries of the continuum case, by using the PQ trick as it was exposed in Section 4.We saw that string theory, section 5.3, was re-written in such a way that the string moveson a lattice in target space, where the lattice basically describes the integer parts of the

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coordinates and the space in between the lattice sites actually correspond to the eigenstates of the displacement operators. Together, they form a continuum, and since theentire system is equivalent to the continuum string theory, it also shares all continuoustranslation and rotation symmetries with that theory.

By allowing the application of this mechanism, string theory appears to be morepowerful than theories of point particles; the commutation rules for the operators in targetspace are fundamentally different, and string theory allows target space to be in a highnumber of dimensions. In PQ theory for ordinary point particles obeying Schrodingerequations, it is much harder to recover continuous symmetries.

In PQ theory, we consider a lattice in position space Qi ∈ Zd , as in Section 5.2,where i = 1, · · · , d . In addition, we have a similar lattice Pi in momentum space. Weshowed in Section 5.2 that it is difficult to generate a model there that is deterministicon these combined lattices yet obtains a reasonably elegant shape in terms of the derivedcontinuum degrees of freedom qi or pi . When limiting oneself to the sector orthogonal tothe edge states (see Section 4 and the next Subsection, 6.3.3), one automatically achievesthe commutation rules

[qi, pj] = iδij , (6.20)

which preserves all the usual symmetry principles of the continuum.

In a deterministic theory then, we wish to identify an evolution law that respects thesesymmetries. As was shown in Section 5.2, it is not difficult to find such an evolution law,but it is difficult, or perhaps impossible, to meet the requirement of a low-energy bound inthe hamiltonian. If that difficulty could be overcome, we might end up with a deterministicsystem being equivalent to a continuum quantum theory having continuous rotation andtranslation symmetries. This has so-far only been achieved in theories without interactionssuch as the non-interacting limit of string theory and superstring theory.

6.3.3. Quantum symmetries and Noether charges

Now let us enter the Noether theorem: every continuous symmetry is associated to aconserved quantity and vice versa. For example, symmetry under time translations isassociated to the conservation of energy, translation symmetry is associated to momentumconservation, and rotation symmetry leads to the conservation of angular momentum. Werefer to these conserved quantities as Noether charges. All these conserved charges areobservable quantities, and therefore, if we wish to investigate them in a quantum theorythat we relate to a deterministic system, then that deterministic system should also exhibitobservable quantities that can be directly related to the Noether charges.

In the PQ formalism, the Noether charges for translation symmetry are built in, ina sense. Translations in the Qi variables are associated to quantities pi , of which theinteger parts Pi are ontological observables. Yet even here, we have a problem: thequantum of momentum is the inverse of the space lattice length, the unit used for theQi variables. This quantum will usually be very large. Are smaller values of momentaontologically unobservable? One might deduce from this difficulty that, if a deterministic

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theory is defined on a lattice with Planckian dimensions, the translation operator mustbe something more complicated than just a shift over one lattice unit. Or is somethingelse going on?

In the case of angular momentum, we may note that angular momentum is quantisedanyway. Can we associate an ontological observable (beable) to angular momentum? Notso easily, because angular momentum consists of non-commuting components. At bestwe will have quantised ontological variables playing the role of “the classical parts” ofangular momentum, supplemented by quantum degrees of freedom (changeables) thatrestore the commutation rules. We observe that, for small particles, angular momentumis only partly observable; sometimes it is a beable, sometimes a changeable. For largesystems, angular momentum is observable, with some margin of error.

This may lead to a powerful method to restore symmetry under continous rotationsin a lattice theory: identify angular momentum operators that are conserved when theevolution law is applied. They must be combinations of beables and changeables. Ifwe find decent candidates, featuring the correct commutation rules, then they shouldgenerate continuous rotations, through Noether’s theorem.

In the PQ formalism, we can just use the continuum definition of angular momentum.Consider the wave function of a single particle living on the product of three P, Q lattices.These lattices generate the three quantum coordinates qi . Its Hilbert space H is theproduct space of three Hilbert spaces H1, H2 and H3 .

Write, as in Eqs. (4.18) and (4.19),

qopi = Qi + aop

i , popi = 2πPi + bop

i , (6.21)

so that the angular momentum operator is (in the 3-dimensional case)

Li = εijkqopj pop

k = εijk(2πQjPk + 2πaopj Pk +Qjb

opk + aop

j bopk ) . (6.22)

If the integers Pi and Qi are assumed to be ontological (beables, which is why we omitthe superscript ‘op’) then the first term, which might be dominant, is a beable and the restare changeables. Again, we must be aware of the fact that the ‘quantum’, or ‘changeable’contributions (the next three terms) need not be very small, so that classical angularmomentum may eventually work out to be something else. So, we see here the samemenace of a difficulty as in the translational symmetries. Note, that indeed the beableparts of this angular momentum operator are quantised in multiples of 2π rather thanone, as one might have expected. Clearly, the additional terms in Eq. (6.22) should notbe neglected.

Let us now inspect the modifications on the commutation rules of these angular mo-mentum operators caused by the edge states. In each of the three Hilbert spaces Hi ,i = 1, 2, 3, , we have Eq. (4.22), while the operators of one of these Hilbert spaces com-mute with those of the others. Writing the indices explicitly:

[q1, p1] = i I2I3 ( I1 − |ψ1e〉〈ψ1

e | ) , [q1, p2] = 0 , and cyclic permutations, (6.23)

where Ii are the identity operators in the ith Hilbert space, and |ψie〉 are the edge stateson the ith P, Q lattice. One then easily derives that the three angular momentum

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operators Li defined in the usual way, Eq. (6.22), obey the commutation rules

[L1, L2] = iL3 I1I2 ( I3 − |ψ3e〉〈ψ3

e | ) , and cyclic permutations. (6.24)

The importance of this result is that now we observe that the operator L3 only acts inHilbert spaces 1 and 2, but is proportional to the identity in H3 (since L3 contains onlyq1, q2, p1, and p2 ). So the projection operator for the edge state |ψ3

e〉 commutes withL3 . This implies that, if we limit ourselves to states that are orthogonal to the edgestates, they will also rotate to states orthogonal to the edge states. In this subspace ofHilbert space the rotations act normally. And we think that this is remarkable, becausecertainly the “ontological” basis defined on the six-dimensional ~P , ~Q lattice has no built-in continuous rotation invariance at all.

These natural rotation symmetries can be readily generalised to more than three di-mensions.

6.3.4. Quantum symmetries and classical evolution

In previous sections it was observed that, when we project classical models on Hilbertspaces, new symmetries emerge. These are the symmetry transformations that map clas-sical states onto superpositions of states. A few examples were shown, the most curious ofwhich being the non-interacting, transverse sector of string and superstring theory. Wherethe classical system possesses only discrete lattice symmetries, the quantum system liveson a continuous target space and indeed exhibits all of the expected continuous transla-tion and rotation symmetries. This feature will play an important role in our discussionsof the physical interpretation of these classical – quantum mappings.

Of course we are most interested in symmetries that are symmetries of the evolutionoperator. The cogwheel model, Section 2.3, for instance has the classical symmetry ofrotations over N steps, if N is the number of cogwheel position states. But if we go tothe energy eigen states |k〉H , k = 0, · · ·N − 1 (Eqs. 2.22 and 2.23), we see that, there, atranslation over n teeth corresponds to multiplication of these states as follows:

|k〉H → e2πikn/N |k〉H . (6.25)

Since these are eigen states of the hamiltonian, this multiplication commutes with Hand hence the symmetry is preserved by the evolution law. But now we can enlarge thesymmetry group by choosing the multiplication factors

|k〉H → e2πikα/N |k〉H . (6.26)

where α now may be any real number, and this also corresponds to a translation in timeover the real number α . This enhances the symmetry group from the group of the cyclicpermutations of N elements to the group of the continuous rotations of a circle.

An other rather trivial yet interesting example is a simple cellular automaton in anynumber d of space dimensions. Consider the boolean variables σ(~x, t) = ±1 distributedover all even sites in a lattice space-time, that is, over all points (~x, t) = (x1, · · · , xd, t)with xi and t all integers, and x1 + · · ·+ xd + t = even.

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Let the evolution law be

σ(~x, t+ 1) =( d∏i=1

σ(~x+ ~ei , t) σ(~x− ~ei , t))σ(~x , t− 1) , (6.27)

where ~ei are the unit vectors in the ith direction in d dimensional space. Or: the productof the data on all direct space-time neighbours of any odd site (~x, t) is +1 . This law ismanifestly invariant under time reversal, and we see that it fixes all variables if the dataare given on a Cauchy surface consisting of two consecutive layers in time t, t− 1 . Theclassical model has the manifest translation symmetry over vectors δx = (a1, · · · , ad, τ)with

∑i ai + τ even.

Now let us introduce Hilbert space, and consider the odd lattice sites. On these oddsites, we define the action of changeables σ1(~x1, t1) as follows:

The data on the time frame t = t1 , are kept unchanged;on the time frame t = t1 − 1 , only σ(~x1, t1 − 1) changes sign, and all othersremain unchanged;consequently, according to the evolution law, also on the time frame t = t1+1 ,only σ(~x1, t1 + 1) changes sign, all others stay the same.

The reason for the notation σ1 is that in a basis of Hilbert space where σ(~x1, t1 − 1) =σ3 = ( 1 0

0 −1) our new operator σ1(~x, t1) = σ1 = (0 1

1 0) , as in the Pauli matrices.

Now, checking how the action of σ1(~x, t) propagates through the lattice, we observethat

σ1(~x, t+ 1) =( d∏i=1

σ1(~x+ ~ei , t) σ1(~x− ~ei , t))σ1(~x , t− 1) , (6.28)

where now the vector (~x, t) is even, while in Eq. (6.27) they were odd. Thus the productof the changeables σ1(~x′, t′) that are direct space-time neighbours of an even site (~x, t)is also one.

Since we recovered the same evolution law but now on the sites that before wereempty, our translation symmetry group now has twice as many elements. Together witha translation over a vector, if the vector is odd, the states in Hilbert space have to undergoa rotation, at every site:

|ψ(~x, t)〉 → U |ψ(~x, t)〉 , Uσ1U−1 = σ3 ; U = 1√

2( 1 1

1 −1) . (6.29)

Since U2 = 1 , this is actually a reflection. This means that the succession of two oddtranslations gives an even translation without further phase changes.

This simple model shows how the introduction of Hilbert space may enhance thesymmetry properties of a theory. In this case it also implies that the Brillouin zone formomentum space becomes twice as large (see Fig. 11).

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ba

κx

κy

Figure 11: a) The Brillouin zone for the lattice momentum ~κ of the ontologicalmodel described by Eq. (6.27), in two space dimensions, with data only on the evenlattice sites (smaller square, tilted by 45 ), and b) the Brillouin zone for the Hilbertspace description of this model (larger square)

6.4. Large symmetry groups in the CAI

We end this chapter with a general view of large symmetry groups, such as translations inspace and in time and the Lorentz group. They have infinite numbers of group elements.Now we imagine our automaton models to have discretised amounts of information spreadover space and time. How can we have infinite symmetry groups act on them?

Our impression from the previous results is that the conventional symmetry generatorswill be operators that always consist of combinations of beables and changeables: theNoether charges, such as angular momentum, energy and momentum, will have classicallimits that are perfectly observable, hence they are beables; yet quantum mechanically,the operators do not commute, and so there must also be changeable parts.

The beable parts will be conjugated to the tiniest symmetry operations such as verytiny translations and rotations. These are unlikely to be useful as genuine transformationsamong the ontological data – of course they are not, since they must commute with thebeables.

The changeable parts of these operators are of course not ontological observables asthey do not commute. The P Q formalism, elaborated in Section 4, is a realisation of thisconcept of splitting the operators: the integer parts here are beables, the fractional partsare changeables. The continuous translation operators pop consist of both ingredients.We suspect that this will have to become a general feature of all large symmetry groups,in particular the hamiltonian itself, and this is what we shall attempt to implement inthe next section, 7.

7. The discretised hamiltonian formalism in PQ theory

Let us now consider translation in time. This must be controlled by the deterministicevolution operator. Since, in many of our models, time is a discrete variable, the eigen-values of the time translation operator (the evolution operator) are periodic; these only

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generate energy values E quant that lie in a segment,

0 ≤ E quant < 2π/δt , (7.1)

where δt is the fundamental time quantum.

In the real world, energy is an additively conserved quantity without any periodicity.In the PQ formalism, we have seen what the best way is to cure such a situation: wemust add a conserved, discrete, integer quantum to this operator: E class = 2πN/δt , sothat we have an absolutely conserved energy,

E = E quant + E class . (7.2)

This also enables us to ensure that our system is stable, by enforcing that the classicalquantity E class is always non-negative, or at least bounded from below.

In principle, it may seem to be easy to formulate a deterministic classical system wheresuch a quantity E class can be defined, but, as we will see, there will be some obstacles ofa practical nature. Note that, if Eq. (7.2) is used to define the total energy, and if E class

reaches to infinity, then time can be redefined to be a continuous variable, since now wecan substitute any value t in the evolution operator U(t) = e−iEt .

One difficulty can be spotted right away: usually, we shall demand that energy be anextensive quantity, that is, for two widely separated systems we expect

E tot = E1 + E2 + E int , (7.3)

where E int can be expected to be small, or even negligible. But then, if both E1 andE2 are split into a classical part and a quantum part, then either the quantum part ofE tot will exceed its bounds (7.1), or E class will not be extensive, that is, it will not exactlybe the sum of the classical parts of E1 and E2 .

An other way of phrasing the problem is that one might wish to write the total energyE tot as

E tot =∑

lattice sites i

Ei →∫

dd~x H(~x) , (7.4)

where Ei or H(~x) is the energy density. It may be possible to spread E totclass over the

lattice, and it may be possible to rewrite E quant as a sum over lattice sites, but then itremains hard to see that the total quantum part stays confined to the interval [0, 2π/δt)while it is treated as an extensive variable at the same time. Can the excesses be stowedin E int ?

This question will be investigated further in our treatment of the technical details ofthe cellular automaton, Chapter 11, Section 11.4.

7.1. Conserved classical energy in PQ theory

If there is a conserved classical energy Eclass(~P , ~Q) , then the set of ~P , ~Q values withthe same total energy E forms closed surfaces ΣE . All we need to demand for a theory

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in (~P , ~Q) space is that the finite-time evolution operator U(δt) generates motion alongthese surfaces[76]. That does not sound hard, but in practice, to generate evolution lawswith this property is not so easy. This is because, at the same time, we demand that ourevolution operator U(δt) be time-reversible: there must exist an inverse, U−1(δt) .

In classical mechanics of continuous systems, the problem of characterising some evo-lution law that keeps the energy conserved was solved by the hamiltonian formalism: letthe continuous degrees of freedom be some classical real numbers qi(t), pi(t) , and takeenergy E to be some function

H(~p, ~q ) = T (~p ) + V (~q ) + ~p · ~A(~q ) , (7.5)

although more general functions that are bounded below are also admitted. The lastterm, describing typically magnetic forces, often occurs in practical examples, but maybe omitted for simplicity to follow the general argument.

Then take as our evolution law:

dqidt

= qi =∂H(~p, ~q )

∂pi, pi = −∂H(~p, ~q )

∂qi. (7.6)

One then derives

dH(~p, ~q )

dt= H =

∂H

∂qiqi +

∂H

∂pipi = piqi − qipi = 0 . (7.7)

This looks so easy in the continuous case that it may seem surprising that this principleis hard to generalise to the discrete case. Yet formally it should be easy to derive someenergy-conserving evolution law:

Take a lattice of integers Pi and Qi , and some bounded, integer energyfunction H(~P , ~Q) . Consider some number E for the total energy. Consider

all points of the surface ΣE on our lattice defined by H(~P , ~Q) = E . Thenumber of points on such a surface could be infinite, but let us take the casethat it is finite. Then simply consider a path Pi(t), Qi(t) on ΣE , where tenumerates the integers. The path must eventually close onto itself. This waywe get a closed path on ΣE . If there are points on our surface that are notyet on a closed path, we repeat the procedure, until ΣE is completely coveredby closed paths. These closed paths then define our evolution law.

At first sight, however, generalising the standard hamiltonian procedure now seems to fail.Since the standard hamiltonian formalism for the continuous case involves just infinitesi-mal time steps and infinitesimal changes in coordinates and momenta, we now need finitetime steps and finite changes. One could think of making finite-size corrections in thelattice equations, but that will not automatically work, since odds are that, after somegiven time step, integer-valued points in the surface ΣE may be difficult to find. Nowwith a little more patience, a systematic approach can be formulated, but we postpone itto Section 7.2.

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7.1.1. Multi-dimensional harmonic oscillator

A superior procedure will be discussed in the next Subsections, but first let us consider thesimpler case of the multi-dimensional harmonic oscillator of Section 5.2, Subsection 5.2.2:take two symmetric integer-valued tensors Tij = Tji , and Vij = Vji . The evolutionlaw alternates between integer and half-odd integer values of the time variable t . SeeEqs, (5.76) and (5.77):

Qi(t+ 1) = Qi(t) + TijPj(t+ 12) ; (7.8)

Pi(t+ 12) = Pi(t− 1

2)− VijQj(t) . (7.9)

According to Eqs. (5.83), (5.84), (5.87) and (5.88), the conserved hamiltonian is

H = 12Tij Pi(t+ 1

2)Pj(t− 1

2) + 1

2Vij Qi(t)Qj(t) =

12Tij Pi(t+ 1

2)Pj(t+ 1

2) + 1

2Vij Qi(t)Qj(t+ 1) =

12~P+T ~P+ + 1

2~P+TV ~Q+ 1

2~QV ~Q =

12(~P+ + 1

2~QV )T (~P+ + 1

2V ~Q) + ~Q(1

2V − 1

8V T V ) ~Q =

~P+(12T − 1

8T V T )~P+ + 1

2( ~Q+ 1

2~P+ T )V ( ~Q+ 1

2T ~P+) , (7.10)

where in the last three expressions, ~Q = ~Q(t) and ~P+ = ~P (t + 12) . The equations

(7.10) follow from the evolution equations (7.8) and (7.9) provided that T and V aresymmetric.

One reads off that this hamiltonian is time-independent. It is bounded from belowonly if not only V and T but also either V − 1

4V TV or T − 1

4TV T are bounded from

below (usually, one implies the other).

Unfortunately, this requirement is very stringent; the only solution where this energyis properly bounded is a linear or periodic chain of coupled oscillators, as in our one-dimensional model of massless bosons. On top of that, this formalism only allows forstrictly harmonic forces, which means that, unlike the continuum case, no non-linearinteractions can be accommodated for. A much larger class of models will be exhibitedin the next Subsections.

Returning first to our model of massless bosons in 1 + 1 dimensions, Section 5, wenote that the classical evolution operator was defined over time steps δt = 1 , and thismeans that, knowing the evolution operator specifies the hamiltonian eigenvalue op tomultiples of 2π . This is exactly the range of a single creation or annihilation operatoraL,R and aL,R † . But these operators can act many times, and therefore the total energyshould be allowed to stretch much further. This is where we need the exactly conserveddiscrete energy function (7.10). The fractional part of H , which we could call Equant ,follows uniquely from the evolution operator U(δt) . Then we can add multiples of 2πtimes the energy (7.10) at will. This is how the entire range of energy values of our2 dimensional boson model results from our mapping. It cannot be a coincidence thatthe angular energy function Equant together with the conserved integer valued energyfunction Eclass taken together exactly represent the spectrum of real energy values forthe quantum theory. This is how our mappings work.

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7.2. More general, integer-valued hamiltonian models with interactions

According to previous subsections, we recuperate quantum models with a continuous timevariable from a discrete classical system if not only the evolution operator over a timestep δt is time-reversible, but in addition a conserved discrete energy beable E class exists,taking values 2π N/δt where N is integer. Again, let us take δt = 1 . If the eigen valuesof U(δt) are called e−iE

quant, with 0 ≤ E quant < 2π then we can define the complete

hamiltonian H to be

H = E quant + 2πE class = 2π(ν +N) , (7.11)

where 0 ≤ ν < 1 (or alternatively, − 1/2 < ν ≤ 1/2 ) and N is integer. The quantityconjugated to that is a continuous time variable. If we furthermore demand that E class isbounded from below then Eq. (7.11) defines a genuine quantum system with a conservedhamiltonian that is bounded from below.

As stated earlier, it appears to be difficult to construct explicit, non-trivial examplesof such models. If we try to continue along the line of harmonic oscillators, perhapswith some non-harmonic forces added, it seems that the standard hamiltonian formalismfails when the time steps are finite, and if we find a hamiltonian that is conserved, it isusually not bounded from below. Such models then are unstable; they will not lead toa quantum description of a model that is stable. Now, we shall show how to cure thissituation, in principle: in the multidimensional models, we had adopted the principle thatwe alternatingly update all variables Qi , then all Pi . That has to be done differently.

To obtain better models, let us phrase our assignment as follows:We wish to formulate a discrete, classical time evolution law for some model with thefollowing properties:

i The time evolution operation must be a law that is reversible in time. Only thenwill we have an operator U(δt) that is unitary and as such can be re-written as theexponent of −i times a hermitean hamiltonian.

ii There must exist a discrete function E class depending on the dynamical variablesof the theory, that is exactly conserved in time.

iii This quantity E class must be bounded from below.

When these first three requirements are met we will be able to map this system on aquantum mechanical model that may be physically acceptable. But we want more:

iv Our model should be sufficiently generic, that is, we wish that it features interac-tions.

v Ideally, it should be possible to identify variables such as our Pi and Qi so thatwe can compare our model with systems that are known in physics, where we havethe familiar hamiltonian canonical variables ~p and ~q .

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vi We would like to have some form of locality ; as in the continuum system, ourhamiltonian should be described as the integral (or sum) of a local hamiltoniandensity, H(~x) , obeying local commutation rules: there is an ε > 0 such that atfixed time t we have

[H(~x), H(~x′)] = 0 , if |~x− ~x′| > ε . (7.12)

We thought it would be hopeless to fulfil all these requirements. Yet there exist beautifulsolutions which we now construct. Only the last condition, (vi) , will lead to (interesting)problems. Let us show how our reasoning goes.

Since we desire an integer-valued energy function that looks like the hamiltonian of acontinuum theory, we start with a hamiltonian that we like, being a continuous functionHcont(~q, ~p ) and take its integer part, when also ~p and ~q are integer. More precisely(with the appropriate factors 2π , as in Eqs. (4.6) and (6.21) in previous chapters): takePi and Qi integer and write19

E class( ~Q, ~P ) = 2πH class( ~Q, ~P ) , H class( ~Q, ~P ) = int( 12πHcont( ~Q, 2π ~P )) , (7.13)

where ‘int’ stands for the integer part, and

Qi = int(qi) , Pi = int(pi/2π) , for all i . (7.14)

This gives us a discrete, classical ‘hamiltonian function’ of the integer degrees of free-dom Pi and Qi . The index i may take a finite or an infinite number of values ( i isfinite if we discuss a finite number of particles, infinite if we consider some version of afield theory).

Soon, we shall discover that not all classical models are suitable for our construction:first of all: the oscillatory solutions must oscillate sufficiently slowly to stay visible in ourdiscrete time variable, but our restrictions will be somewhat more severe than this.

So it is easy to choose a hamiltonian, but what are the Hamilton equations? Sincewe wish to consider discrete time steps ( δt = 1 ), these have to be rephrased with somecare. As is the case in the standard hamiltonian formalism, the primary objective thatour equations of motion have to satisfy is that the function H( ~Q, ~P ) = E class must be

conserved. Unlike the standard formalism, however, the changes in the values ~Q and ~Pat the smallest possible time steps cannot be kept infinitesimal because both time t andthe variables ~Q and ~P contain integer numbers only.

The evolution equations will take the shape of a computer program. At integer timesteps with intervals δt , the evolution law will “update” the values of the integer variablesQi and Pi . Henceforth, we shall use the word “update” in this sense. The entire programfor the updating procedure is our evolution law.

19Later, in order to maintain some form of locality, we will prefer to take our ‘classical’ hamiltonian tobe the sum of many integer parts, as in Eq. (7.26), rather than the floor of the sum of local parts, as inEq. (7.13).

116

As stated at the beginning of this section, it should be easy to establish such a program:compute the total energy E of the initial state, H( ~Q(0), ~P (0) ) = E . Subsequently,

search for all other values of ( ~Q, ~P ) for which the total energy is the same number.

Together, they form a subspace ΣE of the ~Q, ~P lattice, which in general may look like asurface. Just consider the set of points in ΣE , make a mapping ( ~Q, ~P ) 7→ ( ~Q′, ~P ′) thatis one-to-one, inside ΣE . This law will be time-invertible and it will conserve the energy.Just one problem then remains: how do we choose a unique one-to-one mapping?

To achieve this, we need a strategy. Our strategy now will be that we order the valuesof the index i in some given way (actually, we will only need a cyclic ordering), andupdate the (Q,P ) pairs sequentially: first the pair (Q1, P1) , then the pair (Q2, P2) ,and so on, until we arrive at the last value of the index, after which the next sequence ofupdates is performed. Every cycle corresponds to one step δt in time. This reduces ourproblem to that of updating a single Q, P pair, such that the energy is conserved. Thisshould be doable. Therefore, let us first consider a single Q, P pair.

7.2.1. One-dimensional system: a single Q, P pair

While concentrating on a single pair, we can drop the index i . The hamiltonian will bea function of two integers, Q and P . For demonstration purposes, we restrict ourselvesto the case

H(Q,P ) = T (P ) + V (Q) + A(Q)B(P ) , (7.15)

which can be handled for fairly generic choices for the functions T (P ), V (Q), A(Q) andB(P ) . The last term here, the product AB , is the lattice generalisation of the magnetic

term ~p · ~A(~q ) in Eq. (7.5). Many interesting physical systems, such as most many bodysystems, will be covered by Eq. (7.15). It is possible to choose T (P ) = P 2 , or better:12P (P −1) , but V (Q) must be chosen to vary more slowly with Q , otherwise the system

might tend to oscillate too quickly (remember that time is discrete). Often, for sake ofsimplicity, we shall disregard the AB term.

The variables Q and P form a two-dimensional lattice. Given the energy E , thepoints on this lattice where the energy H(Q,P ) = E form a subspace ΣE . We needto define a one-to-one mapping of ΣE onto itself. However, since we have just a single(Q,P ) pair, we encounter a risk: if the integer H tends to be too large, it will oftenhappen that there are no other values of Q and P at all that have the same energy.Then, our system cannot evolve. So, we will find out that some choices of the functionH are better than others. In fact, it is not so difficult to see under what conditions thisproblem will occur, and how we can avoid it: the integer-valued hamiltonian should notvary too wildly with Q and P . What does “too wildly” mean? If, on a small subset oflattice points, a (Q, P ) pair does not move, this may not be so terrible: when embeddedin a larger system, it will move again after the other values changed. But if there are toomany values for the initial conditions where the system will remain static, we will runinto difficulties that we wish to avoid. Thus, we demand that most of the surfaces ΣE

contain more than one point on them – preferably more than two. This means that thefunctions V (Q) , T (P ), A(Q) and B(P ) should not be allowed to be too steep.

117

p

q

V(q)

T(p) +

+

+

+

+

+

−

−

−

−

−−−

−

Figure 12: The QP lattice in the 1+1 dimensional case. Constant energy contoursare here the boundaries of the differently coloured regions. Points shown in whiteare local extrema; they are not on a contour and therefore these are stable restpoints. Black points are saddle points, where two contours are seen to cross oneanother. Here, some unique evolution prescriptions must be phrased, such as: “stickto your right”, and it must be specified which of the two contours contains the blackdot. All these exceptional points are related to local minima (− ) and maxima ( + )of the functions T and V .

We then find the desired invertible mapping as follows. First, extrapolate the functionsT, P, A and B to all real values of their variables. Write real numbers q and p as

q = Q+ α , p = P + β , Q and P integer, 0 ≤ α ≤ 1 , 0 ≤ β ≤ 1 . (7.16)

Then define the continuous functions

V (q) = (1− α)V (Q) + αV (Q+ 1) , T (p) = (1− β)T (P ) + βT (P + 1) , (7.17)

and similarly A(q) and B(p) . Now, the spaces ΣE are given by the lines H(q, p) =T (p)+V (q)+A(q)B(p) = E , which are now sets of oriented, closed contours, see Fig. 12.They are of course the same closed contours as in the standard, continuum hamiltonianformalism.

The standard hamiltonian formalism would now dictate how fast our system runsalong one of these contours. We cannot quite follow that prescription here, because att = integer we wish P and Q to take integer values, that is, they have to be at one ofthe lattice sites. But the speed of the evolution does not affect the fact that energy isconserved. Therefore we modify this speed, by now postulating that

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at every time step t→ t+ δt , the system moves to the next lattice site that ison its contour ΣE .

If there is only one point on the contour, which would be the state at time t , then nothingmoves. If there are two points, the system flip-flops, and the orientation of the contouris immaterial. If there are more than two points, the system is postulated to move in thesame direction along the contour as in the standard hamiltonian formalism. In Fig. 12,we see examples of contours with just one point, and contours with two or more pointson them. Only if there is more than one point, the evolution will be non-trivial.

In some cases, there will be some ambiguity. Precisely at the lattice sites, our curveswill be non-differentiable because the functions T, C, A, and B are non-differentiablethere. This gives sone slight complications in particular when we reach extreme valuesfor both T (P ) and V (q) . If both reach a maximum or both a minimum, the contourshrinks to a point and the system cannot move. If one reaches a minimum and theother a maximum, we have a saddle point, and some extra rules must be added. Wecould demand that the contours “have to be followed to the right”, but we also haveto state which of the two contours will have to be followed if we land on such a point;also, regarding time reversal, we have to state which of the two contours has the latticepoint on it, and which just passes by. Thus, we can make the evolution law unique andreversible. See Fig. 12. The fact that there are a few (but not too many) stationary pointsis not problematic if this description is applied to formulate the law for multi-dimensionalsystems, see Section 7.2.2.

Clearly, this gives us the classical orbit in the correct temporal order, but the readermight be concerned about two things: one, what if there is only one point on our contour,the point where we started from, and two, we have the right time ordering, but do wehave the correct speed? Does this updating procedure not go too fast or too slow, whencompared to the continuum limit?

As for the first question, we will have no choice but postulating that, if there is onlyone point on a contour, that point will be at rest, our system does not evolve. Later, weshall find estimates on how many of such points one might expect.

Let us first concentrate on the second question. How fast will this updating procedurego? how long will it take, on average, to circle one contour? Well, clearly, the discreteperiod T of a contour will be equal to the number of points on a contour (with theexception of a single point, where things do not move20). How many points do we expectto find on one contour?

Consider now a small region on the (Q,P ) lattice, where the hamiltonian H class

approximately linearises:

H class ≈ aP + bQ+ C , (7.18)

with small corrections that ensure that H class is an integer on all lattice points. With alittle bit of geometry, one finds a tilted square with sides of length ε

√a2 + b2 , where the

20But we can also say that, in that case, the period is δt , the time between two updates.

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P

Q

(Q,P) = ε(b,a)

H = C

H = C

H = C + K

H = C + K

(Q,P) = ε(−a,b)

Figure 13: A small region in the QP lattice where the (integer valued) hamiltonianis reasonably smooth. See Eq. (7.18). The sides of the tilted square are ε

√a2 + b2 .

Contours of approximately constant H values are indicated.

values of H class vary between values C and C+K , with K = ε(a2 +b2) . Assuming thatall these integers occur at about the same rate, we find that the total number of latticesites inside the square is ε2(a2 + b2) , and since there are K contours, every contour has,on average,

ε2(a2 + b2)/K = ε (7.19)

points on it. The lengths of the contours in Figure 13 is ε√a2 + b2 , so that, on average,

the distance between two points on a contour is√a2 + b2 .

This little calculation shows that, in the continuum limit, the propagation speed ofour updating procedure will be√(δq

δt

)2

+(δpδt

)2

=

√(∂H∂p

)2

+(∂H∂q

)2

, (7.20)

completely in accordance with the standard Hamilton equations! (Note that the factors2π in Eqs. (7.13) and (7.14) cancel out)

One concludes that our updating procedure exactly leads to the correct continuumlimit. However, the hamiltonian must be sufficiently smooth so as to have more than onepoint on a contour. We now know that this must mean that the continuous motion inthe continuum limit cannot be allowed to be too rapid. We expect that, on the discretelattice, the distance between consecutive lattice points on a contour may vary erratically,so that the motion will continue with a variable speed. In the continuum limit, this mustaverage out to a smooth motion, completely in accordance with the standard Hamiltonequations.

Returning to the question of the contours with only one point on them, we expecttheir total lengths, on average, to be such that their classical periods would correspondto a single time unit δt . These periods will be too fast to monitor on our discrete timescale.

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This completes our brief analysis of the 1+1 dimensional case. We found an evolutionlaw that exactly preserves the discrete energy function chosen. The procedure is unique assoon as the energy function can be extended naturally to a continuous function betweenthe lattice sites, as was realised in the case H = T + V +AB in Eq. (7.17). Furthermorewe must require that the energy function does not vary too steeply, so that most of theclosed contours contain more than one lattice point.

An interesting test case is the choice

T (P ) = 12P (P − 1) ; V (Q) = 1

2Q(Q− 1) , (7.21)

This is a discretised harmonic oscillator whose period is not exactly constant, but thisone is easier to generalise to higher dimensions than the oscillator described in Section 5.2and Subsection 7.1.1.

7.2.2. The multi dimensional case

A single particle in 1 space- and 1 time dimension, as described in the previous section,is rather boring, since the motion occurs on contours that all have rather short periods(indeed, in the harmonic oscillator, where both T and V are quadratic functions of theirvariables, the period will stay close to the fundamental time step δt itself). In higherdimensions (and in multi component oscillators, particularly when they have non-linearinteractions), this will be very different. So now, we consider the variables Qi, Pi, i =

1, · · · , n . Again, we postulate a hamiltonian H( ~Q, ~P ) that, when Pi and Qi are integer,takes integer values only. Again, let us take the case that

H( ~Q, ~P ) = T (~P ) + V ( ~Q ) + A( ~Q )B(~P ) . (7.22)

To describe an energy conserving evolution law, we simply can apply the procedure de-scribed in the previous section n times for each time step. For a unique descriptionhowever, it is now mandatory that we introduce a cyclic ordering for the values 1, · · · , nthat the index i can take. Naturally, we adopt the notation of the values for the index ito whatever ordering might have been chosen:

1 < 2 < · · · < n < 1 · · · . (7.23)

We do emphasise that the procedure described next depends on this ordering.

Let Ui be our notation for the operation in one dimension, acting on the variablesQi, Pi at one given value for the index i . Thus, Ui maps (Pi, Qi) 7→ (P ′i , Q

′i) using the

procedure of Section 7.2.1 with the hamiltonian (7.22), simply keeping all other variablesQj, Pj, j 6= i fixed. By construction, Ui has an inverse U−1

i . Now, it is simple to producea prescription for the evolution U for the entire system, for a single time step δt = 1 :

U(δt) = Un Un−1 · · · U1 . (7.24)

where we intend to use the physical notation: U1 acts first, then U2 , etc., although theopposite order can also be taken. Note, that we have some parity violation: the operators

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Ui and Uj will not commute if i 6= j , and therefore the resulting operator U is not quitethe same as the one obtained when the order is reversed.

Time inversion gives:

U(−δt) = U(δt)−1 = U−11 U−1

2 · · ·U(n)−1 . (7.25)

Finally, the exchange Ui ↔ U−1i might be associated with “particle-antiparticle conju-

gation”, C . Then the product P (parity) T (time inversion) C (conjugation) may stillbe a good symmetry. In the real world, this might lead to a natural explanation of CPTsymmetry, while P , T , or CP are not respected.

7.2.3. Discrete field theories

An important example of an infinite-dimensional (Qi, Pi) system is a local field theory.Suppose that the index i is replaced by a lattice coordinate ~x , plus possibly other indicesj labelling species of fields. Let us rename the variables (Φj(~x ), Pj(~x )) , where Φj arecanonical fields and Pj are their momentum variables (often, in the continuum theory,ddt

Φj ). Now assume that the hamiltonian of the entire system is the sum of local terms:

H int =∑~x

H int(~x ) , H int(~x ) = V (~Φ(~x ), ~Φ(~x ′)) + T (~P (~x )) , (7.26)

where the coordinates ~x ′ are limited to neighbours of ~x only, and all functions V andT are integers. This would be a typical discretisation of a (classical or quantum) fieldtheory (ignoring, for simplicity, magnetic terms).

We can apply our multi-dimensional, discrete hamiltonian equations to this case, butthere is one important thing to remember: where in the previous Sections we statedthat the indices i must by cyclically ordered, this now means that, in the field theory ofEq. (7.26), not only the indices i but also the coordinates ~x must be (cyclically) ordered.The danger of this is that the functions Vi(~x ) also refer to neighbours, and, consequently,the evolution step defined at point ~x affects the evolution at its neighbouring points~x ′ , or: [U(~x ), U(~x ′ )] 6= 0 . Performing the updates in the order of the values of thecoordinates ~x , might therefore produce signals that move much faster than light, possiblygenerating instantaneous non local effects across the entire system over a single time stept→ t+ δt . This we need to avoid, and doing this happens to be easy:

The evolution equations (e.o.m.) of the entire system over one time step δt ,are obtained by ordering the coordinates as follows: first update all evenlattice sites, then update all odd lattice sites.

Since the U operators generated by Hi(~x ) do commute with the evolution operatorsU(~x ′ ) when ~x and ~x ′ are both on an even site or both on an odd site of the lattice, thisordering does not pass on signals beyond two lattice links. Moreover, there is anotherhuge advantage of this law: the order in which the individual even sites of the lattice areupdated is now immaterial, and the same for the set of all odd sites.

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Thus, we obtained a cellular automaton whose evolution law is of the type

U = AB , A =∏

~x= even

A(~x) , B =∏

~y= odd

B(~y) , (7.27)

where the order inside the products over the sites ~x and ~y is immaterial, except thatA(~x) and B(~y) do not commute when ~x and ~y are direct neighbours. Such automataare interesting objects to be studied, see chapter 11.

7.3. From the integer valued to the quantum hamiltonian

A deterministic system obeying a discrete hamiltonian formalism as described in theprevious sections is of particular interest when we map it onto a quantum system followingthe program discussed in this report. This is because we here have two different operatorsthat both play the role of energy: we have the integer valued, discrete hamiltonian H int

that generates the classical equations of motion, and we have the angular, or fractionalvalued hamiltonian Hfract , defined from the eigen states and eigen values of the one-timestep evolution operator U(δt) :

U(δt) = e−iHfract , 0 ≤ Hfract < 2π . (7.28)

We can now uniquely define a total hamiltonian that is a real number operator, by

H = 2πH int +Hfract . (7.29)

The bounds imposed in Eq. (7.28) are important to keep in mind, since Hfract , as defined,is strictly periodic. The hamiltonian (7.29) now defines a quantum theory.

The operator Hfract can be calculated from U(δt) as follows. By Fourier transforma-tions, one easily derives that, if −π < x < π ,

x = 2∞∑n=1

(−1)n−1 sin(nx)

n, (7.30)

Let us, for a moment, write Hfract = ω = x+ π . Then Eq. (7.30) implies

ω = π − 2∞∑n=1

sin(nω)

n, (7.31)

so that

Hfract = π +∞∑n−1

i

n(U(n δt)− U(−n δt) ) . (7.32)

This sum converges nearly everywhere, but it is not quite local, since the evolutionoperator over n steps in time, also acts over n steps in space. Both H int and Hfract are

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uniquely defined, and since Hfract is bound to an interval while H int is bounded frombelow, also H is bounded from below.

The one state where the series (7.32) does not converge to the right value is the caseω = 0 , the lowest energy state, being the vacuum state. We recognise this as an edgestate, because it is at the point where the energy eigen value of Hfract jumps from 2π to0.

Note that demanding a large number of low energy states near the vacuum (the absenceof a large mass gap) implies that U(n δt) be non-trivial in the H int = 0 sector. This isoften not the case in the models described in Subsection 7.2.2, but in principle there isno reason why such models should not exist also. In fact, some of the cellular automatonmodels discussed later in Chapter 11 have no manifest conserved H int , so that all theirstates can be regarded as sitting in the H int = 0 sector of the theory.

Because of the non-locality of Eq. (7.32), the hamiltonian (7.32) does not obey the rulevi , with commutation rule (7.12), but if U(δt) is the product of local evolution operators,the evolution over integer time steps n δt is local, so the theory can be claimed to obeylocality, as long as we refrain from defining its states at time t when t is not an integer.

If we could claim that in the physically relevant sector of Hilbert space the sum (7.32)converges rapidly, we could argue that the ε defined in condition iii can be kept small.However, as we shall show now, the sum does not converge rapidly everywhere in Hilbertspace. We are interested in the hamiltonian as it acts on states very close to the vacuum,in our notation: H int = 0, Hfract = ω , where 0 < ω π . Suppose then that weintroduce a cut-off in the sum (7.30) and (7.32) by multiplying the summand with e−n/R ,where R is also the range of non-locality of the last significant terms of the sum.

The sum with cut-off can be evaluated exactly. For large values of the cut-off R :

ω = π − 2∞∑n=1

sin(nω)e−n/R

n= 2 arctan

(e1/R − cosω

sinω

)= 2 arctan

(1− cosω + 1/Rsinω

)= 2 arctan

( sinω

1 + cosω+

1/Rsinω

), (7.33)

where we replaced e1/R by 1 + 1/R since R is large and arbitrary.

Writing sinα1+cosα

= tan 12α , we see that the approximation becomes exact in the limit

R→∞ . We are interested in the states close to the vacuum, having a small but positiveenergy H = α . Then, at finite R , the cut-off at R replaces the hamiltonian Hfract by

Hfract → Hfract +2

RHfract

, (7.34)

and this is only acceptable if

RMPl/〈Hfract〉2 . (7.35)

Here, MPl is the “Planck mass”, or whatever the inverse is of the elementary time scalein the model. This cut-off radius R must therefore be chosen to be very large, so that,

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indeed, the exact quantum description of our local model generates non-locality in thehamiltonian.

Let us note that better cut-off procedures can be formulated that lead to smallererrors in the hamiltonian eigen values, but, quite generally, the cut-off R must stay largecompared to the inverse of the eigenvalues to be computed. So, it will not be easy torephrase the theory so that the hamiltonian becomes local again. This will have to bean important point in our discussions of the “interpretation” of the resulting quantummodel.

The sum in our definition of the Hfract in Eq. (7.32) converges, but not in the absolute,and precisely for the vacuum state it does not converge. We can do something about this.Consider the Fourier sum

|ω| = π

2− 4

π

∞∑k=0

1

(2k + 1)2cos(2k + 1)ω , −π < ω < π . (7.36)

which implies that we can find an expression for |Hfract| :

|Hfract| =π

2− 2

π

∞∑k=0

1

(2k + 1)2(U((2k + 1)δt) + U(−(2k + 1)δt)) , (7.37)

if |Hfract| < π2

. The reason why this converges faster is the the discontinuity is not inthe function but now in the derivative of the function that is Fourier transformed. Theconvergence is now absolute.

If now we decide to terminate the series after R terms, the error will be much smaller.How large do we now have to take R ? A somewhat tedious calculation gives that if wemake an exponential cutoff with radius R , replacing the sum (7.37) by

f(Hfract, R) =π

2− 2

π

∞∑k=0

1

(2k + 1)2(U(2k + 1)δt) + U(−(2k + 1)δt))e−(2k+1)/R , (7.38)

then the expression for |ω| is smoothened into

f(ω, R)→ 2ω

πarctan(Rω)− 1

πRlog(ω2 + 1/R2

4 e2

)+O(ω2/R, R−3) . (7.39)

This means that the correct hamiltonian is obtained as soon as RHfract 1 , which ismuch better than the condition (7.35) for the sum (7.32).

The fact that we now obtained only |Hfract| in stead of Hfract is somewhat peculiar. Itmeans that we replaced the hamiltonian for a sector of Hilbert space with very high energy(it should have been the domain where π ≤ Hfract < 2π ) by one where the hamiltonian isidentical to the one in the lower energy sector. Hilbert space has doubled! Actually, thisseems not to be harmful for the theory; it is as if a single spin 1

2particle with momentum

zero has been overlooked. As soon as we apply the super selection rule that the lowerenergy states are selected out, then (7.38) can be used.

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Selecting out the low energy sector corresponds to the application of a projectionoperator. This projection operator is obtained from the equation

θ(ω) = 12

+2

π

∞∑k=0

sin(2k + 1)ω

2k + 1, −π < ω < π . (7.40)

and, since the function is discontinuous, it again converges slowly. Here, this might begood. It means that, locally, no harm is done if Hfract is replaced by |Hfract| since thedistinction requires non-local physics.

In this section, we conclude that the hamiltonian can be expressed in terms of localterms, but we need to include the operators U(±∆t) where ∆t is large compared to theinverse of the hamiltonian we wish to calculate. These will develop non localities thatare still serious. This is still an obstacle against the construction of a local hamiltoniandensity. As yet, therefore, condition vi is not yet fulfilled.

8. Standard interpretations of quantum mechanics

This report will not include an exhaustive discussion of all proposed interpretations ofwhat quantum mechanics actually is, but we cannot avoid such discussions completely.Let us briefly list what is still generally accepted as viable attitudes towards quantummechanics. This list is surely quite incomplete. Most existing approaches have someweakness somewhere, and below we indicate how the cellular Automaton interpretationmight address some of these.

8.1. The Copenhagen interpretation

The Copenhagen interpretation slowly took shape in the late 1920s in discussions betweenNiels Bohr, Werner Heisenberg and Wolfgang Pauli[33]. Historically, of course, it tooktime to attain its modern appearance. In the early days, physicists were still strugglingwith the equations and the technical difficulties. Today, we know precisely how to handleall these, so that now we can rephrase the original stand points much more accurately.Originally, quantum mechanics was formulated in terms of wave functions, with whichone referred to the states electrons are in, which, ignoring spin for a moment, were thefunctions ψ(~x, t) = 〈~x |ψ(t)〉 . Now, we may still use the words ‘wave function’ when wereally mean to talk of ket states in more general terms.

Leaving aside who said exactly what in the 1920s, here is what we now call theCopenhagen interpretation. Somewhat anachronistically, we employ Dirac’s notation:

A system is completely described by its ‘wave function’ |ψ(t)〉 , which is anelement of Hilbert space, and any basis in Hilbert space can be used for itsdescription. This wave function obeys a linear first order differential equationin time, to be referred to as Schrodinger’s equation, of which the exact formcan be determined by repeated experiments.

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A measurement can be made using any observable O that one might wantto choose (observables are hermitean operators in Hilbert space). The theorythen predicts the average measured value of O , after many repetitions of theexperiment, to be

〈O〉 = 〈ψ(t)|O|ψ(t)〉 . (8.1)

As soon as the measurement is made, the wave function of the system col-lapses to a state in the subspace of Hilbert space that is an eigen state ofthe observable O , or a probabilistic distribution of eigen states, according toEq. (8.1).When two observables O1 and O2 do not commute, then they cannot bothbe measured accurately. The commutator [O1, O2] indicates how large theproduct of the ‘uncertainties’ δO1 and δO2 should be expected to be.The measuring device itself must be regarded as a classical object, and forlarge systems the quantum mechanical measurement approaches closely theclassical description.

Implicitly included in Eq. (8.1) is the element of probability. If we expand the wavefunction |ψ〉 into eigen states |ϕ〉 of an observable O , then we find that the probabilitythat the experiment on |ψ〉 actually gives as a result that the eigenvalue of the state |ϕ〉is found, will be given by P = |〈ϕ|ψ〉|2 . This is referred to as Born’s probability rule.

An important element in the Copenhagen interpretation is that one may only askwhat the outcome of an experiment will be. In particular, it is forbidden to ask: whatis it that is actually happening? It is exactly the latter question that sparks endlessdiscussions; the important point made by the Copenhagen group is that such questionsare unnecessary. If one knows the Schrodinger equation, one knows everything needed topredict the outcomes of an experiment, no further questions should be asked.

This is a strong point of the Copenhagen interpretation, but it is also a weak one. Ifwe know the Schrodinger equation, we know everything, but what if we do not yet knowthe Schrodinger equation? How does one arrive at the correct equation? In particular,how do we arrive at the correct equation if the gravitational force is involved? Gravityhas been a major focus point of the last 30 years and more in elementary particle theoryand the theory of space and time. Numerous wild guesses have been made. In particular,(super)string theory has made huge advances. Yet no convincing model that unifies gravitywith the other forces has been constructed; models proposed so-far have not been able toexplain, let alone predict, the values of the fundamental constants of Nature, including themasses of many fundamental particles, the fine structure constant, and the cosmologicalconstant. And here it is, according to the author’s opinion, that we do have to ask: Whatis it, or what could it be, that is actually going on?

The Copenhagen approach to quantum theory was strengthened considerably by thecontributions given by Schrodinger, Heisenberg, Dirac, and others. It became clear howa Schrodinger equation can be obtained if the classical limit is known:

If a classical system is described by the (continuum) Hamilton equations, thismeans that we have classical variables pi and qi , for which one can define

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Poisson brackets : any pair of observables A(~q, ~p ) and B(~q, ~p ) , allow for theconstruction of an observable variable called A,B defined by

A,B ≡∑i

(∂A∂qi

∂B

∂pi− ∂A

∂pi

∂B

∂qi

), (8.2)

in particular:

qi, pj = −pi, qj = δij , qi, qj = pi, pj = 0 . (8.3)

A quantum theory is obtained, if we replace the Poisson brackets by commu-tators, which involves a factor i and new constant of Nature, ~ . Basically:

[A,B]→ i~A,B . (8.4)

In the limit ~→ 0 , this quantum theory reproduces the classical system. This very pow-erful procedure allows us more often than not to bypass the question “what is going on?”.We have a theory, we know where to search for the appropriate Schrodinger equation, andwe know how to recover the classical limit.

Unfortunately, in the case of the gravitational force, this ‘trick’ is not good enough togive us ‘quantum gravity’. The problem with gravity is not just that the gravitational forceappears not to be renormalisable, or that it is difficult to define the quantum versions ofspace- and time coordinates, and the physical aspects of non-trivial space-time topologies;some authors attempt to address these problems as merely technical ones, which can behandled by using some tricks. The real problem is that space-time curvature runs out ofcontrol at the Planck scale. We will be forced to turn to a different book keeping systemfor Nature’s physical degrees of freedom there.

A promising approach was to employ local conformal symmetry[77] as a more funda-mental principle than usually thought; this could be a way to make distance and timescales relative, so that what was dubbed as ‘small distances’ ceases to have an absolutemeaning. But such theories also need further polishing, and they too could eventuallyrequire a Cellular Automaton interpretation of the quantum features that they will haveto include.

8.2. Einstein’s view

As is well-known, Einstein was uncomfortable with the Copenhagen interpretation. Hedid ask further questions. Can quantum-mechanical description of physical reality be con-sidered complete? [4], or, does the theory tell us everything we might want to know aboutwhat is going on? In the Einstein-Podolsky-Rosen discussion of a Gedanken experiment,two particles (photons, for instance), are created in a state

x1 − x2 = 0 , p1 + p2 = 0 . (8.5)

Since [x1−x2, p1 +p2] = i− i = 0 , both equations in (8.5) can be simultaneously sharplyimposed.

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What bothered Einstein, Podolsky and Rosen was that, long after the two particlesceased to interact, an observer of particle # 2 might decide either to measure its momen-tum p2 , after which we know for sure the momentum p1 of particle # 1, or its positionx2 , after which we would know for sure the position x1 of particle #1. How can sucha particle be described by a quantum mechanical wave function at all? Apparently, themeasurement at particle # 2 affected the state of particle #1, but how could that havehappened?

In modern terminology, however, we would have said that the measurements proposedin this Gedanken experiment would have disturbed the wave function of the entangledparticles. The measurements on particle # 2 affects the probability distributions forparticle # 1, which in no way should be considered as the effect of a spooky signal fromone system to the other.

In any case, even Einstein, Podolsky and Rosen had no difficulty in computing thequantum mechanical probabilities for the outcomes of the measurements, so that, inprinciple, quantum mechanics emerged unharmed out of this sequence of arguments.

Einstein had difficulties with the relativistic invariance of quantum mechanics (“doesthe spooky information transmitted by these particles go faster than light?”). These,however, are now seen as technical difficulties that have been resolved. Within Copen-hagen’s interpretation, one has to observe that the transmission of information only worksover a distance, if we can identify operators A at position x1 and operators B at po-sition x2 that do not commute: [A,B] 6= 0 . We now understand that, in elementaryparticle theory, all space-like separated observables mutually commute, which precludesany signalling faster than light. It is a built-in feature of the Standard Model, to whichit actually owes much of its success.

So, with the technical difficulties out of the way, we are left with Einstein’s moreessential objections against quantum mechanics: it is a probabilistic theory that does nottell us what actually is going on. As such, it even requires us to modify our (“classical”)sense of logic. This he found unacceptable, and here he may have been right. A theorymust be found that does not force us to redefine logical reasoning. According to thepresent author, the Einstein-Bohr debate is not over.

What Einstein and Bohr did seem to agree about is the importance of the role of anobserver. Indeed, this was the important lesson learned in the 20th century: if somethingcannot be observed, it may not be a well-defined concept – it may even not exist at all. Wehave to limit ourselves to observable features of a theory. It is an important ingredientof our present work that we propose to part from this doctrine, to some extent: Thingsthat are not directly observable may still exist and as such play a decisive role in theobservable properties of an object.

Indeed, there are big problems with the dictum that everything we talk about must beobservable. While observing microscopic objects, an observer may disturb them, even ina classical theory; moreover, in gravity theories, observers may carry gravitational fieldsthat disturb the system they are looking at, so we cannot afford to make an observerinfinitely heavy (carrying large bags full of “data”, whose sheer weight gravitationally

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disturbs the environment), but also not infinitely light (light particles do not transmitlarge amounts of data at all), while, if the mass of an observer would be “somewhere inbetween”, this could entail that our theory will be inaccurate from its very inception.

An interesting blow was given to the doctrine that observability should be central,when quark theory was proposed. Quarks cannot be isolated to be observed individu-ally, and for that reason the idea that quarks would be physical particles was attacked.Fortunately, in this case the theoretical coherence of the evidence in favour of the quarksbecame so overwhelming, and experimental methods for observing them, even while theyare not entirely separated, improved so much, that all doubts evaporated.

In short, we propose to return to classical logic and build models. These modelsdescribe the evolution of large sets of data, which eventually may bring about classicalphenomena that we can observe. The fact that these data themselves cannot be directlyobserved, and that our experiments will provide nothing but statistical information, in-cluding fluctuations and uncertainties, can be fully explained within the settings of themodels; if the observer takes no longer part in the definition of physical degrees of free-dom and their values, then his or her limited abilities will no longer stand in the way ofaccurate formalisms.

We suspect that this view is closer to Einstein’s than it can be to Bohr, but, in asense, neither of them would fully agree. We do not claim the wisdom that our view isobviously superior, but rather advocate that one should try to follow such paths, andlearn from our successes and failures.

8.3. The Many Worlds interpretation

The ‘many worlds interpretation’ was originally conceived by H. Everett[8], and it wasstrongly supported by B.S. DeWitt[14]. Many other physicists, who wanted to do awaywith the difficulties of quantum mechanics, arrived at similar conclusions.

Rather than saying that values are ‘actually taken’ by the physical observables, oneattaches more essential realistic values to the wave functions. The wave functions obey‘classical’ equations such as the Schrodinger equation, so these are real. What theydescribe, however, is not a single world of particles in a 3+1 dimensional space-time, butinfinitely many different worlds.

All these worlds are considered to be equally real, with the only distinction that someare more probable than others. In a sense, this interpretation may be deemed to bedeterministic: The space of all possible worlds is the space in which |ψ(t)〉 takes values,and these values are controlled by an unambiguous equation, the Schrodinger equation.The adherents emphasise that this approach does not require the assumption that a wavefunction ‘collapses’ at any time; only our awareness of the universe we are in automaticallyadjusts to the branch that we are following.

The price one pays is the denial that there would be a single ‘real’ universe, and thatall other options would be virtual. Instead, we have a huge infinity of universes thatall are realised simultaneously. We only consciously experience one of this large class of

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possibilities.

This interpretation is accepted by a sizeable fraction of the researchers as it is prag-matic and it gives us peace of mind as to what we are actually talking about when we doquantum mechanical calculations. Indeed, when we do such calculations, we work on thewave function, and it lives in the world of all particle configurations, called Fock space.That is a universe of all universes. This being the case, one may be tempted to acceptthe fact that all these different universes “exist”.

Nevertheless, these views are also criticised. How often does branching take place?The only reasonable answer to this question would be: branching takes place at timescales that are appropriate for the quantum world, but what is that time scale? We couldassume it to be the Planck scale, once every 10−44 seconds, but one could bring forwardthat the highest frequencies in the quantum world should correspond to the highest energyor mass scales. Since the total mass of all matter (including dark energy) in the universeis some 1023 kg per cubic light year, this frequency would be some 1074 per second percubic light year.

Rather than looking at the frequency of the switches to different worlds, we couldtry to estimate how many different worlds there actually would be. Taking into accountthat there may be something like 1080 particles in our Universe, each of which couldhave been at different positions than where we see them, this may yield numbers such as1010100 different worlds. This gigantic number is sometimes called “googolplex”. It doesindicate that the many world picture may be regarded as extremely inefficient: it is onlyone universe that we wish to describe, not a googolplex of them.

A second objection could be that this interpretation does not add any further physicalinsights; it would not help us understand how the Schrodinger equation could follow frommore basic physics. It is a picture that has no further consequences. Attempts to ‘test’the Many World Interpretation would be tantamount to testing quantum mechanics itself.

Most importantly, in line with the work presented here, we could object that thebranching phenomenon would depend crucially on the choice of basis. In non-relativisticparticle physics, one might say: but of course, we choose the basis of the particle positions|~x 〉 , but it would be less clear what to do in a relativistic theory. Do we take the par-ticles and the holes left by the antiparticles as a basis, or do we pick the fields |Φ(~x )〉 ?In the present work a lesson was learnt: a lot depends on our choice of basis, and wedo not know which basis should be taken (if any) to be used in a conceivable underlyingontological theory.

The Many World Interpretation evaporates completely if one can find an ontologicalbasis, which is one where the Schrodinger equation takes the form of a permutator ofstates. The universe then does not branch anymore. In stead of a practically uncountablenumber of distinct worlds, the interpretation advocated in this report assumes only theexistence of one single world. That one world is not very different from the universethat you find described in undergraduate science books, except for one thing: we havenot (yet?) been able to describe its constituents and evolution laws. The templates (seeSection 9.3.1) we presently use to describe it are wanting; it is these templates that need

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to be superimposed to obtain a picture of the one world, not the other way around.

8.4. The de Broglie Bohm pilot waves

David Bohm’s approach[6] to de Broglie’s pilot waves[2] is often portrayed as a determin-istic but non-local hidden variable theory. As we shall argue, both these adjectives canbe disputed, and actually the picture created in this theory appears to be quite similarto the Many World hypothesis.

The theory appears best when we first consider its formulation for a single, non-relativistic particle. Let ψ(~x, t) = 〈~x|ψ(t)〉 be its wave function. It obeys Schrodinger’sequation,

∂ψ(~x, t)

∂ t= −iHψ(~x, t) =

i

2m~∂ 2ψ(~x, t)− iV (~x)ψ(~x, t)− ~A(~x)~∂ψ(~x, t) , (8.6)

where the last term describes an external magnetic field, with the electric charge of theparticle absorbed in ~A . Furthermore, ~∂ stands short for ∂/∂~x .

Born’s probability interpretation of the wave function is adopted, so we take the non-negative function %(~x, t) = ψ∗(~x)ψ(~x) as the probability distribution of the particle.Now, because of unitarity, the integral∫

d~x %(~x) = 〈ψ|ψ〉 = 1 (8.7)

is conserved in time. Since this must be a consequence of Schrodinger’s equation (8.6),there must exist a local vector quantity ~j(~x, t) such that

∂ %(~x, t)

∂ t= −~∂ ·~j(~x, t) , (8.8)

and indeed, ~j is readily identified to be

~j(~x, t) =i

2m((~∂ ψ∗)ψ − ψ∗~∂ ψ) + ~Aψ∗ψ . (8.9)

Suppose now that the wave function ψ(~x, t) steers a particle around in such a waythat the velocity ~v of the particle, whose coordinates are (~x, t) , is dictated to be

~v =d~x

dt=~j(~x, t)

%(~x, t). (8.10)

It is then not difficult to verify that, if at t = 0 the particle has a probability distribution

W (~x, 0) = %(~x, 0) , (8.11)

its probability distribution at time t will continue to be

W (~x, t) = %(~x, t) . (8.12)

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Notably, this will be true in any type of diffraction experiment where the particle is sentthrough a screen with two or more slits in it. In all respects, this particle seems to dowhat is expected of a “de Broglie particle”.

One might be worried by the question what a particle will do when it approaches azero of the probability function %(~x, t) , but it is easy to verify that it will stay clear ofsuch zeros, and also, if the wave function is complex, the isolated zeros of % will be easyto circumnavigate.

Bohm’s interpretation is now to endow an ontological significance to this wave functionψ(~x, t) . It is a ‘hidden variable’. The particle and its wave obey the combined equations(8.6) and (8.10), and these seem to be deterministic as well as local. The initial condition(8.11) may appear to be somewhat disturbing, and its physical origin somewhat obscure,but one could argue that additional information on the particle in the far past somehowgot lost, and the description one finds at later times is the best one can ask for.

If all particles in the world were non-relativistic single particles interacting only withbackground fields such as V (~x) and ~A(~x) (which may also be allowed to be time-dependent) then this interpretation might be a reasonable one.

However, particles can interact directly with one another, and they can be createdand annihilated. Furthermore, they may have spin. The de Broglie-Bohm theory cancertainly be extended to accommodate for these features, but at a price:

First, consider spin. Particles with spin have wave functions with several components,ψi(~x, t) , where i = 1, · · · , 2s+ 1 . A simple way to extend the previous discussion to thiscase is to observe that now the probability to find a particle at a point ~x at time t is

W (~x, t) =∑i

= %(~x, t) =2s+1∑i=1

|ψi(~x, t)|2 , (8.13)

and again we have a unitarity equation telling us that this density is also the divergenceof a current ~j(~x, t) . So, here also, we know what the probability is to find the particlesomewhere, and this probability obeys Bohm’s theory. However, the question arises, whatis the spin of this particle? How do we see whether the spin is up, down or sideways?

Can we say that %i(~x, t) = ψ∗i (~x, t)ψi(~x, t) , for each value of i , is the probability offinding a particle whose spin is in the i direction? In that case, Bohm will also have toexplain how the particle flips from one spin direction to another one. We could regard thespin index i as a discrete coordinate, for which there is also a current. But this current,being in a discrete coordinate, gives only a finite frequency of spin flips per second. Whendoes a particle flip? One can turn this process into a stochastic law, but this wouldreintroduce indeterminism which we thought we had eliminated.

And there is worse: which spin representation do we choose? Are there differentkinds of particles whose spins are aligned in different directions, the x -direction, the y -direction, etc.? The trouble gets worse if spin is the only property that we will decideto look at, such as in a Bell experiment (see later). Perhaps one should insist that aparticle’s spin can only be observed by detecting the particle’s motion, so that Eqs. (8.13)and (8.10) should suffice, but then one might ask: does this not hold for orbital angular

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momentum? Why should that be treated differently from spin angular momentum?

And then we have to face the fact that particles interact directly. The quantum de-scription of the Standard Model tells us that there is a Fock space of elementary particles,whose combined wave function is in a 3n -dimensional space, where n , the number ofparticles considered, is actually variable and it can approach infinity.

In the many particle case, the de Broglie-Bohm pilot wave is a function in this 3n -dimensional space:

ψ(~x1, · · · , ~xn) . (8.14)

Again, unitarity ensures that this is the divergence of a current in 3n -dimensional space,and so we still have a pilot wave. It obeys a local Schrodinger equation in 3n dimensions.Suppose we disregard the spin problem, can we then call this a local hidden variabletheory?

The answer to this question is generally given as no. It is a “multi-local” theory.Having variables, hidden or not, that depend on many coordinates ~x1, · · · , ~xn , is usuallyregarded as being non-local. A local theory has its physical degrees of freedom eachdepend on only one coordinate ~x in 3-space, and the time dependence at this point isdictated only by its neighbours, not the values the variables take at other coordinates ~x ′ .This is why this interpretation is said to be a non-local one, as soon as several particlesare taken into consideration, something we cannot avoid.

Finally, an important remark: since the pilot wave function depends on the coordinatesof all particles involved, it really is considering all “universes” containing n particles.This number n would eventually count all particles in the universe. So, Bohm’s pilotwave now spans over all universes. This is why Bohm’s theory at the very end is not somuch different from the Many World interpretation. The deeper origin of this remarkis that both interpretations considered in this chapter so-far, do not really address, letalone question, the original dynamical equations of quantum mechanics. Mathematicallytherefore, they are very alike, and the differences between these interpretations may besemantic rather than anything else.

8.5. Schrodinger’s cat. The collapse of the wave function

The following ingredient in the Copenhagen interpretation, Section 8.1, is often the subjectof discussions:

As soon as an observable O is measured, the wave function of the systemcollapses to a state in the subspace of Hilbert space that is an eigen state ofthe observable O , or a probabilistic distribution of eigen states.

This is referred to as the “collapse of the wave function”. It appears as if the action of themeasurement itself causes the wave function to attain its new form. The question thenasked is what physical process is associated to that.

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Again, the official reply from the Copenhagen group is that this question should notbe asked. Do the calculation and check your result with the experiments. However, thereappears to be a contradiction, and this is illustrated by Erwin Schrodinger’s Gedankenexperiment with a cat.[5] The experiment is summarised as follows:

In a sealed box, one performs a typical quantum experiment. It could be aStern Gerlach experiment where a spin 1

2particle with spin up is sent through

an inhomogeneous magnetic field that splits the wave function according tothe values of the spin in the y direction, or it could be a radioactive atomthat has probability 1

2to decay within a certain time. In any case, the wave

function is well specified at t = 0 , while at t = 1 it is in a superposition oftwo states, which are sent to a detector that decides which of the two states isrealised. It is expected that the wave function ‘collapses’ into one of the twopossible final states.The box also contains a live cat (and air for the cat to breathe). Dependingon the outcome of the measurement, a capsule with poison is broken, or keptintact. The cat dies when one state is found, and otherwise the cat stays alive.Then, we open the box and inspect the cat.

Clearly, the probability that we find a dead cat is about 12

, and otherwise we find a livecat. However, we could also regard the experiment from a microscopic point of view. Theinitial state was a pure quantum state. The final state is a superposition. Should the cat,upon opening the box, not be found in a superimposed state: dead and alive?

The collapse axiom tells us that the state should be ‘dead’ or ‘alive’, but the first partsof our description of the quantum mechanical states of Hilbert space, clearly dictates thatif two states, |ψ1〉 and |ψ2〉 are possible in a quantum system, then we can also haveα|ψ1〉 + β|ψ2〉 . According to Schrodinger’s equation, this superposition of states alwaysevolves into a superposition of the final states. The collapse seems to violate Schrodinger’sequation. Something is not quite right.

An answer that several investigators have studied[23], is that, probably, Schrodinger’sequation is only an approximation, and that tiny non-linear ‘correction terms’ bring aboutthe collapse[28][40][41]. The problem with this is that observations can be made at quitedifferent scales of space, time, energy and mass. How big should the putative correctionterms be? Secondly, how do the correction terms know in advance which measurementswe are planning to perform?

This, we believe, is where the cellular automaton interpretation of quantum mechanicswill come to the rescue. It states that there is no wave function, but there are ontologicalstates instead. How this explains the collapse phenomenon will be explained in Chap-ter 9. In summary: quantum mechanics is not the basic theory but a tool to solve themathematical equations. This tool works just as well for superimposed states as for theontological states, but they are not the same thing. The dead cat is in an ontologicalstate and so is the live one. The superimposed cat solves the equations mathematicallyin a perfectly acceptable way, but it does not describe a state that can occur in the realworld. The original atom at t = 0 could not be prepared in that state. All this will

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sound very odd to physicists who have grown up with standard quantum mechanics, butit does form the logical solution to the Schrodinger cat paradox.

8.6. Born’s probability axiom

A question related to the discussion of the previous section, concerns another basic axiomof Copenhagen quantum mechanics. It is the fact the probability that a state |ψ〉 is foundto agree with the properties of another state |ϕ〉 , must be given by

P = |〈ϕ|ψ〉|2 . (8.15)

Where does this come from?

Note, that Born did not have much of a choice.[9] The completeness theorem of linearalgebra implies that the eigen states |ϕ〉 of an hermitean operator span the entire Hilbertspace, and therefore,∑

ϕ

|ϕ〉〈ϕ| = I ;∑ϕ

|〈ϕ|ψ〉|2 =∑ϕ

〈ψ|ϕ〉〈ϕ|ψ〉 = 〈ψ|ψ〉 = 1 , (8.16)

where I stands for the identity operator. If Born would have chosen any other expressionto represent probabilities, they would not have added up to one. The expression (8.15)turns out to be ideally suited to serve as a probability.

Yet this is a separate axiom, and the question why it works so well is not misplaced.In a hidden variable theory, probabilities may have a different origin. The most naturalexplanation as to why some states are more probable than others may be traced to theirinitial states much earlier in time. One can ask which initial states may have lead toa state seen at present. There may be numerous answers to that question. One nowcould attempt to estimate how many different initial states could have lead to the presentone. The probability of a given observed state could then be related to the ratios of thenumbers of different original states. Our question then is, can we explain whether andhow the expression (8.15) is related to these numbers.

8.7. Bell’s theorem, Bell’s inequalities and the CHSH inequalities.

One of the main reasons why ‘hidden variable’ theories are usually dismissed, and em-phatically so when the theory obeys local equations, is the apparent difficulty in suchtheories to represent entangled quantum states. Just because the de Broglie-Bohm the-ory is intrinsically non-local, it is generally concluded that all hidden variable theoriesare either non-local or unable to reproduce quantum features at all. When J.S. Bell wasinvestigating the possibility of hidden variable theories, he hit upon the same difficulties,upon which he attempted to prove that local hidden variable theories are impossible.

As before, we do not intend to follow precisely the historical development of Bell’stheory[21], but limit ourselves to a summary of he most modern formulation of the prin-ciples. Bell designed a Gedanken experiment, and at the heart of it is a pair of quantum-

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entangled particles. They could be spin- 12

particles, which each can be in just two quan-tum states described by the Pauli matrices (1.14), or alternatively spin 1 photons. Thereare a few subtle differences between these two cases, but these are not essential to theargument. The first is that the two orthonormal states for photons are the ones wherethey are polarised horizontally or vertically, while the two spin- 1

2states are polarised up

or down. Indeed, quite generally when polarised particles are discussed, the angles forthe photons are handled as being half the angles for spin- 1

2particles.

The second difference concerns the entangled state, which in both cases has total spin0. For spin- 1

2, this means that (~σ1 + ~σ2)|ψ〉 = 0 , so that

|ψ〉 = 1√2(| ↑↓〉 − | ↓↑〉) . (8.17)

For spin 1, this is different. Let us have these photons move in the ±z -direction.Defining A± = 1√

2(Ax ± iAy) as the operators that create or annihilate one unit of spin

in the z -direction, and taking into account that photons are bosons, the 2 photon statewith zero spin in the z -direction is

|ψ〉 = 1√2(A

(1)+ A

(2)− + A

(1)− A

(2)+ ) | 〉 = 1√

2( |z, −z〉+ | − z, z〉 ) , (8.18)

and since helicity is spin in the direction of motion, while the photons go in oppositedirections, we can rewrite this state as

|ψ〉 = 1√2( |+ +〉+ | − −〉 ) , (8.19)

where ± denote the helicities. Alternatively one can use the operators Ax and Ay toindicate the creators of linearly polarised photons, and then we have

|ψ〉 = 1√2(A(1)

x A(2)x + A(1)

y A(2)y ) | 〉 = 1√

2( |xx〉+ |yy〉 ) . (8.20)

Thus, the two photons are linearly polarised in the same direction.

Since the experiment is mostly done with photons, we will henceforth describe theentangled photon state.

The Bell experiment is now illustrated in Fig. 14. At the point S an atom ε isprepared to be in an unstable J = 0 state at t = t1 , such that it can decay only into another J = 0 state, by simultaneously emitting two photons such that ∆J = 0 , and thetwo photons, α and β , must therefore be in the entangled Stot = 0 state, at t = t2 .

After having travelled for a long time, at t = t3 , photon α is detected by observer A(Alice), and photon β is detected by B (Bob). Ideally, the observers use a polarisationfilter oriented at an angle a (Alice) and b (Bob). If the photon is transmitted by thefilter, it is polarised in the direction of the polarisation filter’s angle, if it is reflected, itspolarisation was orthogonal to this angle. At both sides of the filter there is a detector, soboth Alice and Bob observe that one of their two detectors gives a signal. We call Alice’ssignal A = +1 if the photon passed the filter, and A = −1 if it is reflected. Bob’s signalB is defined the same way.

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III

x

tQP

S

BA

βα

ε

t = t0

t = t1

t = t2

t = t3

t = t4

BobAlice

Figure 14: A bell-type experiment. Space runs horizontally, time vertically. Singlelines denote single bits or qubits travelling; widened lines denote classical informa-tion carried by millions of bits. Light travels over 45 , as indicated by the lightcone on the right. Meaning of the variables: see text.

According to quantum theory, if A = 1 , Alice’s photon is polarised in the directiona , so Bob’s photon must also be polarised in that direction, and the intensity of thelight going through Bob’s filter will be cos2(a − b) . Therefore, according to quantumtheory, the probability that B = 1 is cos2(a− b) . The probability that B = −1 is thensin2(a− b) , and the same reasoning can be set up if A = −1 . The expectation value ofthe product AB is thus found to be

〈AB 〉 = cos2(a− b)− sin2(a− b) = cos 2(a− b) , (8.21)

according to quantum mechanics.

In fact, these correlation functions can now be checked experimentally. Beautifulexperiments[22] confirmed that correlations can come close to Eq. (8.21), in violation ofBell’s inequality.

The point made by Bell is that it seems to be impossible to reproduce this strongcorrelation between the findings A and B in any theory where classical information ispassed on from the atom ε to Alice (I) and Bob(II). All one needs to assume is that theatom emits a signal to Alice and one to Bob, regarding the polarisation of the photonsemitted. It could be the information that both photons α and β are polarised in directionc . Since this information is emitted long before either Alice or Bob decided how to orienttheir polarisation filters, it is obvious that the signals in α and β should not depend onthat. Alice and Bob are free to choose their polarisers. The correlations then directlylead to a contradiction, regardless the nature of the classical signals used.

The contradiction is arrived at as follows. Consider two choices that Alice can make:the angles a and a ′ . Similarly, Bob can choose between angles b and b ′ . Whateverthe signal is that the photons carry along, it should entail an expectation value for thefour observations that can be made: Alice observes A or A′ in the two cases, and Bobobserves B or B′ . If both Alice and Bob make large numbers of observations, every time

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using either one of their two options, they can compare notes afterwards, and measurethe averages 〈AB〉, 〈A′B〉, 〈AB′ 〉 and 〈A′B′ 〉 . Now, they calculate the value of

S = 〈AB〉+ 〈A′B〉+ 〈AB′ 〉 − 〈A′B′ 〉 . (8.22)

Now suppose we assume that, every time photons are emitted, they have well-definedvalues for A, A′, B , and B′ . Whatever the signals are that are carried by the photons,these four terms will at each measurement take the values ±1 , but they can never allcontribute to the quantity S with the same sign (because of the one minus sign in (8.22)).Because of this, it is easy to see that S is always ±2 , and its average value will thereforeobey,

|〈S〉| ≤ 2 ; (8.23)

this modern version of Bell’s original observation is called the CHSH inequality[13]. How-ever, if we choose the angles

a = 22.5 , a′ = −22.5 , b = 0 , b′ = 45 , (8.24)

then, according to Eq. (8.21), quantum mechanics gives for the expectation value

S = 3 cos(45)− cos 135 = 2√

2 > 2 . (8.25)

How can this be? Apparently, quantum mechanics does not give explicit values ±1 forthe measurements of A and A′ ; it only gives a value to the quantity actually measured,which in each case is either A or A′ and also either B or B′ . If Alice measures A ,she cannot also measure A′ because the operator for A′ does not commute with A ; thepolarisers differ by an angle of 45 , and a photon polarised along one of these anglesis a quantum superposition of a photon polarised along the other angle and a photonpolarised orthogonally to that. So the quantum outcome is completely in accordancewith the Copenhagen prescriptions, but it seems that it cannot be realised in a localhidden variable theory.

We say that, if A is actually measured, the measurement of A′ is counterfactual,which means that we imagine measuring A′ but we cannot actually do it, just as weare unable to measure position if we already found out what exactly the momentum ofa particle is. If two observables do not commute, one can measure one of them, but themeasurement of the other is counterfactual.

Indeed, in the arguments used, it was assumed that the hidden variable theory shouldallow an observer actually to carry out counterfactual measurements. This is called real-ism. Local hidden variable theories that allow for counterfactual observations are said tohave local realism. Quantum mechanics forbids local realism.

However, to use the word ‘realism’ for the possibility to perform counterfactual ob-servations is not very accurate. ‘realism’ should mean that there is actually somethinghappening, not a superposition of things; something is happening for sure, while some-thing else is not happening. That is not the same thing as saying that both Alice and

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Bob at all times can choose to modify their polarisation angles, without allowing for anymodification at other places [47].

It is here that the notion of “free will” is introduced[45][46], again, in an imprecisemanner. The real extra assumption made by Bell, and more or less tacitly by many of hisfollowers, is that both Alice and Bob should have the “free will” to modify their settingsat any moment, without having to consult the settings at the system S that produces anunstable atom. If this were allowed to happen in a hidden variable theory, we would getlocal realism, which was ruled out.

How can we deny Alice and/or Bob their free will? Well, precisely in a deterministichidden variable theory, Alice and Bob can only change their minds about the setting oftheir polarisers, if their brains follow different laws than they did before, and, like it ornot, Alice’s and Bob’s actions are determined by laws of physics[67], even if these areonly local laws. Their decisions, logically, have their roots in the distant past, going backall the way to the Big Bang. So why should we believe that they can do counterfactualobservations?

The way this argument is usually countered is that the correlations c between thephotons from the decaying atom and the settings a and b chosen by Alice and Bobhave to be amazingly strong. A gigantically complex algorithm could make Alice an Bobtake their decisions, and yet the decaying atom, long before Alice and Bob applied thisalgorithm, knew about the result. This is called ‘conspiracy’, and conspiracy is said tobe “disgusting”. “One could better stop doing physics than believe such a weird thing”,is what several investigators quipped.

8.7.1. The mouse dropping function

To illustrate how crazy things can get, a polished version of Bell’s experiment was pro-posed: both Alice and Bob carry along with them a mouse in a cage, with food. Everytime they want to set the angles of their polarisers, they count the number of mouse drop-pings. If it is even, they choose one angle, if it is odd, they choose the other. “Now, thedecaying atom has to know ahead of time how many droppings the mouse will produce.Isn’t this disgusting?”

To see what is needed to obtain this “disgusting” result, let us consider a simple model.We assume that there are correlations between the joint polarisations of the two entangledphotons, called c , and the settings a chosen by Alice and b chosen by Bob. All theseangles are taken to be in the interval [0, 180] . We define the conditional probability tohave the photons polarised in the direction c , given a and b , to be given by a functionW (c | a, b) . Next, assume that Alice’s outcome A = +1 as soon as her ‘ontological’photon has |a − c| < 45 or > 135 , otherwise A = −1 . For Bob’s measurement weassume the same.

According to the CHSH inequality, there must be correlations between a, b and c .Let us compute them assuming the quantum mechanical result is right. Everything willbe periodic in a, b, and c with period π (180)) .

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It is reasonable to expect that W only depends on the relative angles c−a and c−b .

W (c | a, b) = W (c− a, c− b) ;

∫ π

0

dcW (c− a, c− b) = 1 . (8.26)

Introduce a sign function s(ϕ) as follows:

s(ϕ) ≡ sign(cos(2ϕ)) ; A = s(c− a) , B = s(c− b) . (8.27)

The expectation value of the product AB is

〈AB〉 =

∫dcW (c | a, b) s(c− a) s(c− b) , (8.28)

How should W be chosen so as to reproduce the quantum expression (8.21)?

Introduce the new variables

x = c− 12(a+ b) , z = 1

2(b− a) , W = W (x+ z, x− z) . (8.29)

Quantum mechanics demands that∫ π

0

dxW (x+ z, x− z) s(x+ z) s(x− z) = cos 4z . (8.30)

Writing

s(x+ z) s(x− z) = sign(cos 2(x+ z) cos 2(x− z)) = sign(cos 4x+ cos 4z) , (8.31)

we see that the equations are periodic with period π/2 , but looking more closely, wesee that both sides of Eq. (8.30) change sign if x and z are shifted by an amountπ/4 . Therefore, one first tries solutions with periodicity π/4 . Furthermore, we have thesymmetry x↔ −x, z ↔ −z .

Eq. (8.30) contains more unknowns than equations, but if we assume W only todepend on x but not on z then the equation can readily be solved. DifferentiatingEq. (8.30) to z , one only gets Dirac delta functions inside the integral, all adding up toyield:

4

∫ π/4

0

W (x)dx(−2δ(x+ z − π/4)) = −4 sin 4z , if 0 < z < 14π (8.32)

(the four parts of the integral in Eq. (8.30) each turn out to give the same contribution,hence the first factor 4). Thus one arrives at

W (c | a, b) = W (x+ z, x− z) = 12| sin 4x| = 1

2| sin(4c− 2a− 2b) | . (8.33)

This also yields a 3-point probability distribution

W (a, b, c) = 12π2 | sin(4c− 2a− 2b)| . (8.34)

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We could call this the ‘mouse dropping function’ (see Fig. 15). If Alice wants toperform a counterfactual measurement, she modifies the angle a , while b and c are keptuntouched. She herewith chooses a configuration that is less probable, or more probable,than the configuration she had before. Taking in mind a possible interpretation of theBorn probabilities, as expressed in Section 8.6, this means that the total number of initialstates where Alice’s mouse produced a different number of droppings, may be more orless probable than the state she had before. In quantum mechanics, we have learned thatthis is acceptable. If we say this in terms of an underlying deterministic theory, manypeople have problems with it.

W

x π0

Figure 15: The mouse dropping function, Eq. (8.34). Horizontally the variablex = c − 1

2(a + b) . Averaging over any of the three variables a , b , or c , gives thesame result as the flat line; the correlations then disappear.

In fact, all we have stated here is that, even in a deterministic theory obeying localequations, configurations of physical states may have non-trivial space-like correlations.It is known that this happens in many physical systems. A liquid close to its thermody-namical critical point shows a phenomenon called critical opalescence: large fluctuationsin the local density. This means that the density correlation functions are non-trivial overrelatively large space-like distances. This does not entail a violation of relativity theoryor any other principle in physics such as causality; it is a normal phenomenon. A liquiddoes not have to be a quantum liquid to show critical opalescence.

It is still true that the effect of the mouse droppings seems to be mysterious, and so wedo return to it, in our final discussion in Sections 14.2 and 14.3. All we can bring forwardnow is that mice guts also obey energy, momentum and angular momentum conservationbecause these are general laws. In our ‘hidden variable’ theory, a general law may haveto be added to this: an ontological state evolves into an ontological state; superpositionsevolve into superpositions. If the mouse droppings are ontological in one description andcounterfactual in another, the initial state from which they came was also ontological orcounterfactual, accordingly. This should not be a mysterious statement.

Actually, there is an even stronger arrangement than the mouse droppings by whichAlice and Bob can make their decisions. They could both monitor the light fluctuationscaused by light coming from different quasars, at opposite points in the sky[48]. Thesequasars, called A and B in Fig. 14, may have emitted their photons shortly after the BigBang, at time t = t0 in the Figure, when they were at billions of light years separationfrom one another. The fluctuations of these quasars should also obey the mouse droppingformula (8.33). How can this be? The only possible explanation is the one offered bythe inflation theory of the early universe: these two quasars do have a common past, and

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therefore their light is correlated.

Note that the correlations generated by the probability distribution (8.34) is a gen-uine three-body correlation. Integrating over any one of the three variables gives a flatdistribution. The quasars are correlated only through the state the decaying atom is in,but not directly with one another. It clearly is a mysterious correlation, but it is not atodds with what we know about the laws of physics, see Sections 10.4 and 10.5.

In fact, these space-like correlations must be huge: who knows where else in theuniverse some alien Alices and Bobs are doing experiments using the same quasars . . .

9. Deterministic quantum mechanics

9.1. Axiomatic approach

Most attempts at formulating hidden variable theories in the presently existing literatureconsist of some sort of modification of the real quantum theory, and of a replacement ofordinary classical physics by some sort of stochastic formalism. The idea behind this isalways that quantum mechanics seems to be so different from the classical regime thatsome deep modification of standard procedures seems to be necessary.

In the present paper, what we call deterministic quantum mechanics is claimed to bemuch closer to standard procedures than usually thought to be possible.

Deterministic quantum mechanics is neither a modification of standard quan-tum mechanics, nor a modification of classical theory. It is a cross section ofthe two.

This cross section is claimed to be much larger and promising than usually thought. Wecan phrase the theory in two ways: starting from conventional quantum mechanics, orstarting from a completely classical setting. We have seen already in many previous partsof this work what this means; here we recapitulate.

Starting from conventional quantum mechanics, deterministic quantum mechanics isa small subset of all quantum theories: we postulate the existence of a very special basisin Hilbert space: the ontological basis. Very likely, there will be many different choices foran ontological basis (often related by a symmetry transformation), and it will be difficultto decide which of these is “the real one”. This is not so much our concern. Any of thesechoices for our ontological basis will serve our purpose. Finding quantum theories thathave an ontological basis will be an important and difficult exercise. Our hope is thatthis exercise might lead to new theories that could help elementary particle physics andquantum gravity theories to further develop. Also it may help us find special theories ofcosmology.

An ontological basis is a basis in terms of which the Schrodinger equation sends basiselements into other basis elements at sufficiently dense moments in time.

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This definition is deliberately a bit vague. We do not specify how dense the momentsin time have to be, nor do we exactly specify how time is defined; in special relativity, wecan choose different frames of reference where time means something different. In generalrelativity, one has to specify Cauchy surfaces that define time slices. What really mattersis that an ontological basis allows a meaningful subset of observables to be defined asoperators that are diagonal in this basis. We postulate that they evolve into one another,and this implies that their eigen values remain sharply defined as time continues. Precisedefinitions of an ontological basis will be needed only if we have specific theories in mind;in the simple examples discussed in earlier chapters it was always clear what this basisis. In some cases, time is allowed to flow continuously, in others, notably when we havediscrete operators, time is also limited to a discrete subset of a continuous time line.

Once such a basis has been identified, we may have in our hands a set of observablesin terms of which the time evolution equations appear to be classical equations. Thisthen links the system to a classical theory. But it was a quantum theory from where westarted, and this quantum theory allows us without much ado to transform to any otherbasis we like. Fock space for the elementary particles is such a basis, and it still allowsus to choose any orthonormal sets of wave functions we like for each particle type. Ingeneral, Fock space will not be an ontological basis. We might also wish to consider thebasis spanned by the field operators φ(~x, t) , Aµ(~x, t) , and so on. This will also not bean ontological basis, in general.

Clearly, an ontological basis for the Standard Model has not yet been found, and itis very dubious whether anything resembling an ontological basis for the Standard Modelexists. More likely, the model will have to be extended to encompass gravity in some way.In the mean time, it might be a useful exercise to identify operators that stay diagonallonger than others, so they may be closer to the ontological variables of the theory thanother operators. In general, this means that we have to investigate commutators ofoperators. Can we identify operators that, against all odds, accidentally commute? Wehave seen a simple example of such a class of operators when we studied our “neutrino”models; “neutrinos” in quotation marks, because these are idealised neutrino-like fermions.We have also seen that anything resembling a harmonic oscillator can be rephrased in anontological basis to describe classical variables that evolve periodically, the period beingthat of the original oscillator.

We also saw that some of the mappings we found are not at all fool-proof. Most ofour examples cease to be linked to a classical system if interactions are turned on, andalso negative-energy states might emerge, as we shall also see later (Subsection 11.4.2).Furthermore, we have the so-called edge states. These are states that form a subsetof Hilbert space with measure zero, but their contributions may still spoil the exactcorrespondence.

Alternatively, one may define deterministic quantum mechanics by starting with somecompletely classical theory. Particles, fields, and what not move around following classicallaws. These classical laws could resemble the classical theories we are already familiarwith: classical mechanics, classical field theories such as the Navier Stokes equations, andso on. A very interesting class of models is what we call “cellular automata”. A cellular

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automaton is a system with localised, classical, discrete degrees of freedom, typicallyarranged in a lattice, which obey evolution equations. The evolution equations take theshape of a computer program, and could be investigated in computers exactly, by runningthese programs in a model. A typical feature of a cellular automaton is that the evolutionlaw for the data in every cell only depends on the data in the adjacent cells, and noton what happens at larger distances. This is a desirable form of locality, which indeedensures that information cannot spread faster than some limiting speed, usually assumedto be the local speed of light.

These classical theories may also be chosen to be more general than the classical modelsmost often used in physics. There is for instance no need for a classical Newtonian actionprinciple; without such a principle, our theories can still be set up. A very importantlimitation is the time reversibility. If a classical model is not time reversible, it seemsthat our procedures will fail. For instance, we wish our evolution operator to be unitary,so that the hamiltonian will turn out to be an hermitean operator. But, as we shallsee, it may be possible to relax even this condition. The Navier Stokes equations, forinstance, although time reversible at short time scales, do seem to dissipate informationaway. When a Navier Stokes liquid comes to rest, due to the viscosity terms, it cannotbe followed back in time anymore. Nevertheless, time non reversible systems could be ofinterest for our theories anyway, as will be discussed in Section 13.2

Starting from any classical system, we consider its book keeping procedure, and iden-tify a basis element for every state the system can be in. These basis elements are declaredto be orthonormal. In this artificial Hilbert space, the states evolve, and it appears to bea standard exercise first to construct the evolution operator that describes the postulatedevolution, and then to identify a hamiltonian that generates this evolution operator, byexponentiation.

As soon as we have our Hilbert space, we are free to perform any basis transformationwe like. Then, in a basis where quantum calculations can be done to cover long distancesin space and time, we find that the states we originally called “ontological” now indeedare quantum superpositions of the new basis elements, and as such, they can generateinterference phenomena. The central idea behind deterministic quantum mechanics is,that at this stage our transformations might tend to become so complex that the originalontological states can no longer be distinguished from any other superposition of states,and this is why, in conventional quantum mechanics, we treat them all without distinction.We lost our ability to identify the ontological states in today’s ‘effective’ quantum theories.

9.1.1. The double role of the hamiltonian

In earlier investigations, the author frequently hit upon an important problem: the hamil-tonian thus obtained does not seem to have a ground state, or, if it has, there seems tobe nothing special about this ground state, while in the real world this one state is thevacuum state, and that is very special indeed. It is the only state that transforms intoitself under Lorentz transformations, while Lorentz invariance is notoriously difficult toestablish in discretised theories.

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Now when time is assumed to be a discrete variable, there may be an elegant wayout of this problem. In discrete time, the hamiltonian that generates the time evolutionis only defined on a finite segment of the energy scales, in terms of which it is periodic.This means that this hamiltonian can be limited to the interval [0, 2π/δt] , where δt isthe time quantum.

Now, in the real world there is no such restriction on the total energy, but we havefound a quite different expression for a classical hamiltonian that can also generate timeevolution, the discretised hamiltonian of Section 7.2. This hamiltonian appears to beentirely complementary to the ‘quantum hamiltonian’, and invites us to combine the twointo one, as was shown in Section 7.3. Now this hamiltonian has a natural lower bound,and could be used in theories of thermodynamics. However, it appears to be not yetsufficiently local. This question needs to be further investigated; the observed lack oflocality of the induced hamiltonian (even if the theory itself is entirely local!) could playa role in the ‘mysterious’ long range correlations that are responsible for the emergenceof entangled particles.

The discussion of the lower bound of the hamiltonian brings us to observe more closelya peculiar feature of the physical world: the hamiltonian, which is a very essential functionboth in classical and in quantum theories, appears to have a double function in controllingthe dynamical laws:

- On the one hand, the hamiltonian Hop generates the equations of motion. Theenergy of a state is a function of its dynamical variables. The classical hamiltonformalism then tells us that this same function can serve to define how the dynamicalvariables evolve in time. This continues to be true in quantum mechanics; all weneed is the hamiltonian, which is now an operator, to determine how things evolve.Thus, the hamiltonian generates all laws of physics.

- At the same time, of course, the hamiltonian determines the total energy of a state.That energy is conserved, but it is also bounded from below. This lower bound istotally essential for a theory to work properly.

Without lower bound in the energy, the world as we know it would be unstable. Considerany approximate solution of the equations, but now add a tiny perturbation. Total energyis conserved, but due to this perturbation, some energy could flow from one region tosomewhere else. If the energy somewhere else would be allowed to be negative, the energyof the rest of the solution could begin to increase. The law of equipartition would tendto dictate that all allowed states would be roughly equally probable. Now in our worldthere are vastly more states with high energy than with lower energy, and therefore onewould expect the upwards flow of energy to be never-ending: this configuration would endwith all states, with all possible energies, being equally probable. That’s the completelychaotic state. The only stationary solutions would be the entirely chaotic ones.

We can state this differently: solutions of the equations of motion are stationary ifthey are in thermal equilibrium (possibly with one or more chemical potentials added).

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In a thermal equilibrium, we have the Boltzmann distribution:

Wi = C e−βEi+∑j µjRji , (9.1)

where β = 1/kT , the inverse of the temperature T with Boltzmann constant k , andi labels the states; µj are chemical potentials, and Rji the corresponding conservedquantities.

If the energies Ej were not properly bounded from below, the lowest energies wouldcause this expression to diverge, particularly at low temperatures.

What is needed is a lower bound of the energies Ei so as to ensure stability of ourworld. In addition, having a ground state is very important to construct systematic ap-proximations to solutions of the time-independent Schrodinger equation, using extremumprinciples. This is not just a technical problem, it would raise doubt on the mere exis-tence of correct solutions to Schrodinger’s equation, if no procedure can be described thatallows one to construct such solutions systematically. In Chapter 7, we saw that energyconservation is such an important feature in physical law, that it is used to generatethe evolution equations, so that this conservation is guaranteed. What one concludes is,that both in classical theories and in quantum theories, a hamiltonian is needed that isbounded from below and at the same time generates the equations of motion. This evenholds for our beable theories: they must have a hamilton principle; the point is, that in asector where the discrete hamiltonian is fixed, the number of states seems to be finite, sothat further stability might not be required, and the evolution within such a sector wouldnot be controlled b a hamiltonian anymore. This would be wrong, we need a hamiltonianthat generates all of the evolution. The existence of stationary states with extremely lowtemperature in this world proves that the hamiltonian is much more precisely definedthan the course-grained hamiltonian of Chapter 7, and this is why the hamiltonian de-fined there must be combined with the hamiltonian that determines all other features ofthe beable evolution, into one hamiltonian. This is a difficult thing to accomplish.

In Section 13.1, we return to this important problem.

9.2. The classical limit revisited

Now there are a number of interesting issues to be discussed. One is the act of measure-ment, and the resulting ‘collapse of the wave function’. What is a measurement?[16]

The answer to this question may be extremely interesting. A measurement allows asingle bit of information at the quantum level, to evolve into something that can be recog-nised and identified at large scales. Information becomes classical if it can be magnifiedto arbitrary strength. Think of space ships that react on the commands of a computer,which in turn may originate in just a few electrons in its memory chips. A single cos-mic ray might affect these data. The space ship in turn might affect the course of largesystems, eventually forcing planets to alter their orbits, first in tiny ways, but then thesemodifications might get magnified.

Now we presented this picture for a reason: we define measurement as a process thatturns a single bit of information into states where countless many bits and bytes react on

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it. Imagine a planet changing its course. Would this be observable in terms of the original,ontological variables? It would be very hard to imagine that it would not be. The interiorof a planet may have its ontological observables arranged in a way that differs ever soslightly from what happens in the vacuum state. Whatever these minute changes are,the planet itself is so large that the tiny differences can be added together statisticallyso that the classical orbit parameters of a planet will be recognisable in terms of theontological degrees of freedom. This is a very important point, because it means that ata large scale, all other classical observables of our world must also be diagonal in termsof the ontological basis: large scale observables, such as the orbits of planets, and then ofcourse also the classical data shown in a detector, are beables. They commute with ourmicroscopic beable operators. See also Figure 16.

Now we must address the nature of the wave functions, or states |ψ〉 , that representreal observed phenomena. We explained earlier that, in terms of the basis we wouldnormally use in quantum mechanics, these states will be complicated quantum super-positions. In terms of the original, ontological basis they will just be elementary basiselements. And what we just argued is that they will also be elementary elements of theclassical observables at large scales! What this means is that the states |ψ〉 that weactually produce in our laboratories, will automatically collapse into states that are dis-tinguishable classically. There will be no need to modify Schrodinger’s equation to realisethe collapse of the wave function; it will happen automatically.

This does away with Schrodinger’s cat problem. The cat will definitely emerge eitherdead or alive, but never in a superposition. This is because not all states |ψ〉 are onto-logical. The state actually started from in Schrodinger’s Gedanken experiment, was anontological state, and for that reason could only evolve into either a dead cat or a live cat.If we would have tried to put the superimposed state, α|dead〉+ β|alive〉 in our box, wewould not have had an ontological state. We can’t produce such a state! What we cando is repeat the experiment; in our simplified description of it, using our effective but notontological basis, we might have thought to have a superimposed state as our initial state,but that of course never happens, all states we ever realise in the lab are the ontologicalones, that later will collapse into states where classical observables take definite values,even if we cannot always predict these.

9.3. Born’s probability rule

9.3.1. The use of templates

For the approach advocated here it is very useful to introduce the notion of templates.We argued that conventional quantum mechanics is arrived at if we perform some quitecomplicated basis transformation on the ontological basis states. These new basis elementsso obtained will all be quite complex quantum superpositions of the ontological states. Itis these states that we shall call “template states”; they are the recognisable states wenormally use in quantum mechanics.

Upon inverting this transformation, one finds that, in turn, the ontological states will

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be complicated superpositions of the template states. The superpositions are complicatedbecause they will involve many modes that are hardly visible to us. For instance, thevacuum state, our most elementary template state, will be a superposition of very manyontic (short for ontological) states. Why this is so, is immediately evident, if we realisethat the vacuum is the lowest eigen state of the hamiltonian, while the hamiltonian is nota beable but a ‘changeable’ (see Section 2.1.1). Of course, if this holds for the vacuum, itwill surely also hold for all other template states.

statesmacrostates

sub-micro

statesmacro

templatesmicrostates

sub-micro

b)a)

Figure 16: Classical and quantum states. a) The sub-microscopic states are the“hidden variables”. Atoms, molecules and fields are templates, defined as quantumsuperpositions of the sub-microscopic states, and used at the conventional micro-scopic scale. The usual “classical” objects, such as planets and people, are macro-scopic, and again superpositions of the micro-templates. The lines here indicatequantum matrix elements; the second set is essential, as it refers to the environ-ment, see the main text. b) The classical, macroscopic states are probabilisticdistributions of the sub-microscopic states. Here, the lines therefore indicate prob-abilities. All states are astronomical in numbers, but the microscopic templates aremore numerous than the classical states, while the sub-microscopic states are evenmore numerous.

The macroscopic states, which are the classical states describing people and planets,but also the pointers of a measuring device, and of course live and dead cats, are again su-perpositions of the template states, in general, but they are usually not infinitely preciselydefined, since we do not observe every atom inside these objects. Each macroscopic stateis actually a composition of very many quantum states, but they are well-distinguishablefrom one another.

In Figure 16, the ontological states are the sub-microscopic ones, then we see themicroscopic states, which are the quantum states we usually consider, that is, the templatestates, and finally the macroscopic or classical states. The matrix elements relating thesevarious states are given by lines of variable thickness.

What was argued in the previous Section was that the macroscopic states are diagonalin terms of the sub-microscopic states, so these are all ontic states. It is a curious fact

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of Nature that the states that are most appropriate for us to describe atoms, moleculesand sub-atomic particles are the template states, requiring superpositions. So, whenwe observe a classical object, we are also looking at ontological things, which is why atemplate state we use to describe what we expect to see, “collapses” into delta peekedprobability distributions in terms of the classical states.

9.3.2. Probabilities

At first sight, it may seem that the notion of probability is lost in our treatment of quantummechanics. Our theory is ontic, it describes certainties, not probabilities. However, wecan re-establish the notion of probabilities when we transform from the ontic states tothe template states, and eventually to the macroscopic states.

Suppose that a quantum calculation is done by using one of our template states.This means that the initial state does not represent a single ontological situation, but itcovers many. Those ontological states may all evolve differently. When, at the end ofa calculation, again one of the template configurations is arrived at, which we want tocompare with one of the classical states. The latter are again ontological. The Born rulenow states that the absolute squares of the inner products represent probabilities. Howdid that enter?

Even in classical calculations, the use of statistical methods may be very useful. Onedoes not know exactly which ontological state represents the initial state. Think of ascattering experiment where two beams of particles cross one another and some of theparticles are expected to collide. We never know exactly the orbits in the initial state, soof course, one takes the average configurations and does the calculation over the entireensemble. The outcome of such calculations will always be that the scattered particlesform a distribution, which is a probabilistic function over all conceivable scattering pa-rameters, such as the values of the exchanged energy and momentum. In the classicalcalculation then, the origin of the uncertainty of the exact outcome lies in our lack ofexact knowledge concerning the initial state.

Our template states form a very tiny subset of all ontological states, so that every timewe repeat an experiment, the actual ontic state is a different one. The initial templatestate now does represent the probabilities of the initial ontic states, and because these areprojected into the classical final states, the classical final states obey the Born rule if theinitial states do. Therefore, we can prove that our theory obeys the Born rule if we knowthat the initial state does that regarding the ontic modes. If we now postulate that thetemplate states that we use always reflect the relative probabilities of the ontic states ofthe theory, then the Born rule appears to be an inevitable consequence.[72]

Most importantly, there is absolutely no reason to attempt to incorporate deviationsfrom Born’s probability interpretation of the Copenhagen interpretation into our theory.Born’s rule will be exactly obeyed; there cannot be systematic, reproducible deviations.

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10. Quantum Field Theory

We have seen that producing quantum Schrodinger equations starting from non quantummechanical systems is essentially straightforward. However, to employ this observationas a viable ontological interpretation of quantum mechanics, more is needed. The mainobjection against these concepts has always been that theories obeying locality are muchmore difficult to reproduce with classical systems, and impossible according to many.Locality means that interaction at a distance can only occur with signals that undergosome delay.

The relativistic quantised field theories employed in the Standard Model do have therequired locality properties. It is important to recall here the essential features of thesesystems.

Relativistic quantum field theories with a proper continuum limit can only incorporateelementary fields with spin 0, 1

2and 1. As is well-known, gravity would be propagated

by gravitons with spin 2, and supergravity would add one or more gravitino species withspin 3/2 , but then, if we want these fields to interact, we would have to be close to thePlanck scale, and this would require discretisation due to micro states, which is why weomit those fields for the moment.

On the other hand, one could argue that special relativity is not our first priority,and ignoring special relativity would imply no rigorous constraint on spin. If we ignorespecial as well as general relativity, we could just as well ignore rotation invariance21.What is left then is a theory of quantised fields enumerated by an index that may or maynot represent spin. Imposing Poincare invariance, at least at the quantum side of theequation, will then be left as an important exercise for the (hopefully near) future.

What we want to keep however is a speed limit for signals that describe interactions,so that the notion of locality can be addressed. In practice, an elegant criterion can begiven that guarantees this kind of locality. Consider the quantum system in its Heisenbergnotation. We have operators Oi(~x, t) where both the space coordinates ~x and the timecoordinate t may be either continuous or discrete. The discrete index i enumeratesdifferent types of operators. If the two space-time points (~x, t) and (~x ′, t′) are outsidethe reach of signals, or, in the relativistic case,

(~x− ~x ′)2 − (t− t ′)2 > 0 , (10.1)

then we must require all commutators to vanish:

[Oi(~x, t) , Oj(~x ′, t ′) ] = 0 . (10.2)

In the continuous case, this means that the hamiltonian must be the integral of a

21Ignoring such important symmetries in considering certain models does not mean that we believe thesesymmetries to be violated, but rather that we wish to focus on simple models where these symmetriesdo not, or not yet, play a role. In more sophisticated theories of Nature, of course one has to obey allknown symmetry requirements.

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hamiltonian density:

H =

∫d3~x H(~x, t) , [H(~x, t) , H(~x ′, t)] = 0 if ~x 6= ~x ′ , (10.3)

while the commutator [H(~x, t) , H(~x ′, t) ] may contain derivatives of Dirac delta dis-tributions. Note that, here, we kept equal times t so that these space-time points arespace-like separated unless they coincide.

On a discrete space-like lattice, it is tempting to replace Eqs. (10.3) by the latticeexpressions

H =∑~x

H(~x, t) , [H(~x,′ , t), H(~x ′, t)] = 0 if |~x− ~x ′| > a, (10.4)

where a is the link size of the lattice. The Hamilton densities at neighbouring sites,|~x− ~x ′| = a , will, in general, not commute. Although Eqs. (10.4) may well serve as agood definition of locality, they do not guarantee that signals are subject to a speed limit.Only in the continuum limit, one may recover commutation at space-like separations,(10.1), if special relativity holds.

Cellular automata are typically lattice theories. In general, these theories are difficultto reconcile with Lorentz invariance. This does not mean that we plan to give up Lorentzinvariance; quite possibly, this important symmetry will be recovered at some stage. Butsince we want to understand quantum mechanics as a reflection of discreteness at a scalecomparable to the Planck scale, we are unable at present to keep Lorentz invariance inour models, so this price is paid, hopefully temporarily.

For simplicity, let us now return our attention to continuum quantum field theories,which we can either force to be Lorentz invariant, or replace by lattice versions at somelater stage. The present chapter is included here just to emphasise some important fea-tures.

10.1. General continuum theories – the bosonic case

Let the field variables be real number operators Φi(~x, t) and their canonical conjugatesPi(~x, t) . Here, i is a discrete index counting independent fields. The commutation rulesare postulated to be

[Φi(~x, t), Φj(~x′, t)] = [Pi(~x, t), Pj(~x

′, t)] = 0 ,

[Φi(~x, t), Pj(~x′, t)] = δijδ

3(~x− ~x ′) (10.5)

(for simplicity, space was taken to be 3-dimensional).

In bosonic theories, when writing the hamiltonian as

H =

∫d~xH(~x) , (10.6)

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the hamiltonian density H(~x) typically takes a form such as

H(~x) = 12

∑i

(P 2i (~x) + (~∂Φi(~x))2 + V (~Φ(~x)) . (10.7)

If we are in 3+1 dimensions, and we want the theory to be renormalisable, V (~Φ(~x)) must

be a polynomial function of ~Φ , of degree 4 or lower. Typically, one starts with

V (~Φ) = 12

∑i

m2i~Φ2i + V4(~Φ) , (10.8)

where mi are the (unrenormalised) masses of the particles of type i , and V4 is a homo-

geneous quartic expression in the fields ~Φi(~x) such as a self interaction 14!λΦ4 .

However, when ~Φ(~x) contains components that form a vector in 3-space, Lorentz-invariance dictates a deviation from Eq. (10.7). We then have local gauge invariance,

which implies that a constraint has to be imposed. Writing these vector fields as ~A(~x) ,

and their associated momentum fields as ~E(~x) , one is forced to include a time componentA0(~x) .

Gauge-invariance must then be invoked to ensure that locality and unitarity of thetheory are not lost, but the resulting hamiltonian deviates a bit from Eq. (10.7). Thisdeviation is minimal if we choose the space-like radiation gauge

3∑i=1

∂iAi(~x) = 0 , (10.9)

since then the hamiltonian will have the quadratic terms

H2(~x) = 12~E2(~x) + 1

2(~∂Ai(~x))2 − 1

2(~∂A0(~x))2 . (10.10)

Notably, the field A0(~x) does not have a canonical partner that would have been calledE0(~x) , and therefore, the field A0 can be eliminated classically, by extremising the hamil-tonian (10.10), which leads to the Coulomb force. This Coulomb force is instantaneousin time, and would have destroyed locality (and hence Lorentz invariance) if we did nothave local gauge invariance.

This, of course, is a description of quantised gauge theories in a nut shell. How thenthe Schrodinger equation is solved by perturbation expansion in powers of the couplingconstant(s) λ and/or gauge coupling parameters g , is well-known and discussed in thestandard text books.[20][49][66]

The most important point we need to emphasise is that the above formulation ofquantised field theories is easy to replace by discretised versions. All we need to do isreplace the partial derivatives ∂i = ∂/∂xi by lattice derivatives:

∂iΦ(~x)→ 1

a

(Φ(~x+ ~eia)− Φ(~x)

), (10.11)

where a is the lattice link size, and ~ei is the unit vector along a lattice link in the idirection. The continuum limit, a ↓ 0 , seems to be deceptively easy to take; in particular,

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renormalisation will now only lead to finite correction terms. Note however, that symme-tries such as rotation symmetry and Lorentz invariance will be lost. How to recover suchsymmetries in more sophisticated models (without taking a continuum limit) is beyondour abilities just now – but notice that our treatment of string theory, Section 5.3, appearsto be heading in the right direction.

10.2. Fermionic field theories

Fermionic field systems are also an essential element in the Standard Model. The funda-mental variables are the Dirac fields ψi(~x) and their canonical associates ψ†i (~x) . Theyare spinor fields, so that i contains a spinor index. The fields anticommute. The anti-commutation rules are

ψ†i (~x), ψj(~x′) = δijδ(~x− ~x ′) , ψi, ψj = ψ†i , ψ

†j = 0 , (10.12)

where a, b ≡ ab+ ba . Note, that these rules are typical for operators of the form (0 10 0

)

and (0 01 0

) , so these rules mean that ψi(~x) is to be regarded as an operator that annihilates

an object i at position ~x , and ψ†i (~x) creates one. The rules imply that ψ2i (~x) = 0 , so

that we cannot create two objects i at the same spot ~x . Similarly, (ψ†i (~x))2 = 0 . Astate containing two (or more) particles of different type, and/or at different positions~x , will always be antisymmetric under interchange of two such fermions, which is Pauli’sprinciple.

In conventional quantum field theory, one now proceeds to the Lagrange formalism,which works great for doing fast calculations of all sorts. For our purpose, however, weneed the hamiltonian. The quantum hamiltonian density for a fermionic field theory is

HF (~x) = ψ (m+W (~Φ) + ~γ · ~∂)ψ , (10.13)

where W (~Φ) stands short for the Yukawa interaction terms that we may expect, andψ = ψ†γ4 .

The matrices γµ, µ = 1, 2, 3, 4 , need to obey the usual anti-commutation rule

γµ, γν = 2δµν , (10.14)

which requires them to be at least 4× 4 matrices, so that the spinors are 4 dimensional.One can, however, reduce these to 2 component spinors by using a constraint such as

ψ = C ψ = Cγ4ψ† , ψ† = C∗γ4 ψ , (10.15)

where ˜ stands for transposition, and C is a spinor matrix obeying 22, 23

γµC = −Cγ∗µ , C†C = 1 , C = C† . (10.16)

22In Section 3.3, we also used 2 dimensional spinors. The two-component spinor field used thereis obtained by using one of the projection operators P± = 1

2 (1 ± γ5) . The mass term can be madecompatible with that.

23The reason why Dirac needed a four-dimensional representation is that the constraint (10.15) wouldnot allow coupling to an electromagnetic field as it is would violate gauge-invariance (in particular themass term).

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Just as in the bosonic case, we may consider replacing the continuum in space by aspace-like lattice, using expressions such as Eq. (10.11), at the price of (hopefully tem-porarily) giving up Lorentz invariance.

The Yukawa term W (Φ) in Eq. (10.13) may include interactions with gauge fields inthe usual way. The question addressed in this work is to what extent hamiltonians suchas the sum of Eqs. (10.7) and (10.13) can be obtained from deterministic theories.

10.3. Perturbation theory

How to compute the effects of these hamiltonians in perturbation theory, such as thespectra, cross sections and lifetimes of the quantised particles that it contains, is standardtext material, and not the subject of this treatise, but we do need to know about someessential features for our further discussion.

One routinely splits up the hamiltonian into a “free” part H0 and an interactionpart H int , where the free part only contains terms that are bilinear in the field variablesΦi(~x), Pi(~x), ψi(~x) and ψi(~x) (possibly after having shifted some of the fields by a vac-uum value, such as in the Brout-Englert-Higgs mechanism). The interaction hamiltonianH int may also contain bilinear terms, needed to renormalise divergent effective interac-tions. There is some freedom as to whether we put parts of these so-called counter termsin H0 or in H int , which in fact is a choice concerning the book keeping process of theperturbative expansion. The fact that final calculational results should be independentof these choices is an important ingredient of what is called the renormalisation group ofthe theory.

H int is assumed to depend on the coupling parameters λi, gi, etc., of the theory,such that H int vanishes if these parameters are set to zero.

The analysis is facilitated by the introduction of auxiliary terms in the hamiltonian,called source terms, which are linear in the fields:

HJ(~x, t) =∑i

Ji(~x, t)Φi(~x) +∑i

ηi(~x, t)ψi(~x) +∑i

ψi(~x)ηi(~x, t) , (10.17)

where the “source functions” Ji(~x, t), ηi(~x, t) and ηi(~x, t) are freely chosen functions ofspace and time, to be replaced by zero at the end of the computation (the time dependenceis not explicitly mentioned here in the field variables, but, in a Heisenberg representation,of course also the fields are time dependent).

One can then find all amplitudes one needs to know, by computing at any desiredorder in perturbation theory, that is, up to certain powers of the coupling parameters andthe source functions, the vacuum-to-vacuum amplitude:

t=∞〈 ∅ | ∅ 〉t=−∞ = 1− i2

∫ ∫d4xd4x′ Ji(x)Jj(x

′)Pij(x− x′) +16

∫ ∫ ∫d4xd4x′d4x′′Wijk(x, x

′, x′′)Ji(x)Jj(x′)Jk(x

′′) + · · · , (10.18)

where x stands short for the space-time coordinates (~x, t) , and the correlation functionsPij(x − x′), Wijk(x, x

′, x′′) and many more terms of the sequence are to be calculated.

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Physically, this means that we compute expectation values of the products of operatorsΦi(~x, t), ηi(~x, t) and ηi(~x, t) of Eq. (10.17) in a Heisenberg representation. Local productsof these operators also follow from the expressions (10.18), if we take space-time pointsx and x′ to coincide.

The algorithm for the calculation of the two-point functions P , the three-point func-tions W , etc., is conveniently summarised in the so-called Feynman rules, for which werefer to the standard text books.[27][49][66]

10.4. Vacuum fluctuations, correlations and commutators

Because all contributions to our hamiltonian are translation invariant, one expects thecorrelation functions to be translation invariant as well, and this is a good reason toconsider their Fourier transforms, so, instead of x space, one considers k space:

Pij(x(1) − x(2)) = (2π)−4

∫d4kPij(k) eik(x(1)−x(2)) , (10.19)

where we will often omit the caret ( ).

Disregarding factors 2π for the moment, one finds that the two-point functions arebuilt from elementary expressions such as the Feynman propagator,

∆Fm(x) ≡ −i

∫d4k

eikx

k2 +m2 − iε, x = x(1) − x(2) , k2 = ~k2 − k2

0 , (10.20)

where ε is an infinitesimal positive number, indicating how one is allowed to arrangethe complex contour when k0 is allowed to be complex. This propagator describes thecontribution of a single, non interacting particle to the two-point correlation function. Ifthere are interactions, one finds that, quite generally, the two-point correlation functionstake the form of a dressed propagator :

∆F (x) =

∫ ∞0

dm%(m) ∆Fm(x) , (10.21)

where %(m) is only defined for m ≥ 0 and it is always non-negative. This property isdictated by unitarity and positivity of the energy, and always holds exactly in a relativisticquantum field theory[66]. The function %(m) can be regarded as the probability that anintermediate state emerges whose centre-of-mass energy is given by the number m . Inturn, %(m) can be computed in terms of Feynman diagrams. Diagrams with more externallegs (which are usually the contributions to the scattering matrix with given numbers offree particles asymptotically far away in the in-state and the out-state), can be computedwith these elementary functions as building blocks.

The two-point function physically corresponds to the vacuum expectation value of atime-ordered product of operators:

Pij(x(1) − x(2)) = 〈 ∅ |T (Φi(x

(1)),Φj(x(2))| ∅ 〉 , (10.22)

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where

T (A(t1), B(t2)) = A(t1)B(t2) , if t1 > t2 ,

= B(t2)A(t1) , if t2 > t1 (10.23)

(for fermions, this is to be replaced by the P product: a minus sign is added if twofermions are exchanged).

We shall now show how, in explicit calculations, it is always found that two operatorsO1(x(1)) and O2(x(2)) commute when they both are local functions of the fields Fi(~x, t) ,and when their space-time points are space-like separated:

(~x )2 − (x0)2 > 0 , x = x(1) − x(2) . (10.24)

To this end, one introduces the on-shell propagators :

∆±m(x) = 2π

∫d4k eikx δ(k2 +m2)θ(±k0) ; k x = ~k · ~x− k0x0 . (10.25)

By contour integration, one easily derives:

∆Fm(x) = ∆+

m(x) if x0 > 0 ;

= ∆−m(x) = ∆+m(−x) if x0 < 0 ;

∆F∗m (x) = ∆−m(x) if x0 > 0 ;

= ∆+m(x) if x0 < 0 . (10.26)

Here, ∆F∗m is obtained from the Feynman propagator DF

m in Eq. (10.20) by replacing iwith −i .

Now we can use the fact that the expressions for ∆Fm(x) and ∆±m(x) are Lorentz-

invariant. Therefore, if x is space-like, one can always go to a Lorentz frame wherex0 > 0 or a Lorentz frame where x0 < 0 , so then,

∆Fm(x) = ∆+

m(x) = ∆−m(x) = ∆F∗m (x) = ∆F

m(−x) . (10.27)

This implies that, in Eq. (10.22), we can always change the order of the two operatorsO(x(1)) and O(x(2)) if x(1) and x(2) are space-like separated. Indeed, for all two-pointfunctions, one can derive from unitarity that they can be described by a dressed propagatorof the form (10.21), where, due to Lorentz invariance, %(m) cannot depend on the signof x0 . The only condition needed in this argument is that the operator O1(x(1)) is alocal function of the fields Φi(x

(1)) , and the same for O2(x(2)) . To prove that compositefields have two-point functions of the form (10.21), using unitarity and positivity of thehamiltonian, we refer to the literature.[49] To see that Eqs. (10.27) indeed imply thatcommutators between space-like separated operators vanish, and that this implies thenon existence of information carrying signals between such points, if further shown inSection 10.5.

The fact that in all quantised field theories such as the Standard Model all space-likeseparated operators commute, implies that, in these models, no signal can be exchanged

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that goes faster than light, no matter how entangled the particles are that one looks at,a fact sometimes overlooked in studies of peculiar quantum phenomena.

Of course, if we replace the space-time continuum by a lattice in space, while time stayscontinuous, we lose Lorentz invariance, so that signals can go much faster, in principle(they still cannot go backwards in time).

Now it is crucial to notice that the Feynman propagator ∆Fm(x) itself does not vanish

at space-like separations. In general, one finds for free fields with mass m , at vanishingx(1)0 − x(2)0 , and writing ~r = ~x(1) − ~x(2) ,

〈 ∅ |T (Φ(x(1)), Φ(x(2)) | ∅ 〉 = (2π)−4∆Fm(0, ~r ) =

∫d3~k

1

2(2π)3√~k2 +m2

ei~k·~r

=1

(2π)2

∫ ∞0

k2

√k2 +m2

eik|~r | , (10.28)

but, since the fields here commute, we can omit the T symbol. For large m |~r| , thisvanishes rapidly. But when m vanishes, we have long-range correlations:

〈 ∅ |Φ(0, ~r) Φ(0,~0 ) | ∅ 〉 =1

(2π)2 ~r 2. (10.29)

For instance, for the photon field, the vacuum correlation function for the two-pointfunction is, in the Feynman gauge,

〈 ∅ |Aµ(0, ~r)Aν(0,~0 ) | ∅ 〉 =gµν

(2π)2 ~r 2. (10.30)

This means that we do have correlations over space-like distances. We attribute thisto the fact that we always do physics with states that are very close to the vacuum state.The correlations are non-vanishing in the vacuum, and all states close to the vacuum(such as all n -particle states, with n finite). One may imagine that, at very high orinfinite temperature, all quantum states will contribute with equal probabilities to theintermediate states, and this may wipe out the correlations, but today’s physics alwaysstays restricted to temperatures that are very low compared to the Planck scale, most ofthe time, at most places in the Universe.

There is even more one can say. Due to the special analytic structure of the propagators∆Fm(x) , the n point functions can be analytically continued from Minkowski space-time

to Euclidean space-time and back. This means that, if the Euclidean correlation functionsare known, also the scattering matrix elements in Minkowski space-time follow, so that theentire evolution process at a given initial state can be derived if the space like correlationfunctions are known. Therefore, if someone thinks there is “conspiracy” in the space-likecorrelations that leads to peculiar phenomena later or earlier in time, then this mightbe explained in terms of the fundamental mathematical structure of a quantum fieldtheory. The author suspects that this explains why “conspiracy” in “unlikely” space-likecorrelations seems to invalidate the Bell and CHSH inequalities, while in fact this may beseen as a natural phenomenon.

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10.5. Commutators and signals

How does information spread over space and time? Consider a field φ(x) , where x is apoint in space-time. Let the field be self-adjoint:

φ(x) = φ†(x) . (10.31)

In conventional quantum field theories, fields are operators in the sense that they measurethings and at the same time modify the state, all at one space-time point x . Usually, thefield averages in vacuum are zero:

〈∅|φ(x)|∅〉 = 0 . (10.32)

Can a signal arrive at a point x(1) when transmitted from a point x(2) ? To find out, takethe field operators φ(x(1)) and φ(x(2)) . Let us take the case t(1) ≥ t(2) . In this case,consider the propagator

〈∅|T (φ(x(1)), φ(x(2)))|∅〉 = 〈∅|φ(x(1))φ(x(2))|∅〉 =

= ∆Fm(x(1) − x(2)) = ∆+

m(x(1) − x(2)) . (10.33)

It tells us what the correlations are between the field values at x(1) and at x(2) . Thisquantity does not vanish, as is typical for correlation functions, even when points arespace-like separated.

The question is now whether the operation of the field at x(2) can affect the stateat x(1) . This would be the case if the result of the product of the actions of the fieldsdepends on their order, and so we ask: to what extent does the expression (10.33) differfrom

〈∅|φ(x(2))φ(x(1))|∅〉 = (〈∅|φ†(x(1))φ†(x(2))|∅〉)∗ = (〈∅|T (φ(x(1)), φ(x(2)))|∅〉)∗ =

= ∆F ∗m (x(1) − x(2)) = ∆−m(x(1) − x(2)) . (10.34)

In stead of ∆m(x) we could have considered the dressed propagators of the interactingfields, which, from general principles, can be shown to take the form of Eq. (10.21). Wealways end up with the identity (10.27), which means that the commutator vanishes:

〈∅|[φ(x(1)), φ(x(2))]|∅〉 = 0 , (10.35)

if x(1) and x(2) are space-like separated. Thus, it makes no difference whether we actwith φ(x(1)) before or after we let φ(x(2)) act on the vacuum. This means that no signalcan be sent from x(2) to x(1) if it would have to go faster than light.

Since Eqs. (10.27) can be proved to hold exactly in all orders of the perturbationexpansion in quantum field theory, just by using the general properties (10.26) of thepropagators in the theory, one concludes that conventional quantum field theories neverallow signals to be passed on faster than light. This is very important since less rigorousreasoning starting from the possible production of entangled particles, sometimes makeinvestigators believe that there are ‘spooky signals’ going faster than light in quantumsystems. Whatever propagates faster than light, however, can never carry information.This holds for quantum field theories and it holds for cellular automata.

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10.6. The renormalization group

A feature of quantum field theories that plays a special role in our considerations is therenormalization group. This group consists of symmetry transformations that in theirearliest form were assumed to be associated to the procedure of adding renormalizationcounter terms to masses and interaction coefficients of the theory. These counter termsare necessary to assure that higher order corrections do not become infinitely large whensystematic (perturbative) calculations are performed. The ambiguity in separating inter-action parameters from the counter terms can be regarded as a symmetry of the theory.[7]

In practice, this kind of symmetry becomes important when one applies scale trans-formations in a theory: at large distances, the counter terms should be chosen differentlyfrom what they are at a small distance scale, if in both cases we require that higher or-der corrections are kept small. In practice, this has an important consequence for mostquantum field theories: a scale transformation must be accompanied by small, calculablecorrections to all mass terms and interaction coefficients. This then adds ‘anomalous di-mensions’ to the mass and coupling parameters[66]. In lowest order, these anomalies areeasy to calculate, and the outcome is typically:

d

dµλ(µ) = βλ λ(µ)2 ,

d

dµm(µ) = βm λ(µ)m(µ) , (10.36)

with dimensionless coefficients βλ and βm . Here, µ represents the mass scale at whichthe coupling and mass parameters are being considered.

In gauge theories such as quantum electrodynamics, it is the charge squared, e(µ)2 ,or the fine structure constant α(µ) , that plays the role of the running coupling parameterλ(µ) . A special feature for non-Abelian gauge theories is that, there, the coefficient βg2receives a large negative contribution from the gauge self couplings, so, unless there aremany charged fields present, this renormalization group coefficient is negative.

It is important to observe, that the consideration of the renormalization group wouldhave been quite insignificant if there had not been large scale differences that are relevantfor the theory. These differences originate from the fact that we have very large and verytiny masses in the system. In the effective hamiltonians that we might be able to obtainfrom a cellular automaton, it is not quite clear how such large scale differences could arise.Presumably, we have to work with different symmetry feautures, each symmetry beingbroken at a different scale. Here we just note that this is not self-evident. The problemthat we encounter here is the hierarchy problem, the fact that enormously different length-,mass- and time scales govern our world. This is not only a problem for our theory here,it is a problem that will have to be confronted by other theories as well.

The mass and coupling parameters of the theory are not the only quantities thatare transformed in a non-trivial way under a scale transformation. All local operatorsO(~x, t) will receive finite renormalizations when scale transformations are performed.When composite operators are formed by locally multiplying different kinds of fields, theoperator product expansion requires scale dependent counter terms. What this meansis that operator expressions obtained by multiplying fields together undergo thorough

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changes and mixtures upon large scale transformations. The transformation that leadsus from the Planck scale to the Standard Model scale is probably such a large scaletransformation24, so that not only the masses and couplings that we observe today, butalso the fields and operator combinations that we use will be quite different from whatthey may look like at the Planck scale.

Note that, when a renormalization group transformation is performed, couplings, fieldsand operators re-arrange themselves according to their canonical dimensions. When go-ing from high mass scales to low mass scales, coefficients with highest mass dimensions,and operators with lowest mass dimensions, become most significant. This implies that,seen from a large distance scale, the most complicated theories simplify, as complicated,composite fields, as well as the coefficients they are associated with, will rapidly becomeinsignificant. This is generally assumed to be the technical reason why all our ‘effective’theories at the present mass scale are renormalizable field theories. Non-renormalizablecoefficients have become insignificant. Even if our local hamiltonian density may be quiteugly at the Planck scale, it will come out as a clean, renormalizable theory at scales suchas the Standard Model scale, exactly as the model we arrived at by fitting all observedphenomena.

11. The cellular automaton

The fundamental notion of a cellular automaton was briefly introduced in Section 3.We here resume the discussion of constructing a quantum hamiltonian for these classicalsystems, with the intention to arrive at some expression that may be compared withthe hamiltonian of a quantum field theory[71], resembling Eq. (10.6), with hamiltoniandensity (10.7), and/or (10.13). In this chapter, we show that one can come very close,although, not surprisingly, we do hit upon difficulties that cannot be completely resolved.

11.1. Local reversability by switching from even to odd sites and back

Time reversibility is important for allowing us to perform simple mathematical manipu-lations. Without time reversibility, one would not be allowed to identify single states ofan automaton with basis elements of a Hilbert space. Now this does not invalidate ourideas if time reversibility is not manifest; in that case one has to identify basis states inHilbert space with information equivalence classes, as will be explained in Section 13.2.The author does suspect that this more complicated situation might well be inevitable inour ultimate theories of the world, but we have to investigate the simpler models first.They are time reversible. Fortunately, there are rich classes of time reversible models thatallow us to sharpen our analytical tools, before making our lives more complicated.

Useful models are obtained from systems where the evolution law U consists of twoparts: one, UA , prescribes how to update all even lattice sites, and one, UB , gives the

24Unless several extra dimensions show up just beyond the TeV domain, which would bring the Planckscale closer to the Standard Model scale.

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updates of the odd lattice sites. So we have U = UA · UB .

11.1.1. The Margolus rule

In Section 3.1, the Margolus rule was introduced. The way it was phrased there, the dataon two consecutive time layers were required to define the time evolution in the futuredirection as well as back towards the past – these automata are time reversible.[26] Since,quite generally, most of our models work with single time layers that evolve towards thefuture or the past, we shrink the time variable with a factor 2. Then, one complete timestep for this automaton consists of two procedures: one that updates all even sites only,in a simple, reversible manner, leaving the odd sites unchanged, while the procedure doesdepend on the data on the odd sites, and one that updates only the odd sites, whiledepending on the data at the even sites. The first of these operators will be called UA . Itis the operator product of all operations UA(~x) , where ~x are all even sites, and we takeall these operations to commute:

UA =∏~x even

UA(~x) ; [UA(~x), UA(~x ′)] = 0 , ∀ ~x, ~x ′ . (11.1)

The commutation is assured if A(~x) depends only on its neighbours, which are odd,but not on the next-to-nearest neighbours, which are even again. Similarly, we have theoperation at the odd sites:

UB =∏~y odd

UB(~y) ; [UB(~y), UB(~y ′)] = 0 , ∀ ~y, ~y ′ , (11.2)

while [UA(~x), UB(~y)] 6= 0 only if ~x and ~y are direct neighbours.

In general, UA(~x) and UB(~y) at any single site are sufficiently simple (often they arefinite-dimensional, orthogonal matrices) that they are easy to write as exponentials:

UA(~x) = e−iA(~x) , [A(~x), A(~x ′)] = 0 ;

UB(~y) = e−iB(~y) , [B(~y), B(~y ′)] = 0 . (11.3)

A(~x) and B(~y) are defined to lie in the domain [0, 2π) , or sometimes in the domain(−π, π] .

The advantage of this notation is that we can now write25

UA = e−iA , A =∑~x even

A(~x) ; UB = e−iB , B =∑~y odd

B(~y) , (11.4)

and the complete evolution operator for one time step δt = 1 can be written as

U(δt) = e−iH = e−iA e−iB . (11.5)

Let the data in a cell ~x be called Q(~x) . In the case that the operation UA(~x) consistsof a simple addition (either a plane addition or an addition modulo some integer N ) by

25The sign in the exponents is chosen such that the operators A and B act as hamiltonians themselves.

162

a quantity δQ(Q(~yi)) , where ~yi are the direct neighbours of ~x , then it is easy to writedown explicitly the operators A(~x) and B(~y) . Just introduce the translation operator

UP (~x) = eiP (~x) , UP |Q(~x)〉 ≡ |Q− 1 modulo N〉 , (11.6)

to find

UA(~x) = e−iP (~x) δQ(~yi) , A(~x) = P (~x) δQ(~yi) ; B(~y) = P (~y) δQ(~xi) . (11.7)

The operator P (~x) is not hard to analyse. Assume that we are in a field of additionsmodulo N , as in Eq. (11.6). Go the basis of states |k〉U , with k = 0, 1, · · · , N− 1 , wherethe subscript U indicates that they are eigen states of UP and P (at the point ~x ):

〈Q|k〉U ≡1√Ne2πikQ/N . (11.8)

We have

〈Q|UP |k〉U = 〈Q+ 1|k〉U = e2πik/N〈Q|k〉U ; UP |k〉 = e2πik/N|k〉U (11.9)

(if −12N < k ≤ 1

2N ), so we can define P by

P |k〉U =2π

Nk|k〉U , 〈Q1|P |Q2〉 =

∑k

〈Q1|k〉U(2πN k) U〈k|Q2〉

=2π

N2

∑|k|< 1

2N

k e2πik(Q1−Q2)/N =4πi

N2

12N∑

k=1

k sin(2πk(Q1 −Q2)/N) . (11.10)

Now δQ(~yi) does not commute with P (~yi) , and in Eq. (11.7) our model assumes thesites ~yi to be only direct neighbours of ~x and ~xi are only the direct neighbours of ~y .Therefore, all A(~x) also commute with B(~y) unless |~x − ~y | = 1 . This simplifies ourdiscussion of the hamiltonian H in Eq. (11.5).

11.1.2. The discrete classical hamiltonian model

In subsection 7.2.3, we have seen how to generate a local discrete evolution law from aclassical, discrete hamiltonian formalism. Starting from a discrete, non negative hamil-tonian function H , typically taking values N = 0, 1, 2, · · · ,one searches for an evolutionlaw that keeps this number invariant. This classical H may well be defined as a sum oflocal terms, so that we have a non negative discrete hamiltonian density. It was decidedthat a local evolution law U(~x) with the desired properties can be defined, after whichone only has to decide in which order this local operation has to be applied to definea unique theory. In order to avoid spurious non-local behaviour, the following rule wasproposed:

The evolution equations (e.o.m.) of the entire system over one time step δt ,are obtained by ordering the coordinates as follows: first update all evenlattice sites, then update all odd lattice sites

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(how exactly to choose the order within a given site is immaterial for our discussion).The advantage of this rule is that the U(~x) over all even sites ~x can be chosen all tocommute, and the operators on all odd sites ~y will also all commute with one another;the only non-commutativity then occurs between an evolution operator U(~x) at an evensite, and the operator U(~y) at an adjacent site ~y .

Thus, this model ends up with exactly the same fundamental properties as the Cellularautomaton with Margolus’ rule: we have UA as defined in Eq. (11.1) and UB as in (11.2),followed by Eqs. (11.3)–(11.5).

We conclude that, for model building, splitting a space-time lattice into the even andthe odd sub lattices is a trick with wide applications. It does not mean that we shouldbelieve that the real world is also organised in a lattice system, where such a fundamentalrole is to be attributed to the even and odd sub lattices; it is merely a convenient tool formodel building. We shall now discuss why this splitting does seem to lead us very closeto a quantum field theory.

11.2. The Baker Campbell Hausdorff expansion

To compute the hamiltonian H , we may consider using the Baker Campbell Hausdorffexpansion[24]:

eP eQ = eR ,

R = P +Q+ 12[P,Q] + 1

12[P, [P,Q]] + 1

12[[P,Q], Q] + 1

24[[P, [P,Q]], Q] + · · · , (11.11)

a series that continues exclusively with commutators. Replacing P by −iA , Q by −iBand R by −iH , we find a series for H in the form of an infinite sequence of commutators.Now we noted at the end of the previous subsection that the commutators between thelocal operators A(~x) and B(~x ′) are non-vanishing only if ~x and ~x ′ are neighbours,|~x− ~x ′| = 1 . Therefore, if we insert the sums (11.4) into Eq. (11.11), we obtain again asum. Writing

K(~r ) = A(~r ) if ~r is even, and B(~r ) if ~r is odd,

L(~r ) = A(~r ) if ~r is even, and −B(~r ) if ~r is odd, (11.12)

so that

A(~r) = 12

(K(~r) + L(~r)

)and B(~r) = 1

2

(K(~r)− L(~r)

), (11.13)

we find

H =∑~r

H(~r ) ,

H(~r ) = H1(~r ) +H2(~r ) +H3(~r ) + · · · , (11.14)

where

H1(~r ) = K(~r ) ,

164

H2(~r ) = 14i∑~s

[K(~r ), L(~s)] ,

H3(~r ) = 124

∑~s1, ~s2

[L(~r ) , [K(~s1), L(~s2)]] , etc. (11.15)

The sums here are only over close neighbours, so that each term here can be regarded asa local hamiltonian density term.

Note however, that as we proceed to collect higher terms of the expansion, more andmore distant sites will eventually contribute; Hn(~r ) will receive contributions from sitesat distance n− 1 from the original point ~r .

Note furthermore that the expansion (11.14) is infinite, and convergence is not guar-anteed; in fact, one may suspect it not to be valid at all, as soon as energies larger thanthe inverse of the time unit δt come into play. We will have to discuss that problem. Butfirst an important observation that improves the expansion.

11.3. Conjugacy classes

One might wonder what happens if we change the order of the even and the odd sites.We would get

U(δt) = e−iH?= e−iB e−iA , (11.16)

in stead of Eq. (11.5). Of course this expression could have been used just as well. Infact, it results from a very simple basis transformation: we went from the states |ψ〉 tothe states UB|ψ〉 . As we stated repeatedly, we note that such basis transformations donot affect the physics.

This implies that we do not need to know exactly the operator U(δt) as defined inEqs. (11.5) or (11.16), we just need any one element of its conjugacy class. The conjugacyclass is defined by the set of operators of the form

GU(δt)G−1 , (11.17)

where G can be any unitary operator. Writing G = eF , where F is anti-hermitean, wereplace Eq. (11.11) by

eR = eF eP eQ e−F , (11.18)

so that

R = R + [F,R] + 12[F, [F,R]] + 1

3![F, [F, [F,R]]] + · · · . (11.19)

We can now decide to choose F in such a way that the resulting expression becomesas simple as possible. For one, interchanging P and Q should not matter. Secondly,replacing P and Q by −P and −Q should give −R , because then we are looking atthe inverse of Eq. (11.19).

165

The calculations simplify if we write

S = 12(P +Q) , D = 1

2(P −Q) ; P = S +D , Q = S −D (11.20)

(or, in the previous section, S = −12iK , D = −1

2iL ). With

eR = eF eS+D eS−D e−F , (11.21)

we can now demand F to be such that:

R(S,D) = R(S,−D) = −R(−S,−D) , (11.22)

which means that R contains only even powers of D and odd powers of S . We can fur-thermore demand that R only contains terms that are commutators of D with something;contributions that are commutators of S with something can be removed iteratively byjudicious choices of F .

Using these constraints, one finds a function F (S,D) and R(S,D) . First let usintroduce a short hand notation. All our expressions will consist of repeated commutators.Therefore we introduce the notation

X1, X2, · · · , Xn ≡ [X1, [X2, [· · · , Xn]] · · ·] . (11.23)

Subsequently, we even drop the accolades . So if we write

X1X2X23 X4 , we actually mean: [X1, [X2, [X3, [X3, X4]]]] .

Then, with F = −12D + 1

24S2D + · · · , one finds

R(S,D) = 2S − 112DSD + 1

960D(8S2 −D2)SD+

160480

D(−51S4 − 76DSDS + 33D2S2 + 44SD2S − 38D4)SD + O(S,D)9 . (11.24)

There are three remarks to be added to this result:

(1) It is much more compact than the original BCH expansion; already the first twoterms of the expansion (11.24) corresponds to all terms shown in Eq. (11.11).

(2) The coefficients appear to be much smaller than the ones in (11.11), considering thefactors 1/2 in Eqs. (11.20) that have to be included. We found that, in general,sizeable cancellations happen between the coefficients in (11.11).

(3) There is no reason to suspect that the series will actually converge any better. Thedefinitions of F and R may be expected to generate singularities when P and Q ,or S and D reach values where eR obtains eigen values that return to one, so, forfinite matrices, the radius of convergence is expected to be of the order 2π .

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In this representation, all terms Hn(~r ) in Eq. (11.14) with n even, vanish. Using

S = −12iK , D = −1

2iL , R = −iH , (11.25)

one now arrives at the hamiltonian in the new basis:

H1(~r ) = K(~r ) ,

H3(~r ) = 196

∑~s1,~s2

[L(~r ), [K(~s1), L(~s2)]] , (11.26)

H5(~r ) = 130720

∑~s1,···,~s4 [L(~r ),

(8[K(~s1), [K(~s2)− [L(~s1), [L(~s2)

), [K(~s3), L(~s4)]]]] ,

and H7 follows from the last line of Eq. (11.24).

All these commutators are only non-vanishing if the coordinates ~s1 , ~s2 , etc., areall neighbours of the coordinate ~r . It is true that, in the higher order terms, next-to-nearest neighbours may enter, but still, one may observe that these operators are all localfunctions of the ‘fields’ Q(~x, t) , and thus we arrive at a hamiltonian H that can beregarded as the sum over d -dimensional space of a hamiltonian density H(~x) , which hasthe property that

[H(~x), H(~x ′)] = 0 , if |~x, ~x ′| 1 . (11.27)

At every finite order of the series, the hamiltonian density H(~x) is a finite-dimensionalhermitean matrix, and therefore, it will have a lowest eigenvalue h . In a large but finitevolume V , the total hamiltonian H will therefore also have a lowest eigenvalue, obeying

E0 > hV . (11.28)

However, it was tacitly assumed that the Baker-Campbell-Hausdorff formula con-verges. This is usually not the case. One can argue that the series may perhaps convergeif sandwiched between two eigenstates |E1〉 and |E2〉 of H , where E1 and E2 are theeigenvalues, that obey

|E1 − E2| < 2π , (11.29)

We cannot exclude that the resulting effective quantum field theory will depend quitenon-trivially on the number of Baker-Campbell-Hausdorff terms that are kept in theexpansion.

The hamiltonian density (11.26) may appear to be quite complex and unsuitable toserve as a quantum field theory model, but it is here that we actually expect that therenormalization group will thoroughly cleanse our hamiltonian, by invoking the mecha-nism described at the end of Subsection 10.6.

11.4. An other approach to obtain a local hamiltonian density

The hamiltonian constructed in the previous chapter looks very similar to the genuinequantum field theory hamiltonians that we normally work with in elementary particle

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physics. Of course, we found a lattice hamiltonian, and of course it does not show allthe continuous symmetry features that characterise our Universe, but this may not seeman issue that should affect our arguments concerning its fundamental quantum nature,its ability to allow for entangled states, and thus to overcome the Bell and the CHSHinequalities.

There are two more serious problems however. One is that the convergence of theBCH series, even the series for the conjugacy class, Eq. (11.24), reflected in Eq. (11.26),is not at all guaranteed. No matter how many terms we include, this may never lead to ahamiltonian that properly represents the model. Possibly, the hamiltonian will representparts of all states, but which parts? Do we have to require some smearing of the ontologicalstates, so as to wipe out the highest order terms in the expansion? This is not known, sowe do not wish to rely on that.

This then leads us also to a second difficulty: if we carry along too many terms ofthe expansion, we introduce more and more fundamentally non-local contributions tothe hamiltonian, see also the expansions discussed in Section 7.3. Paradoxically, themodel itself is entirely local, quantum field theories require a local hamiltonian density tosafeguard locality features, but now we see that our technical construction does not allowus to reflect this kind of locality. What did we do wrong?

Critics will now say that this is a simple consequence of an incorrect philosophy.Quantum mechanics has nothing to do with classical cellular automata, so just give it up.

It may well be so that our automaton models are too simplistic to demand that theywork so well, but it seems far more likely that the problems mentioned before requiremore physical input, rather than totally abandoning the procedure. We think that allsmall, partial successes reported in this work warrant a continuation along this path. Forinstance, one obvious physical feature has been left aside completely, while it is extremelylikely that it cannot be ignored: quantum gravity.

In this section we demonstrate how certain concepts from gravity will sneak in, evenif we did not ask for them. One feature is obvious: a local concept of hamiltonian densityis needed in gravity theory to act as the generator of a local diffeomorphism in the timedirection. Indeed, sooner or later we also might wish to identify a local Pointing vector, ormomentum density, which is the generator of local displacements in space. Thus, generalcoordinate transformations may well be expected to play a decisive role in our search fora local quantum formalism of classical systems.

Quantum gravity is still an open issue of physics; indeed, one motivation for thisstudy is to get a better handle on that fundamental problem; we believe that a completeunderstanding of the principles of quantum gravity will come hand in hand with a completeunderstanding of quantum mechanics.

Let us now go back to our calculations.

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11.4.1. Locality in fractional time evolution

Suppose we continue the BCH expansion to include very large numbers of terms. Aswe saw, the hamiltonian density (11.26) develops more and more non-locality. We ac-tually find sequences of K and L operators that are adjacent, but stretch over longer‘trajectories’ in space.

Such behaviour of a hamiltonian in physical models is not uncommon, if we think ofour hamiltonian as an effective hamiltonian; it may have a very simple physical origin:a forgotten physical variable. Suppose a field exists that obeys local equations, but it isintegrated out to obtain an ‘effective’ hamiltonian. This effective hamiltonian will thenfeature non-local behaviour very similar to what we see here. So, let us see if we canidentify a candidate for such a field variable.

The idea is to take a field θ(~x ) that takes some predefined values, for instance θ(~x) =0 , at integer times ( t = n δt ), it evolves to some values during a fraction of the time,but it returns to the initial values ( θ(~x) = 0 ) at integer times. We could require a localevolution law for θ(~x) , but when we integrate it out, the hamiltonian may develop somedeceptive non-locality with no physical consequences. Whether the θ field will actually beobservable is immaterial, it suffices to describe the entire system in terms of a hamiltonianthat, when θ(~x) is included, acquires the desired locality properties.

Let us start with constructing a partial solution to the question how to do this. Theoperator U(δt) is beautifully local only at integer time steps δt , but now take any integerN . If we try to rewrite U(δt) as the Nth power of an evolution operator U(δt/N) , thenU(δt/N) is non-local. Now here, we can identify the ‘hidden’ field variable θ(~x) . Wemake the classical theory local for time steps δt/N in the following way.

Our field variable θ at a site ~x takes N possible values:

τ(~x) = 0, 1, · · · ,N− 1 . (11.30)

The operator U(δt/N) is now defined as follows:

U(~x, δt/N) =

e−iK(~x) e−iPθ(~x) , if θ(~x) = 0 ,e−iPθ(~x) , if θ(~x) 6= 0 ,

(11.31)

where K(~x) = A(~x) on even sites and B(~x) on odd sites, while e−iPθ(~x) is the stepoperator for the θ field at the point ~x , that is, e−iP (~x)|θ(~x)〉 = |θ(~x) + 1 mod N〉 .

As our initial state we do choose this θ(~x ) field to vanish, but only at the even sites.At the odd sites it takes a value ≈ N/2 .

We immediately see that if we apply this operator N times, the field θ(~x ) returnsto its initial values, while the operations UA and UB take place alternatingly, as theyshould.

It is important to note that generalisations exist that make the transition operatorsmoother: there is no need to have the operators e−iK(~x) act only at one value of θ(~x) .In stead of Eq. (11.31), one may choose:

U(~x, δt/N) = e−iK(~x) g(θ(~x)) e−iPθ(~x) , (11.32)

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where the function g(θ) is chosen such that

N−1∑0

g(θ) = 1 , g(θ) = 0 for 12N ≤ θ < N , (11.33)

but we may want to choose this function to be as smooth as possible, see Fig. 17.

g(θ) g(θ + N/2)

0 θ N − 1

Figure 17: The function g(θ) used in Eq. (11.32), It is chosen such that it doesnot overlap with g(θ + N/2) (dashed curve).

We see that, again, if we start with an initial state where θ(~x) alternates between 0on the even and N/2 on the odd sites (assuming N to be even at the moment), then theevolution will exactly reproduce the alternating applications of UA and UB .

11.4.2. Taking the continuous time limit. A local hamiltonian?

In many of our models the operators UA(~x) and UB(~x) act in a finite-dimensional vectorspace, because our automaton may only allow for a finite number of different states ateach site. This means that the operator K(~x) acts in a finite dimensional vector space,and so, as a matrix, it will certainly have a lower bound. If also the operator θ(~x) canonly take a finite number of values, we may be well on our way to construct a finite,bounded hamiltonian density, if, somehow, we can construct the continuum limit of theprocedure sketched in the previous subsection.

What we wish to construct is a hamiltonian of the general form (10.6), where H(~x)obeys

[H(~x), H(~x ′)] = 0 if |~x− ~x ′| 6= 0 . (11.34)

This does not guarantee that signals will have a bounded velocity, because we have noexact Lorentz invariance as yet, but it does reproduce the desired locality conditions inthe sense that interactions must be associated to signals transmitted through space andtime in between.

Now the hamiltonian generates time translations over infinitesimal distances, whereasU(δt/N) generates the evolution over a finite, but arbitrarily small time stretch. Unfortu-nately, this is not quite the same thing. We can take the continuum limit of Eqs. (11.32)–(11.33), but there is a price to be paid: in the continuum limit, the hamiltonian density

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will act in an infinite dimensional Hilbert space, and it will not be automatically boundedfrom below.

In the continuum limit, the operator θ/N turns into the continuum angular operatorϕ/2π . It can actually be interpreted as the fractional part of time, turned into an (aux-iliary) operator. It is easy to see how Eq. (11.32) can be exponentiated: with δt ≡ 1 ,and

θ =N2πϕ , Pθ =

2π

NPϕ , g(θ) =

2π

Nh(ϕ) , (11.35)

so that now ∫ 2π

0

h(ϕ)dϕ = 1 , (11.36)

we have

U(~x, δt/N) = exp(− iH(~x)

N

), H(~x) = 2π

(K(~x)h(ϕ(~x)) + P op

ϕ (~x)). (11.37)

Indeed, this hamiltonian is completely local, obeying Eq. (11.34). However, if we takethe Hilbert space of all functions of the variable ϕ(~x) on the periodic interval [0, 2π] ,then the eigen values Pϕ of P op

ϕ (~x) run over all positive and negative integers. It makesH(~x) unbounded from below.

One may be tempted to use the mathematics displayed in Subsection 2.6.1. There,the periodically rotating angle ϕ was linked to the harmonic oscillator, which does havea hamiltonian bounded from below. Only the non-negative values of the integer Pϕ areused there. Can we do the same thing in the present case?

To see what happens, let us consider the Fourier expansion of the function h(ϕ) . Sinceh(ϕ) vanishes over an entire stretch of the variable ϕ , it cannot be analytically continuedwithout singularities, which means that there will always be non vanishing amplitudes forthe positive and the negative Fourier modes, or, we will definitely encounter the Fouriermodes e±ikϕ with both signs in the exponent. Now, from Eqs. (2.50) we see that thesemodes can be recast into creation and annihilation operators of the harmonic oscillator,but they will also include operators of the form

|N− k〉〈 ` | and | ` 〉〈N− k| , (11.38)

where the integers k and ` may emerge when higher Fourier coefficients are encountered.

Regardless how large N , the effects from these contributions will not go away, inparticular if combinations of both sides of (11.38) occur. They will, because we justargued that Fourier coefficients with both signs are inevitable.

One could investigate whether approximations to the function h(ϕ) can be madewhich have only positive or only negative Fourier coefficients. But then we cannot keep hreal-valued, so that our hamiltonian will include non hermitean parts, again a questionableway to proceed.

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What we conclude from the above exercise is that there seem to be many ways torewrite a cellular automaton in terms of a theory with a local hamiltonian density, butthere are always some obstacles, usually being the absence of a lower bound on thehamiltonian. What we saw in this last subsection is that it may be helpful to introduce alocal time variable ϕ(~x) that is periodic; it extrapolates between two integer time values,t = Nδt and t = (N + 1)δt . Being defined locally, it can be linked to a local energydensity, H(~x) , and this is what we think we have to require for a healthy quantumdescription of a cellular automaton. Our problem was the fact that the operator P op

ϕ isneeded, and its eigen values are integers running from −∞ to ∞ .

At the same time, we saw in Chapter 7 that we can construct cellular automata startingfrom a discrete classical Hamilton function that is bounded from below by construction.Perhaps these two ideas have to be combined into one, but we do not know exactly howto do this; it seems that the group of general space-time automorphisms will play animportant role, or, the problem of quantising gravity. It may well be that our failure toclose these last gaps is due to the simple fact that we still do not understand quantumgravity.

12. Concise description of the CA interpretation

We finally arrived at describing as precisely as possible what the Cellular AutomatonInterpretation (CAI) entails. It is exactly the thing usually claimed categorically to beimpossible. One could characterise it as a ‘hidden variable’ theory, although in most ofour models, nothing is hidden (In Section 13.2 we do discuss a refinement of the theorywhere the physical variables do tend to get hidden from us, due to information loss, butit is not obvious that such a refinement will be necessary, as yet).

The CAI actually has more in common with the original Copenhagen doctrine thanmany other approaches, as will be explained. It will do away with the ‘many worlds’, moreradically than the de Broglie-Bohm interpretation. The CAI assumes the existence of oneor more models of Nature that have not yet been discovered, although, in the above, manytoy models were discussed. These toy models are not good enough to come anywhere closeto the Standard Model, but there is reason to hope that one day such a model will befound. In any case, the CAI gives only a tiny sub class of all quantum mechanical modelsusually considered to explain the observed world, and for this reason we hope it might behelpful to pin down the right procedure to arrive at the correct theory.

Most of the models in previous chapters were exposed in this work just to display theset of tools that one might choose to use.

12.1. Motivation

It is not too far-fetched to expect that, one day, quantum gravity will be completelysolved, that is, a concise theory will be phrased that shows an air-tight description of therelevant physical degrees of freedom, and a simple model will be proposed that shows how

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these physical variables evolve. We might not even need a conventional time variable forthat, but what we do need is an unambiguous prescription telling us how the physicaldegrees of freedom will look in a region of space-time that lies in the future of a regiondescribed earlier.

A complete theory explaining quantum mechanics can probably not be formulatedwithout also addressing quantum gravity, but what we can do is to formulate our proposal,and to establish the language that will have to be employed. Today, our description ofmolecules, atoms, fields and relativistic subatomic particles is interspersed with wavefunctions and operators. Our operators do not commute with operators describing otheraspects of the same world, and we have learned not to be surprised by this, but just tochoose a set as we please, guess a reasonable looking Schrodinger equation, and calculatewhat we should find when we measure things. We were told not to ask what reality is, andthis turned out to be a useful advice: we calculate, and we observe that the calculationsmake sense. It is unlikely that any other aspects of fields and particles can ever becalculated, it will never be more than what we can derive from quantum mechanics. Forexample, given a radio-active particle, we cannot calculate exactly at what moment it willdecay. The decay is controlled by a form of randomness that we cannot control, and thisrandomness seems to be far more perfect than what can be produced using programmedpseudo-random sequences. We are told to give up any hope to outsmart Nature in thisrespect.

The Cellular Automaton Interpretation (CAI) is suggesting to us what it is that weactually do when we solve a Schrodinger equation. We thought that we are followingan infinite set of different worlds, each with some given amplitude, and the final eventsthat we deduce from our calculations depend on what happens in all these worlds. This,according to the CAI, is an illusion. There is no infinity of different worlds, there isjust one, but we are using the “wrong” basis to describe it. The word “wrong” is hereused not to criticise the founding fathers of quantum mechanics, who made marvellousdiscoveries, but to repeat what they of course also found out, which is that this basis isnot an ontological one. The terminology used to describe that basis does not disclose tous exactly how our world, our single world, ‘actually’ evolves in time.

Many other proposed interpretations of quantum mechanics exist. These may eitherregard the endless numbers of different worlds all to be real, or they require some sort ofmodification, mutilation rather, to understand how a wave function can collapse to pro-duce measured values of some observable without allowing for mysterious superpositionsat the classical scale.

The CAI proposes to use the complete mathematical machinery that has been de-veloped over the years to address quantum mechanical phenomena. It includes exactlythe Copenhagen prescriptions to translate the calculations into precise predictions whenexperiments are done, so, at this point, definitely no modifications are required.

There is one important way in which we deviate from Copenhagen however. Accordingto Copenhagen, certain questions are not to be asked:

Can it be that our world is just a single world where things happen, according

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to evolution equations that might be fundamentally simpler than Schrodinger’sequation, and are there ways to find out about this? Can one remove theprobability element from quantum mechanics?

According to Copenhagen, there exist no experiments that can answer such questions,and therefore it is silly even to ask them. But what I have attempted to show in thiswork is that not experimentally, but theoretically, we may find answers. We may beable to identify models that describe a single classical world, even if forbiddingly complexcompared to what we are used to, and we may be able to identify its physical degrees offreedom with certain quantum variables that we already know about.

The cellular automaton described in previous chapters would be the prototype exam-ple; it is complicated yet quite possibly not complicated enough. It has symmetries, butin the real world there are much larger symmetry groups, such as the Lorentz or Poincaregroup, showing relations between different kinds of events, and these are admittedly diffi-cult to implement. The symmetry groups, think of space-time translation symmetry, mayactually be at the roots of the mysterious features that were found in our quantum world,so that these may have natural explanations.

Why should we want just a single world with classical equations describing its evo-lution? What’s wrong with obeying Copenhagen’s dictum about not asking questions, ifthey will not be experimentally accessible anyway?

According to the author, there will be overwhelming motivations: If a classical modeldoes exist, it will tremendously simplify our view of the world, it will help us once andforever to understand what really happens when a measurement is made and a wavefunction ‘collapses’. It will explain the Schrodinger cat paradox once and for good.

But more importantly, the quantum systems that allow for a classical interpretationform an extremely tiny subset of all quantum models. If it is indeed true that our worldfalls in that class, which one may consider to be likely after having read all this, then thisrestricts our set of allowable models so much that it may enable us to guess the right one,a guess that otherwise could never have been made. So indeed, what we are really after isa new approach towards guessing what the ‘Theory of the World’ is. We strongly suspectthat, without this superb guide, we will never even come close. Thus, our real motivationis not to be able to better predict the outcomes of experiments, which may not happensoon, but rather to predict which class of models we should scrutinise to find out aboutthe truth.

Let us emphasise once more, that this means that the CAI will primarily be of impor-tance if we wish to decipher nature’s laws at the most fundamental time- and distancescales, that is, the Planck scale. Thus, an important new frontier where the empire of thequantum meets the classical world, is proclaimed to be near the Planckian dimensions.As we also brought forward repeatedly, this idea requires a reformulation of our standardquantum language also when describing this other important border: that between thequantum empire and the ‘ordinary’ classical world at distance scales larger than the sizesof atoms and molecules.

Of course we assume that the reader is familiar with the ‘Schrodinger representation’

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as well as the ‘Heisenberg representation’ of conventional quantum mechanics. We oftenuse these intermittently.

12.1.1. The wave function of the universe

Standard quantum mechanics can confront us with a question that appears to be difficultto answer: Does the universe as a whole have a wave function? Can we describe that wavefunction? Several responses to this question can be envisaged:

(1) I do not know, and I do not care. Quantum mechanics is a theory about observationsand measurements. You can’t measure the entire universe.

(2) I do care, but I do not know. Such a wave function might be so much entangledthat it will be impossible to describe. Maybe the universe has a density matrix, nota wave function.

(3) The universe has no fixed wave function. Every time an observation or measurementis made, the wave function collapses, and this collapse phenomenon does not followany Schrodinger equation.

(4) Yes, the universe has a wave function. It obeys the Schrodinger equation, and atall times the probability that some state |ψ〉 is realised is given by the norm of theinner product squared. Whenever any ‘collapse’ takes place, it always obeys theSchrodinger equation.

The agnostic answers (1) and (2) are scientifically of course defensible. We should limitourselves to observations, so don’t ask silly questions. However, they do seem to admitthat quantum mechanics may not have universal validity. It does not apply to the universeas a whole. Why not? Where exactly is the limit of the validity of quantum mechanics?The ideas expressed in this work were attacked because they allegedly do not agree withobservations, but all observations ever made in atomic and subatomic science appear toconfirm the validity of quantum mechanics, while the answers (1) and (2) suggest thatquantum mechanics should break down at some point. In the CAI, we assume that themathematical rules for the application of quantum mechanics have absolute validity. This,we believe, is not a crazy assumption.

In the same vein, we also exclude answer (3). The collapse should not be regardedas a separate axiom of quantum mechanics, one that would invalidate the Schrodingerequation whenever a collapse takes place. Therefore, according to our theory, the onlycorrect answer is answer (4). An apparent problem with this would be that the collapsewould require ‘conspiracy’, a very special choice of the initial state because otherwise wemight accidentally arrive at wave functions that are quantum superpositions of differentcollapsed states. This is where the ontological basis comes as a rescue. If the universe isin an ontological state, its wave function will collapse automatically, whenever needed.

Now, let us go back to Copenhagen, and formulate the rules. As the reader cansee, in some respects we are even more conservative than the most obnoxious quantumdogmatics.

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12.2. The rules

As for the Copenhagen rules that we keep, we emphasise the ones most important to us:

i To describe a physical phenomenon, the use of any basis is as legitimate as any other.We are free to perform any transformation we like, and rephrase the Schrodingerequation, or rather, the hamiltonian employed in it, accordingly. In each basis wemay find a useful description of variables, such as positions of particles, or thevalues of their momenta, or the energy states they are in, or the fields of whichthese particles are the energy quanta. All these descriptions are equally ‘real’ .

But none of the usually employed descriptions are completely real. We often see thatsuperpositions occur, and the phase angles of these superpositions can be measured, oc-casionally. In that case, the basis used is not an ontological one. In practice, we havelearned that this is just fine; we use all of these different basis choices in order to havetemplates. We shall impose no restrictions on which template is ‘allowed’, or which ofthem may represent the truth ‘better’ than others. Since these are mere templates, realitymay well emerge to be a superposition of different templates.

Curiously, even among the diehards of quantum mechanics, this was often thought notto be self-evident. “Photons are not particles, protons and electrons are”, it was claimed.Photons must be regarded as energy quanta. They certainly thought it was silly to regardthe phonon as a particle, it is merely the quantum of sound. Sometimes it is claimed thatelectric and magnetic fields are the ”true” variables, while photons are merely abstractconcepts. There were disputes on whether a particle is a true particle in the positionrepresentation or in the momentum representation, and so on. We do away with all this.All basis choices are equivalent. They are nothing more than a coordinate frame in Hilbertspace. As long as the hamiltonian and all finite-time evolution operators contain non-diagonal elements, these basis choices are non-ontological; none of them describes whatis really happening. As for the energy basis, see Subsection 12.3.4.

Note, that this means that it is not really the hamiltonian that we will be interested in,but rather its conjugacy class. If a hamiltonian H is transformed into a new hamiltonianby the transformation

H = GH G−1 , (12.1)

where G is a unitary operator, then the new hamiltonian H , in the new basis, is justas valid as the previous one. In practice, we shall seek the basis, or its operator G , thatgives the most concise possible expression for H .

ii The probability for the outcome of a measurement to be described by a given statein Hilbert space, is exactly given by the absolute square of the inner product of thegiven state with the computed state.

This is the well-known Born rule. We also accept it without any modifications. The Bornrule if often portrayed as a separate axiom of the Copenhagen Interpretation. In our view

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it is an inevitable consequence of the mathematical nature of quantum mechanics as atool to perform calculations, see Section 8.6.

The most important point where we depart from Copenhagen is that we make afundamental assumption:

a We postulate the existence of an ontological basis. It is an orthonormal basis ofHilbert space that is truly superior to the basis choices that we are familiar with. Interms of an ontological basis, the evolution operator for a sufficiently fine mesh oftime variables, does nothing more than permute the states.

How exactly to define the mesh of time variables we do not know at present, and may wellbecome a subject of debate, particularly in view of the known fact that space and timehas a general coordinate invariance built in. We do not claim to know how to fabricatethis construction – it is too difficult. In any case, the system is expected to behaveentirely as a classical construction. Our basic assumption is that a classical evolutionlaw exists, dictating exactly how space-time and all its contents evolve. The evolution isdeterministic in the sense that nothing is left to chance. Probabilities enter only if, dueto our ignorance, we seek our refuge in some non-ontological basis.

b What we do when we perform a conventional quantum mechanical calculation isthat we employ a set of templates for what we thought the wave function is like.These templates, such as the orthonormal set of solutions of the hydrogen atom,just happen to be the states for which we know how they evolve. However, they arein a basis that is a rather complicated rotation of the ontological basis.

Humanity discovered that these templates obey Schrodinger equations, and we employthese to compute probabilities for the outcomes of experiments. These equations arecorrect to very high accuracies, but they falsely suggest that there is a ‘multiverse’ ofmany different worlds that interfere with one another. Today, our templates are the bestwe have.

c Very probably, there are more than one different choices for the ontological basis,linked to one another by Nature’s continuous symmetry transformations such asthe elements of the Poincare group, but possibly also by the local diffeomorphismgroup used in General Relativity. Only one of these ontological bases will be ‘truly’ontological.

Which of them will be truly ontological will be difficult or impossible to determine. Thefact that we shall not be able to distinguish the different possible ontological bases, willpreclude the possibility of using this knowledge to perform predictions beyond the usualquantum mechanical ones. This was not our intention anyway. The motivation for thisinvestigation has alway been that we are searching for new clues in constructing modelsmore refined than the Standard Model.

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The symmetry transformations that link different (but often equivalent) ontologicalbasis choices are likely to be truly quantum mechanical: operators that are diagonal inone of these ontological bases may be non diagonal in an other. However, in the very endwe shall only use the ‘real’ ontological basis. This will be evident in axiom e.

d Classical states are ontological, which means that classical observables are alwaysdiagonal in the ‘truly’ ontological basis.

This would be more difficult to ‘prove’ from first principles, so we introduce it indeed asan axiom. However, it seems to be very difficult to avoid: it is hard to imagine that twodifferent classical states, whose future evolution will be entirely different in the end, couldhave non-vanishing inner products with the same ontological state.

e From the very beginning onwards, the Universe was, and it always will be, in a single,evolving ontological state. This means that not only the observables are diagonal inthe ontological basis, but the wave function always takes the simplest possible form:it is one of the elements of the basis itself, so this wave function only contains asingle one and for the rest zeros.

Note that this singles out the ‘true’ ontological basis from other choices, where the physicaldegrees of freedom can also be represented by ‘beables’, that is, operators that at alltimes commute. So, henceforth, we only refer to this one ‘true’ ontological basis as ‘the’ontological basis.

Most importantly, the last two axioms completely resolve the measurement problem[16], the collapse question and Schrodinger’s cat paradox. The argument is now simplythat Nature is always in a single ontological state, and therefore it has to evolve into asingle classical state.

12.3. Features of the Cellular Automaton Interpretation (CAI)

A very special feature of the CAI is that ontological states do never form superpositions.From day and time zero onwards, the universe must have been in a single ontological statethat evolves. This is also the reason why it never evolves into a superposition of classicalstates. Now remember that the Schrodinger equation is obeyed by the ontological statesthe universe may be in. This then is the reason why this theory automatically generates‘collapsed wave functions’ that describe the results of a measurement, without ever partingfrom the Schrodinger equation. For the same reason, ontological states can never evolveinto a superposition of a dead cat and a live cat. Regarded from this angle, it actuallyseems hard to see how any other interpretation of quantum mechanics could have survivedin the literature: quantum mechanics by itself would have predicted that if states |ψ 〉 and|χ 〉 can be used as initial states, so can the state α|ψ 〉 + β|χ 〉 . Yet the superpositionof a dead cat and a live cat cannot serve to describe the final state. If |ψ 〉 evolves intoa live cat and |χ 〉 into a dead one, then what does the state α|ψ 〉+ β|χ 〉 evolve into?the usual answers to such questions cannot be correct.

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The Cellular Automaton Interpretation adds some notions to quantum mechanics thatdo not have any distinguished meaning in the usual Copenhagen view. We introduced theontological basis as being, in some sense, superior to any other choice of basis. One mightnaturally argue that this would be a step backwards in physics. Did we not emphasise inthe previous section that all choices of basis are equivalent? Why would one choice standout?

Indeed, what was stated in rule #i in Section 12.2 was that all basis choices areequivalent, but what we really meant was that all basis choices normally employed areequivalent. Once we adopt the Copenhagen doctrine, it does not matter anymore whichbasis we choose. Yet there is one issue in the Copenhagen formalism that has been heavilydisputed in the literature and now is truly recognised as a weakness: the collapse of thewave function and the treatment of measurements. At these points the superpositionaxiom fails. As soon as we admit that one superior basis exists, this weakness disappears.All wave functions may be used to describe a physical process, because this is what wehave to tolerate if we do not work in the ontological basis.

The CAI allows us to use a basis that stands out. It stands out because, in thisbasis, we recognise wave functions that stand out: the ontological wave functions. In theontological basis, the ontological wave functions are the wave functions that correspondto the basis elements; these wave functions contain a one and for the rest zeros. Theontological wave functions cannot be superimposed to produce another ontological wavefunction. In this basis, there is no room for chance anymore.

The evolution is deterministic. However, this term must be used with caution. “De-terministic” cannot imply that the outcome of the evolution process can be foreseen. Nohuman, nor even any other imaginable intelligent being, will be able to compute fasterthan Nature itself. The reason for this is obvious: our intelligent being would also haveto employ Nature’s laws, and we have no reason to expect that Nature can duplicate itsown actions more efficiently than itself. This is how one may restore the concept of “freewill”: whatever happens in our brains is unique and unforeseeable by anyone or anything.

12.3.1. Beables, changeables and superimposables

Having a special basis and special wave functions, also allows us to distinguish specialobservables or operators. In standard quantum mechanics, we learned that operators andobservables are indistinguishable, so we use these concepts interchangeably. Now, we willhave to learn to restore the distinctions. We repeat what was stated in Subsection 2.1.1,operators can be of three different forms:

(I) beables Bop : these denote a property of the ontological states, so that beables arediagonal in the ontological basis |A〉, |B〉, · · · of Hilbert space:

B aop|A〉 = B a(A)|A〉 , (beable) . (12.2)

Beables will always commute with one another, at all times :

[B aop(~x1, t1), B bop(~x2, t2)] = 0 ∀ ~x1, ~x2, t1, t2. (12.3)

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Quantised fields, copiously present in all elementary particle theories, do obey Eq. (12.3),but only outside the light cone (where |t1 − t2| < |~x1 − ~x2| ), not inside that cone, whereEqs. (10.27) do not hold, as can easily be derived from explicit calculations. Therefore,quantised fields are altogether different from beables.

(II) changeables Cop : operators that replace an ontological state by another ontologicalstate, such as a permutation operator:

Cop|A〉 = |B〉 , (changeable) . (12.4)

Changeables do not commute, but they do have a special relationship with beables; theyinterchange them:

B(1)op C(1)

op = C(2)op B(2)

op . (12.5)

We do make an exception for infinitesimal changeables, such as the hamiltonian Hop :

[Bop, Hop] = −i ∂∂tBop . (12.6)

(III) superimposables Sop : these map ontological states onto any other, more generalsuperpositions of ontological states:

Sop|A〉 = λ1|A〉+ λ2|B〉+ · · · . (12.7)

All operators normally used are superimposables, even the simplest position or momentumoperators of classroom quantum mechanics. This is easily verified by checking the time-dependent commutation rules (in the Heisenberg representation):

[~x(t1), ~x(t2)] 6= 0 , if t1 6= t2 . (12.8)

12.3.2. Observers and the observed

Standard quantum mechanics taught us a number of important lessons. One was thatwe should not imagine that an observation can be made without disturbing the observedobject. This warning is still valid in the CAI. If measuring the position of a particle means

checking the wave function whether perhaps ~x?= ~x(1) , this may be interpreted as having

the operator Pop(~x(1)) act on the state:

|ψ〉 → Pop(~x(1))|ψ〉 , Pop(~x(1)) = δ(~xop − ~x(1)) . (12.9)

This modifies the state, and consequently all operators acting on it after that may yieldresults different from what they did before the “measurement”.

However, when a genuine beable acts on an ontological state, the state is simplymultiplied with the value found, but will evolve in the same way as before (assuming we

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chose the ‘true’ ontological basis, see axiom # c in Section 12.2). Thus, measurements ofbeables are, in a sense, classical measurements.

Other measurements, which seem to be completely legal according to conventionalquantum mechanics, will not be possible in the CAI. In the CAI, as in ordinary quantummechanics, we can consider any operator and study its expectation value. But sincethe class of physically realisable wave functions is now smaller than in standard QM,certain states can no longer be realised, and we cannot tell what the outcome of such ameasurement could possibly be. Think of any (non infinitesimal) changeable Cop . Allontological states will give the ‘expectation value’ zero to such a changeable, but we canconsider its eigen values, which in general will not yield the value zero. The correspondingeigen states are definitely not ontological (see Subsection 12.3.5).

Does this mean that standard quantum mechanics is in conflict with the CAI? Weemphasise that this is not the case. It must be realised that, also in conventional quantummechanics, it may be considered acceptable to say that the Universe has been, and willalways be, in one and the same quantum state, evolving in time in accordance with theSchrodinger equation (in the Schrodinger picture), or staying the same (in the Heisenbergpicture). If this state happens to be one of our ontological states then it behaves exactlyas it should. Ordinary quantum mechanics makes use of template states, most of whichare not ontological, but then the real world in the end must be assumed to be in asuperposition of these template state so that the ontological state resurfaces anyway, andour apparent disagreements disappear.

12.3.3. Inner products of physical states

When technical calculations are performed, we perform transformations and superposi-tions that lead to large sets of quantum states which we now regard as templates, orcandidate models for (sub) atomic processes that can always be superimposed at a laterstage to describe the phenomena we observe. Inner products of template can be studiedthe usual way.

A state used to serve as a model of some actually observed phenomenon will be called aphysical state. A physical state |ψ〉 is defined to be any quantum superposition of ontolog-ical states |A〉 . The inner product expressions |〈A|ψ〉|2 then represent the probabilitiesthat ontological state |A〉 is actually realised.

According to the Copenhagen rule #ii, Section 12.2, the probability that the physicalstate described by |ψ1〉 is found to be equal to the state |ψ2〉 , is given by | 〈ψ1|ψ2〉|2 .However, already at the beginning, Section 2.1, we stated that the inner product 〈ψ1|ψ2〉should not be interpreted this way. Even if their inner product vanishes, physical states|ψ1〉 and |ψ2〉 may both have a non vanishing coefficient with the same ontological state|A〉 . This does not mean that we depart from Copenhagen rule #ii, but that the truewave function cannot be just a generic physical state; it is always an ontological state|A〉 . This means that the inner product rule is only true if either |ψ1〉 or |ψ2〉 is atemplate while the other may be a physical state. The template state is assumed tocoincide exactly with one of the ontological states. We may use the Born interpretation

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of the inner product if the template state in question is used to describe only a smallsubspace of all physical degrees of freedom. How the unobserved degrees of freedom areentangled with this state is then immaterial, which is why the template state then can beassumed to represent the ontological state of the universe.

We see that the inner product rule can be used in two ways; one is to describe theprobability distribution of the initial states of a system under consideration, and one isto describe the probability that a given classical state is reached at the end of a quantumprocess. If the Born rule is used to describe the initial probabilities, the same rule can beused to calculate the probabilities for the final states.

The density matrix of the universe must also be diagonal in the ontological basis.

12.3.4. The energy basis

Beables are operators that commute at all times, and ontological states are eigen statesof these beables. There is a trivial example of such operators and such states in the realworld: the hamiltonian and its eigen states. According to our definition they form a set ofbeables, but unfortunately, they are trivial: there is only one hamiltonian, and the energyeigen states do not change in time at all. This describes a static classical world. What iswrong with it?

Since we declared that the ontological states do not allow for superpositions, thissolution also tells us that no superpositions are allowed, while here all conventional physicsonly re-emerges when we do consider superpositions of the energy eigen states, so yes, thisis a solution but no, it is not the solution we want. The energy basis solution emerges forinstance if we take the model of Section 2.3.1, Figures 2 and 3, where we replace all loopsby trivial loops, having only a single state, while putting all the physics in the arbitrarynumbers δEi . It is in accordance with the rules but not useful.

The same may happen in Subsection 7.2.1, Figure 12, if we take the classical energyfunction too steep: then all loops may contain just a single site, and all (classical) motioncomes to a halt. Clearly, our beable models must be constructed with more care; themost productive approaches will contain different closed loops of states evolving into oneanother, but the numbers of sites in one loop must be very large. Indeed, in the real worldwe expect contours whose lengths equal the Poincare recursion time of the Universe, inshort, gigantically big. The models with finite or very small contours will only be usefulas intermediate stages for the construction of more realistic theories.

12.3.5. The Earth–Mars interchange operator

The CAI surmises that quantum models exist that can be regarded as classical systemsin disguise. If one looks carefully at these classical systems, it seems as if any classicalsystem can be rephrased in quantum language: we simply postulate an element of abasis of Hilbert space to all classical physical states that are allowed in the system. Theevolution operator is the permutator that replaces a state by its successor in time, atwhich we may or may not decide to consider the continuous time limit.

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Naturally, therefore, we ask the question whether one can reverse the CAI, and con-struct quantum theories for systems that are normally considered classical. The answer isyes. To illustrate this, let us consider the planetary system. It is the prototype of a classi-cal theory. We consider large planets orbiting around a sun, and we ignore non-Newtoniancorrections such as special and general relativity, or indeed the actual quantum effects,all of which being negligible corrections. We regard the planets as point particles, evenif they may feature complicated weather patterns, and even life; we just look at theirclassical, Newtonian equations of motion.

The ontological states are defined by listing the positions ~xi and velocities ~vi of theplanets (which commute), and the observables describing them are the beables. Yet alsothis system allows for the introduction of changeables and superimposables. The quantumhamiltonian here is not the classical thing, but

Hquant =∑i

(~p opx, i · ~vi + ~p op

v, i · ~Fi(x)/mi

), (12.10)

where

~p opx, i = −i ∂

∂~xi, ~p op

v, i = −i ∂∂~vi

, and x = ~xi . (12.11)

here, ~Fi(x) are the classical forces on the planets, which depend on all positions.Eq. (12.10) can be written more elegantly as

Hquant =∑i

(~p opx, i ·

∂Hclass

∂~pi− ~p op

p, i ·∂Hclass

∂~xi

). (12.12)

Clearly, ~p opx i , ~p op

v i and ~p opp i are infinitesimal changeables, and so is, of course, the

hamiltonian Hquant . The planets now span a Hilbert space and behave as if they werequantum objects.

We can continue to define more changeables, and ask how they evolve in time. One ofthe author’s favourites is the Earth–Mars interchange operator. It puts the Earth whereMars is now, and puts Mars where planet Earth is. The velocities are also interchanged.If Earth and Mars had the same mass then the planets would continue to evolve as ifnothing happened. Since the masses are different however, this operator will have rathercomplicated properties when time evolves. It is not conserved in time.

The eigen values of the Earth–Mars interchange operator XEM are easy to calculate:

XEM = ±1 , (12.13)

simply because the square of this operator is one. In standard QM language, XEM is anobservable. It does not commute with the hamiltonian because of the mass differences,but, at a particular moment, t = t1 , we can consider one of its eigen states and ask howit evolves.

Now why does all this sound so strange? How do we observe XEM ? No-one canphysically interchange planet Earth with Mars. But then, no-one can interchange two

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electrons either, and yet, in quantum mechanics, this is an important operator. Theanswer to these questions is, that regarding the planetary system, we happen to knowwhat the beables are: they are the positions and the velocities of the planets, and thatturns them into ontological observables. The basis in which these observables are diagonaloperators is our preferred basis. The elements of this basis are the ontological states ofthe planets. If, in a quantum world, investigators one day discover what the ontologicalbeables are, everything will be expressed in terms of these, and any other operators areno longer relevant.

12.3.6. Rejecting local counterfactual realism and free will

The arguments usually employed to conclude that local hidden variables cannot exist,begin with assuming that such hidden variables should imply local counterfactual realism.One imagines a set-up such as the EPR-Bell experiment that we exposed in Section 8.7.Alice and Bob are assumed to have the ‘free will’ to choose the orientation of theirpolarisation filter anytime and anyway they wish, and there is no need for them to consultanyone or anything to reach their (arbitrary) decision. The quantum state of the photonthat they are about to investigate should not depend on these choices, nor should thestate of a photon depend on the choices made at the other side, or the outcome of thosemeasurements.

This means that the outcomes of measurements should already be determined bysome algorithm long before the actual measurements are made, and also long before thechoice was made what to measure. It is this algorithm that generates conflicts with theexpectations computed in a simple quantum calculation. It is counterfactual, which meansthat there may be one ‘factual’ measurement but there would have been many possiblealternative measurements, measurements that are not actually carried out, but whoseresults should also have been determined. This is what is usually called counterfactualreality, and it has essentially been disproven by simple logic.

Now, as has been pointed out repeatedly, the violation of counterfactual reality is notat all a feature that is limited to quantum theory. In our example of the planetary system,Subsection 12.3.5, there is no a priori answer to the question which of the two eigen statesof the Earth-Mars exchange operator, the eigen value +1 or −1 , we are in. This is aforbidden, counterfactual question. But in case of the planetary system, we know whatthe beables are (the positions and velocities of the planets), whereas in the StandardModel we do not know this. There, the illegitimacy of counterfactual statements is notimmediately obvious. In essence, we have to posit that Alice and Bob do not have the freewill to change the orientation of their filters; if they do, their decisions have their roots inthe past and as such affect the possible states a photon can be in. In short, Alice and Bobmake decisions that are correlated with the polarisations of the photons, as explained inSection 8.7.

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12.3.7. Entanglement

An objection often heard against the general philosophy advocated in this work, is thatone would never be able to accommodate for entangled particles this way. The carefulreader, however, must realise by now that there should be no problem of this sort. Anyquantum state can be considered as a template, and the evolution of these templates willbe governed by the real Schrodinger equation. If the relation between the ontologicalbasis and the more conventional basis choices is sufficiently complex, we will encountersuperimposed states of all sorts, so one definitely also expects states where particles behaveas being ‘quantum-entangled’.

There are some problems and mysteries, however. The EPR paradox and Bell’s the-orem are examples. As explained in Section 8.7, the apparent contradictions can onlybe repaired if we assume rather extensive correlations among the ‘hidden variables’ ev-erywhere in the Universe. It seems as if conspiracy takes place: due to some miraculouscorrelations in the events at time = 0 , a pair of photons ‘knows in advance’ what thepolarisation angles of the filters will be that they will encounter later, and how theyshould pass through them. Where and how did this enter in our formalism, and how doesa sufficiently natural system, without any conspiracy at the classical level, generate thisstrange behaviour?

It is not only a feature among entangled particles that may appear to be so problematic.The conceptual difficulty is already manifest at a much more basic level. Consider a singlephoton, regardless whether it is entangled with other particles or not. Our description ofthis photon in terms of the beables dictates that these beables behave classically. Whathappens later, at the polarisation filter(s), is also dictated by classical laws. These classicallaws in fact dictate how myriads of variables fluctuate at what we call the Planck scale, ormore precisely, the smallest scale where distinguishable physical degrees of freedom canbe recognised, which may or may not be close to what is usually called the Planck scale.We claim that the quantum templates, presently represented by the laws of the StandardModel, are the only way to deal with these highly complex an uncontrollable phenomena,in a statistical manner. These templates include entangled states.

But this is not the answer to the question posed. The usual way to phrase the questionis to ask how ‘information’ is passed on. Is this information classical or quantum? If itis true that the templates are fundamentally quantum templates, we are tempted to say,well, the information passed on is quantum information. Yet it does turn into classicalinformation at the Planck scale, and this was generally thought not to be possible.

That must be a mistake. As we saw in Section 8.7, the fundamental technical contra-diction goes away if we assume strong correlations between the vacuum fluctuations at alldistance scales (including correlations between the fluctuations generated in quasars thatare separated by billions of light years). We think the point is the following. When weuse templates, we do not know in advance which basis one should pick to make them looklike the ontological degrees of freedom as well as possible. For a photon going through apolarisation filter, the basis closest to the ontological one is the basis where coordinatesare chosen to be aligned with the filter. But this photon may have been emitted by a

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quasar billions of years ago, how did the quasar know what the ontological basis is?

The answer is that indeed the quasar knows what the ontological basis is, because ourtheory extends to these quasars as well. The information, ‘this is the ontological basis,and any set of superimposed states is not’, is a piece of information that, according to ourtheory, is absolutely conserved in time. So, if that turns out to be the basis now, it wasthe basis a billion years ago. The quasars seem to conspire in a plot to make fools of ourexperimenters, but in reality they just observe a conservation law: the information as towhich quantum states form the ontological basis, is conserved in time. Much like angularmomentum, this conservation law tells us what this variable is in the future if it is knownin the past, and vice-versa.

12.3.8. The superposition principle in Quantum Mechanics

What exactly happened to the superposition principle in the CA Interpretation of quan-tum mechanics? Critics of our work brought forward that the CAI disallows superposition,while obviously the superposition principle is serving quite well as a solid back bone ofquantum mechanics. Numerous experiments confirm that if we have two different states,also any superposition of these states can be realised. Although the reader should haveunderstood by now how to answer this question, let us attempt to clarify the situationonce again.

At the most basic level of physical law, we assume only ontological states to occur, andany superposition of these, in principle, does not correspond to an ontological state. Atbest, a superposition can be used to describe probabilistic distributions of states (we callthese “physical states”, since in physics, we often do not have the exact information athand to determine with absolute certainty which ontological state we are looking at). Inour description of the Standard Model, or any other known physical system such as atomsand molecules, we do not use ontological states but templates, which can be regarded assuperpositions of ontological states. The hydrogen atom is a template, all elementaryparticles we know about are templates, and this means that the wave function of theuniverse, which is an ontological state, must be a superposition of our templates. Whichsuperposition? Well, we will encounter many different superpositions when doing repeatedexperiments. This explains why we were led to believe that all superpositions are alwaysallowed.

But not literally all superpositions can occur. Superpositions are man-made. Ourtemplates are superpositions, but that is because they represent only the very tiny sectorof Hilbert space that we understand today. The entire universe is in only one ontologicalstate at the time, and it of course cannot go into superpositions of itself. This factnow becomes manifestly evident when we consider the “classical limit”. In the classicallimit we again deal with certainties. Classical states are also ontological. When we do ameasurement, by comparing the calculated “physical state” with the ontological classicalstates that we expect in the end, we again recover the probabilities by taking the normsquared of the amplitudes. Classical states also never go into superpositions of classicalstates. Such superpositions never occur.

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It appears that for many scientists this is difficult to accept. During a whole century wehave been brainwashed with the notion that superpositions occur everywhere in quantummechanics. At the same time we were told that if you try to superimpose classical states,you will get probabilistic distributions instead. It is here that our present theory is moreaccurate: if we knew the wave function of the universe exactly, we would find that it alwaysevolves into one classical state only, without uncertainties and without superpositions.

Of course this does not mean that standard quantum mechanics would be wrong. Ourknowledge of the template states, and how these evolve, is very accurate today. It is onlybecause it is not yet known how to relate these template states to the ontological statesthat we have to perform superpositions all the time when we do quantum mechanicalcalculations. They do lead to statistical distributions in our final predictions, rather thancertainties. This could only change if we would find the ontological states, but since eventhe vacuum state is expected to be a template, and as such a complicated superpositionof uncountably many ontic states, we should expect quantum mechanics to stay with usforever – but as a mathematical tool, not as a mystic departure from classical logic.

13. Alleys to be further investigated, and open questions

The Cellular Automaton Interpretation is not entirely free of problems. As was explainedextensively in previous chapters, we are not so much concerned about interference phe-nomena, entanglement, the Bell - CHSH inequalities, and related issues. We reproducequantum mechanics exactly, so also the numerous peculiarities and counter intuitive char-acteristics of quantum mechanics will be duly reproduced.

There are however other reasons why the most explicit model constructions sketchedhere are not, or not yet, sufficiently refined to serve as models where we can explain alltypically quantum mechanical consequences. A critical reader may rightly point to theseobstacles.

13.1. Positivity of the hamiltonian

The most obnoxious, recurring question is that the hamiltonians reproduced in our models,more often than not, appear to lack a lower bound. This problem was already emphasisedin Section 9.1.1.

In the real world we have (apart from a freely adjustable additive constant)

〈ψ |H |ψ 〉 ≥ 0 , (13.1)

for all states |ψ 〉 in Hilbert space. At the same time, the hamiltonian generates the timeevolution:

d

dt|ψ 〉 = −iH |ψ 〉 . (13.2)

If we take just any classical model, we usually have no difficulty finding an operator H

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that generates the evolution law (13.2), but then it often does not automatically obey(13.1).

Both these equations are absolutely crucial for understanding quantum mechanics. Inparticular, the importance of the bound (13.1) is sometimes underestimated. If the boundwould not have been there, none of the familiar solutions of the Schrodinger equationwould be stable; any, infinitesimally tiny, perturbation in H would make a solution withenergy E decay into a combination of two solutions with energies E + δE and −δE .

All solutions of the Schrodinger equation would have such instabilities, so that wouldnot be the quantum world we are used to.

This is the reason why we always try to find an expression for H such that a lowerbound exists (which we can subsequently normalise to be exactly zero). From a formalpoint of view, it should be easy to find such a hamiltonian. Every classical model shouldallow for a description in the form of the simple models of Section 2.3, a collection ofcogwheels, and we see in Fig. 3 that we can subsequently adjust the constants δEi sothat the bound (13.1) exists, even if there are infinitely many cogwheels with unlimitednumbers of teeth.

However, we then would like to impose a third condition: The hamiltonian should beextensive, that is,

H =

∫d3~xH(~x ) , (13.3)

where H(~x ) is the hamiltonian density. Locality then amounts to the demand that, atdistances exceeding some close distance limit, these Hamilton densities must commute(Eq. (11.27)). This is where our problem becomes hard. Using special elements of theconjugacy classes, leading to expansions such as (11.24) and (11.26), we get quite rapidlydecreasing values for these commutators, but this convergence was found not to be goodenough.

As stated in Subsection 11.4.2, one may suspect that the ultimate solution that obeysall our demands will come from quantising gravity, where we know that there must exista local hamiltonian density that generates local time diffeomorphisms. In other treatiseson the interpretation of quantum mechanics, this important role that might be played bythe gravitational force is often not mentioned.

Some authors do suspect that gravity is a new elementary source of ‘quantum deco-herence’, but such phrases are hardly convincing. In these arguments, gravity is treatedperturbatively (Newton’s law is handled as an additive force, while black holes and scatter-ing between gravitational waves are ignored). As a perturbative, long-range force, gravityis in no fundamental way different from electromagnetic forces. Anyway, decoherence[34]is a vague concept that we completely avoid here.

Since we have not solved the hamiltonian positivity problem completely, we have nosystematic procedure to control the kind of hamiltonians that can be generated fromcellular automata. Ideally, we should try to approximate the hamiltonian density ofthe Standard Model. Having approximate solutions that only marginally violate our

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requirements will not be of much help, because of the hierarchy problem: the StandardModel hamiltonian (or what we usually put in its place, its lagrangian), requires finetuning. This means that very tiny mismatches at the Planck scale will lead to very largeerrors at the Standard Model scale. The hierarchy problem has not been solved, so thisindeed gives us an other obstacle.

13.2. Information loss

Gravity is perhaps not the only refinement that may help us towards building bettermodels. An other interesting modification might help us; perhaps it will have to be addedin combination with the adaptations required for the addition of gravity. We shall nowdiscuss information loss.[39][62]

Let us return to the Cogwheel Model, discussed in Section 2.2. The most generalautomaton may have the property that two or more different initial states evolve into thesame final state. For example, we may have the following evolution law involving 5 states:

(4)→ (5)→ (1)→ (2)→ (3)→ (1) . (13.4)

The diagram for this law is a generalisation of Fig. 1, now shown in Fig. 18. We see thatin this example states #3 and state #5 both evolve into state #1.

a)

54

3

21

b)

1

3,5

2,4

c)

2

1

0

E

Figure 18: a) Simple 5-state automaton model with information loss. b) Its threeequivalence classes. c) Its three energy levels.

At first sight, one might imagine to choose the following operator as its evolutionoperator:

U(δt)?=

0 0 1 0 11 0 0 0 00 1 0 0 00 0 0 0 00 0 0 1 0

. (13.5)

However, since there are two states that transform into state #1 whereas there are nonethat transform into state #4, this matrix is not unitary, and it cannot be written as theexponent of −i times an hamiltonian.

One could think of making tiny modifications in the evolution operator (13.5), sinceonly infinitesimal changes suffice to find some sort of (non hermitean) hamiltonian ofwhich this then would be the exponent. This turns out not to be such a good idea. It

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is better to look at the physics of such models. Physically, of course, it is easy to seewhat will happen. States #4 and 5 will only be realised once if ever. As soon as we areinside the cycle, we stay there. So it makes more sense simply to delete these two ratherspurious states.

The problem with that is that, in practice, it might be quite difficult to decide whichstates are inside a closed cycle, and which states descend from a state with no past(“gardens of Eden”). Here, it is the sequence #4, #5. but in many more realisticexamples, the gardens of Eden will be very far in the past, and difficult to trace. Ourproposal is, instead, to introduce the concept of information equivalence classes (info-equivalence classes for short):

Two states (a) and (b) at time t0 are equivalent if there exists a time t1 > t0such that, at time t1 , state (a) and (b) have evolved into the same state (c) .

This definition sends states #5 and #3 in our example into one equivalence class, andtherefore also states #4 and #2 are in the same equivalence class. Our example hasjust 3 equivalence classes, and these classes do evolve with a unitary evolution matrix,since, by construction, their evolution is time-reversible. Info-equivalence classes willshow some resemblance with gauge equivalence classes, and they may well actually bequite large. Also, the concept of locality will be a bit more difficult to understand, sincestates that locally look quite different may nevertheless be in the same class. Of course,the original underlying classical model may still be completely local. Our pet example isConway’s game of life[15]: an arbitrary configuration of ones and zeros arranged on a two-dimensional grid evolve according to some especially chosen evolution law. The law is nottime-reversible, and information is lost at a big scale. Therefore, the equivalence classesare all very big, but the total number of equivalence classes is also quite large, and themodel is physically non-trivial. An example of a more general model with informationloss is sketched in Figure 19. We see many equivalence classes that each may containvariable numbers of members.

The advantage of this treatment is that it can be established whether two states arein the same equivalence class also if we cannot follow the evolution very far back intime. Also, we now see how such models might be made time reversible anyway: we keepthe same equivalence classes but try to establish which larger system now possess theseclasses, when evolved backwards in time. This can be done by considering the sites wheredifferent strands merge (“merge sites”). Time reversal is then achieved by interchangingthe merge sites with the gardens of Eden, see Fig. 19.

Thus, we find how models with information loss may still be connected with quantummechanics:

Each info-equivalence class corresponds to an element of the ontological basisof a quantum theory.

The mathematics of these procedures, which may resemble the mathematics of thegauge-equivalence classes in local gauge theories, is presently still beyond the scope ofthis work.

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Figure 19: Example of a more generic finite, deterministic, time non reversiblemodel. Largest (pink and blue) dots: these also represent the equivalence classes.Smallest (green) dots: “gardens of Eden”. Heavier dots (blue): equivalence classesthat have “merge sites” among their members. The info-equivalence classes and theenergy spectrum are as in Figs. 2 and 3.

Can information loss be helpful? Intuitively, the idea might seem to be attractive.Consider the measurement process, where bits of information (“qubits”) that originallywere properties of single particles, are turned into macroscopic observables. These maybe considered as being later in time, and all mergers that are likely to happen musthave taken place. In other words, the classical states are obviously represented by theequivalence classes. However, when we were still dealing with individual qubits, themergers have not yet taken place, and the equivalence classes may form very complex, ina sense “entangled”, sets of states. Locality is then difficult to incorporate in the quantumdescription, so, in these models, it may be easier to expect some rather peculiar featuresregarding locality – just the thing we need.

13.3. Holography

There is an other reason to expect information loss, which concerns microscopic blackholes. Classically, black holes absorb information, so, one may have to expect that ourclassical, or ‘pre-quantum’ system also features information loss. Even more compellingis the original idea of holography [36][57]. It was Stephen Hawking’s important deriva-tion that black holes emit particles[19], due to quantum fluctuations near the horizon.However, his observation appeared to lead to a paradox:

The calculation suggests that the particles emerge in a thermal state, withperfect randomness, regardless how the black hole was formed. Not only indeterministic theories, but also in pure quantum theories, the arrangement ofthe particles coming out should depend to some extent on the particles thatwent in to form a black hole.

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In fact, one expects that the state of all particles emerging from the black hole shouldbe related to the state of all particles that formed the black hole by means of a unitarymatrix, the scattering matrix S .[51][54]

Properties of this matrix S can be derived using physical arguments.[60] One usesthe fact that particles coming out must have passed by all particles that went in, andthis involves the conventional scattering matrix determined by their “Standard Model”interactions. Now since the centre-of-mass energies involved here are often much largerthan anything that has been tested in laboratories, much stays uncertain about this theory,but in general one can make some important deductions:

The only events that are relevant to the black hole’s reaction upon the particlesthat went in, take place in the region very close to the event horizon. Thishorizon is two-dimensional.

This means that all information that is processed in the vicinity of a black hole, must beeffectively represented at the black hole horizon. Hawking’s expression for the black holeentropy resulting from his analysis of the radiation clearly indicates that it leaves onlyone bit of information on every surface element of roughly the Planck length squared. Innatural units:

S = πR2 = log(W ) = (log 2) 2logW ; W = 2Σ/4 log 2 , (13.6)

where Σ is the surface area of the horizon.

Where did the information of the surrounding bulk of space-time go? We think itwas lost. If we phrase the situation this way, we can have holography without losinglocality of the physical evolution law. This evolution law apparently is extremely effectivein destroying information.26

Information loss in a local field theory must then be regarded in the following way(“holography”):

In any finite, simply connected region of space, the information contained inthe bulk gradually disappears, but what sits at the surface will continue to beaccessible, so that the information at the surface can be used to characterisethe info-equivalence classes.

At first sight, using these info-equivalence classes to represent the basis elements of aquantum description may seem to be a big departure from our original theory, but we haveto realise that, if information gets lost at the Planck scale, it will be much more difficultto lose any information at much larger scales; there are so many degrees of freedom thaterasing information completely is very hard and improbable; rather, we are dealing with

26Please do not confuse this statement with the question whether quantum information is lost near ablack hole horizon. According to the hypothesis phrased here, quantum information is what is left if weerase all redundant information by combining states in equivalence classes. The black hole micro statesthen correspond to these equivalence classes. By construction, equivalence classes do not get lost.

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the question how exactly information is represented, and how exactly do we count bits ofinformation in the info-equivalence classes.

In short: information loss is mostly a local feature; globally, information is preserved.This does mean that our identification of the basis elements of Hilbert space with info-equivalence classes appears to be not completely local. On the other hand, both theclassical theory and the quantum theory naturally forbid information to be spread fasterthan the speed of light.

13.4. More about edge states

Edge states were encountered frequently in this investigation. Usually, they are completelydelocalised in space-time, or in the space of field variables. To require that we limitourselves to quantum states that are orthogonal to edge states means that we are makingcertain restrictions on our boundary conditions. What happens at the boundary of theuniverse? What happens at the boundary of the hamiltonian (that is, at infinite energies)?This seems to be hardly of relevance when questions are asked about the local laws ofour physical world. In Chapter 4, we went out of our way to reduce our edge state to asingle point on the (κ, ξ) torus. There, we were motivated by the desire to obtain moreconvergent expressions. Edge states generate effective non-locality, which we would liketo see reduced to a minimum.

We also had to confront the edge states in our treatment of the constraints for thelongitudinal modes of string theory (Subsection 5.3.5). Intuitively, these edge state effectsseem to be more dangerous; quite conceivably, the edge states will not always be asinnocent as they look.

13.5. Invisible hidden variables

In the simplest examples of models that we discussed, for instance those in Chapter 2, therelation between the deterministic states and the quantum basis states is mostly straight-forward and unambiguous. However, when we reach more advanced constructions, we findthat, given the quantum hamiltonian and the description of the Hibert space in which itacts, there is a multitude of ways in which one can define the ontological states. Thishappens when the quantum model possesses symmetries that are broken in the ontologicaldescription. Look at our treatment of string theory in Section 5.3: the quantum theoryhas the entire continuous, D dimensional Poincare group as a symmetry, whereas, inthe deterministic description, this is broken down to the discrete lattice translations androtations in the D − 2 dimensional transverse space.

Since most of our deterministic models necessarily consist of discretised variables, theywill only, at best, have discrete symmetries, which means the all continuous symmetriesC of the quantum world that we attempt to account for, must be broken down to discretesubgroups D ⊂ C . This means that there is a group C/D of non-trivial transformations ofthe ontological variables into themselves. We can never know which of these are the ‘true’ontological variables, and this means that the ‘true’ ontological variables are hidden from

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us. Thus, which operators exactly are to be called true beables, which are changeablesand which are superimposables, will be hidden from our view. This is why we are happyto adopt the phrase ‘hidden variables’ to describe our ontological variables. Whether ornot we can call them ‘invisible’ depends on the question whether any quantum states canbe invisible. That phrase might be misleading.

14. Conclusions

It may seem odd that our theory, unlike most other approaches, does not contain anystrange kinds of stochastic differential equation, no “quantum logic”, not an infinity ofother universes, no pilot wave, just completely ordinary equations of motion that wehave hardly been able to specify, as they could be almost anything. Our most essentialpoint is that we should not be deterred by ‘no go theorems’ if these contain small printand emotional ingredients in their arguments. The small print that we detect is theassumed absence of strong local as well as non-local correlations in the initial state. Ourmodels show that there should be such strong correlations. Correlations do not requiresuperluminal signals.

The emotional ingredient is the idea that the conservation of the ontological natureof a wave function would require some kind of ‘conspiracy’, as it is deemed unbelievablethat the laws of nature themselves can take care of that. Our point is that they obviouslycan. Once we realise this, we can consider studying very simple local theories.

In principle it is almost trivial to obtain “quantum mechanics” out of classical theories.We demonstrated how it can be done with a system as classical as the Newtonian planetsmoving around the sun. But then difficulties do arise, which of course explain why ourproposal is not so simple after all. The positivity of the hamiltonian is one of the primestumbling blocks. We can enforce it, but then the plausibility of our models needs to bescrutinised. At the very end we have to concede that the issue will involve the mysteries ofquantum gravity. Our present understanding of quantum gravity suggests that discretisedinformation is spread out in a curved space-time manifold; that’s difficult to reconcile withnature’s various continuous symmetry properties such as Lorentz invariance. So, yes, it isdifficult to get these issues correct, but we suggest that these difficulties will only indirectlybe linked to the problem of interpreting quantum mechanics.

This report is about these questions, but also about the tools needed to address them;they are the tools of conventional quantum mechanics, such as symmetry groups andNoether’s theorem.

A distinction should be made between on the one hand explicit theories concerningthe fate of quantum mechanics at the tiniest meaningful distance scale in physics, and onthe other hand proposals for the interpretation of today’s findings concerning quantumphenomena.

Our theories concerning the smallest scale of natural phenomena are still very incom-plete. Superstring theory has come a long way, but seems to make our view more opaquethan desired; in any case, here we investigated only rather simplistic models, of which it

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is at least clear what they say.

14.1. The CAI

What we did find, however, seems to be more than sufficient to extract a succinct in-terpretation of what quantum mechanics really is about. The technical details of theunderlying theory do not make much difference here. All one needs to assume is thatsome ontological theory exists; it will be a theory that describes phenomena at a verytiny distance scale in terms of evolution laws that process bits and bytes of information.These evolution laws may be “as local as possible”, requiring only nearest neighbours tointeract directly. The information is also strictly discrete, in the sense that every “Planck-ian” volume of space may harbour only a few bits and bytes. We also suspect that thebits and bytes are processed as a function of local time, in the sense that only a finiteamount of information processing can take place in a finite space-time 4-volume. On theother hand, one might suspect that some form of information loss takes place such thatinformation may be regarded to occupy surface elements rather than volume elements,but this we could not elaborate very far.

In any case, this local theory of information being processed, does not require anyHilbert space or superposition principles to be properly formulated. The bits and byteswe discussed are classical bits and bytes; at the most basic level of physics (but onlythere) qubits do not play any role, in contrast with more standard approaches considered intoday’s literature. Hilbert space only enters when we wish to apply powerful mathematicalmachinery to address the question how these evolution laws generate large scale behaviour,possibly collective behaviour, of the data.

Our theory for the interpretation of what we observe is now clear: humanity discoveredthat phenomena at the distance and energy scale of the Standard Model (which comprisesdistances vastly larger, and energies far smaller, than the Planck scale) can be capturedby postulating the effectiveness of templates. Templates are elements of Hilbert spacethat form a basis that can be chosen in numbers of ways (particles, fields, entangledobjects), which allow us to compute the collective behaviour of solutions to the evolutionequations that do require the use of Hilbert space and linear operations in that space. Theoriginal observables, the beables, can all be expressed as superpositions of our templates.Which superpositions one should use, differs form place to place. This is weird but notinconceivable. Apparently there exists a powerful scheme of symmetry transformationsallowing us to use the same templates under many different circumstances. The rule fortransforming beables to templates and back is complex and not unambiguous; exactlyhow the rules are to be formulated, for all objects we know about in the universe, is notknown or understood, but must be left for further research.

Most importantly, the original ontological beables do not allow for any superposition,just like we cannot superimpose planets, but the templates, with which we compare thebeables, are elements of Hilbert space and require the well-known principles of superpo-sition.

The second element in our CAI is that objects we normally call classical, such as

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planets and people, but also the dials and all other detectable signals coming from mea-surement devices, can be directly derived from beables, without use of the templates.

Of course, if we want to know how our measurement devices work, we use our tem-plates, and this is the origin of the usual “measurement problem”. What is often portrayedas mysteries in quantum theory: the measurement problem, the ‘collapse of the wave func-tion’, and Schrodinger’s cat, is completely clarified in the CAI. All wave functions thatwill ever occur in our world, may seem to be superpositions of our templates, but they arecompletely peaked, ‘collapsed’, as soon as we use the beable basis. Since classical devicesare also peaked in the beable basis, their wave functions are collapsed. No violation ofSchrodinger’s equation is required for that, on the contrary, the templates, and indirectly,also the beables, exactly obey the Schrodinger equation.

In short, it is not nature’s degrees of freedom themselves that allow for superposition, itis the templates we normally use that are man-made superpositions of nature’s ontologicalstates. It is due to our intuitive thinking that our templates represent reality in someway, that we hit upon the apparently inevitable paradoxes concerning superposition ofthe natural states. If, instead, we start from the ontological states that we may one daysucceed to characterise, the so-called ‘quantum mysteries’ will disappear.

14.2. Counterfactual reality

Suppose we have an operator Oop1 whose value cannot be measured since the value of an-

other operator, Oop2 , has been measured, while [Oop

1 , Oop2 ] 6= 0 . Counterfactual reality

is the assumption that, nevertheless, the operator Oop1 takes some value, even if we don’t

know it. It is often assumed that hidden variable theories imply counterfactual reality. Weshould emphasise categorically that no such assumption is made in the Cellular Automa-ton Interpretation. In this theory, the operator Oop

2 , whose value has been measured,apparently turned out to be composed of ontological observables (beables). OperatorOop

1 is, by definition, not ontological and therefore has no well-defined value, for the samereason why, in the planetary system, the Earth-Mars interchange operator, whose eigenvalues are ±1 , has neither of these values; it is unspecified, in spite of the fact that plan-ets evolve classically, and in spite of the fact that the Copenhagen doctrine would dictatethat these eigen values are observable!

The tricky thing about the CAI when applied to atoms and molecules, is that one oftendoes not know a priori which of our operators are beables and which are changeables orsuperimposables (as defined in Subsections 2.1.1 and 12.3.1). One only knows this aposteriori, and one might wonder why this is so. We are using templates to describeatoms and molecules, and these templates give us such a thoroughly mixed up view ofthe complete set of observables in a theory that we are left in the dark, until someonedecides to measure something.

It looks as if the simple act of a measurement sends a signal backwards in time and withsuperluminal speed to other parts of the universe, to inform observers there which of theirobservables can be measured and which not. Of course, that is not what happens. Whathappens is that now we know what can be measured accurately and which measurements

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will give uncertain results. The Bell and CHSH inequalities are violated as they shouldbe in quantum field theory, while nevertheless quantum field theory forbids the possibilityto send signals faster than light or back to the past.

14.3. Superdeterminism and Conspiracy

Superdeterminism may be defined to imply that not only all physical phenomena aredeclared to be direct consequences of physical laws that do not leave anything anywhereto chance (which we refer to as ‘determinism’), but it also emphasises that the observersthemselves behave in accordance with the same laws. They also cannot perform anywhimsical act without any cause in the near past as well as in the distant past. By itself,this statement is so obvious that little discussion would be required to justify it, but whatmakes it more special is that it makes a difference. The fact that an observer cannotreset his or her measuring device without changing physical states in the past is usuallythought not to be relevant for our description of physical laws. The CAI claims that it is.

It is often argued that, if we want any superdeterministic phenomenon to lead to vio-lations of the Bell-CHSH inequalities, this would require conspiracy between the decayingatom observed and the processes occurring in the minds of Alice and Bob, which wouldbe a conspiracy of a kind that should not be tolerated in any decent theory of naturallaw. The whole idea that a natural mechanism could exist that drives Alice’s and Bob’sbehaviour is considered unacceptable.

In the CAI, however, natural law forbids the emergence of states where beables aresuperimposed. Neither Alice nor Bob will ever be able to produce such states by rotatingtheir polarisation filters. Indeed, the state their minds are in, are ontological in terms ofthe beables, and they will not be able to change that.

Superdeterminism has to be viewed in relation with correlations over space-like dis-tances. We claim that not only there are correlations, but the correlations are also ex-tremely strong. The state we call ‘vacuum state’ is full of correlations. Quantum fieldtheory (Section 10), which should be a direct consequence of the underlying ontologicaltheory, explains these correlations. All 2-particle expectation values, called propagators,are non-vanishing outside the light cone. Also the many-particle expectation values arenon-vanishing there; indeed, by analytic continuation, these amplitudes are seen to turninto the complete scattering matrix, which encapsulates all laws of physics that are im-plied by the theory. In chapter 8.7, it is shown how a 3-point correlation can, in principle,generate the violation of the CHSH inequalities required by quantum mechanics.

In view of these correlation functions, and the law that says that beables will neversuperimpose, we can now say that this law forbids Alice and Bob to both change theirminds in such a way that these correlation functions would no longer hold.

It may take years, decades, perhaps centuries to arrive at a comprehensive theory ofquantum gravity, combined with a theory of quantum matter that will be an elaborateextension of the Standard Model. Only then will we have the opportunity to isolate thebeables that are the basic ingredients for an ontological theory; the templates of that

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theory will be mapped onto the beable basis. Only then can we tell whether the CAIreally works. Conceivably however, we may be able to use the CAI as a guide to arriveat such theories. That was the main motivation for writing this report.

Appendix

A. Some remarks on gravity in 2+1 dimensions

Gravity in 2+1 dimensions is a special example of a classical theory that is difficult toquantise properly, at least if we wish to admit the presence of matter27. One can think ofscalar (spin zero) particles whose inly interactions consist of the exchange of a gravitationalforce. The classical theory[50][58][59] suggests that it can be quantised, but somethingvery special happens[61].

The Einstein equations for regions without matter particles read

Rµν = 0 , (A.1)

but in 2+1 dimensions, we can write

Sαβ = Sβα = 14εαµνεβκλRµνκλ , Rµνκλ = εµναεκλβS

αβ ,

Rµν = gµνSαα − Sµν , Sµν = −Rµν + 1

2Rgµν . (A.2)

This implies that Eq. (A.1) also implies that the Riemann tensor Rµνκλ vanishes. There-fore, matter-free regions are flat pieces of space-time.

When a particle is present, however, Rµν does not vanish, and therefore a particle is alocal, topological defect. One finds that a particle cuts out a wedge from its surroundingspace, turning the surrounding space in a cone, and the deficit angle of the excised regionis proportional to the mass: In convenient choices of the units, the total wedge angle isexactly twice the mass µ of the particle.

When the particle moves, we choose to orient the deficit angle such that the particlemoves in the direction of the bisectrix of the angle. Then, if we ask for the effect of theassociated Lorentz transformation, we see that the wedge is Lorentz contracted. This isillustrated in Figure 20, where the crosses indicate which points are identified when wefollow a loop around the particle. We see that, because we chose the particle to movealong the bisectrix, there is no time shift at this identification, otherwise, there wouldhave been. This way we achieve that the surrounding space can be handled as a Cauchysurface for other particles that move around.

Some arithmetic shows that, if the particle’s velocity is defined as tanh ξ , the Lorentzcontraction factor is cosh ξ , and the opening angle H of the moving particle is given by

tanH = cosh ξ tanµ . (A.3)

27In 2+1 dimensions, gravity without particles present can be quantised[31][32][58][59], but it is a ratheresoteric topological theory.

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contractionLorentz

Hµ

η

η

ξ

Figure 20: The angle cut out of space when a particle moves with velocity ξ . See text.

If η is the velocity of the seam between the two spaces (arrow in Fig. 20), then we find

tanh η = sinH tanh ξ , (A.4)

cosµ = cosH cosh η (A.5)

sinh η = sinµ sinh ξ . (A.6)

Eq. (A.3) gives a relation between H and µ that turns into the usual relation betweenmass in motion and energy in the weak gravity limit, while deficit angles such as H areadditive and conserved. Therefore, we interpret H as the energy of the moving particle,in the presence of gravity.

In Ref. [61], we argued that, taking µ to be constant, d(cosH cosh η) = 0 , so that

− sinH cosh η dH + cosH sinh η dη = 0 ,∂H

∂η=

tanh η

tanH. (A.7)

Furthermore, it was derived that we can make tessellations of Cauchy surfaces usingconfigurations such as in Fig. 20 in combination with vertices where no particles areresiding, so that the Cauchy surface is built from polygons. The edges Li where onepolygon is connected to another either end in one of these auxiliary vertices or one of thephysical particles. We can then calculate how the lengths of the edges Li grow or shrink.

Both end points of a boundary line make the line grow or shrink with independentvelocities, but the orthogonal components are the same. To define these unambiguously,we take a point such as the small circle in Figure 20, which moves in a direction orthogonalto the seam. We now see that our particle gives a contribution to dL/dt equal to

dL

dt

∣∣∣ξ

= tanh ξ cosH . (A.8)

Combining this with Eqs. (A.4) and (A.7), one finds

dL

dt

∣∣∣ξ

=tanh η

tanH=

∂H

∂η. (A.9)

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Furthermore, the hamiltonian does not depend on L , while η does not depend on time,so that

dη

dt= −∂H

∂L= 0 . (A.10)

These last two equations can be seen as the Hamilton equations for L and η . This meansthat η and L are canonically associated with one another. If there are many polygonsconnected together with seams Li , moving with transverse velocities tanh ηi , then weobtain hamiltonian equations for their time dependence, with Poisson brackets

Li, ηj = δij . (A.11)

Thus, the lengths Li are like positions and the ηi are like the associated momenta.This clearly suggests that all one needs to do to obtain the quantum theory is to postulatethat these Poisson brackets are replaced by commutators.

A.1. Discreteness of time

There is, however, a serious complication, lying in the nature of our hamiltonian. If wehave many particles, all adding deficit angles to the shape of our Cauchy surface, one caneasily see what might happen:

If the total energy due to matter particles exceeds the value 2π , the universewill close into itself, allowing only the value 4π for its total hamiltonian,

assuming that the universe is simply connected. Thus, the question arises what itmeans to vary the hamiltonian with respect to η ’s, as in Eq. (A.9).

There is a way to handle this question: consider some region X in the universe Ω ,and ask how it evolves with respect to data in the rest of the universe, Ω\X . The problemis then to define where the boundary is.

An other peculiar feature of this hamiltonian is that it is defined as an angle (evenif it might exceed the value 2π ; it cannot exceed 4π ). In the present work, we becamequite familiar with hamiltonians that are actually simple angles: this means that theirconjugate variable, time, is discrete. The well-defined object is not the hamiltonian itself,but the evolution operator over one unit of time: U = e−iH . Apparently, what we aredealing with here, is a world where the evolution goes in discretised steps in time.

The most remarkable thing however, is that we cannot say that the time for the entireuniverse is discrete. Global time is a meaningless concept, because gravity is a diffeo-morphism invariant theory. Time is just a coordinate, and physical states are invariantunder coordinate transformations, such as a global time translation. It is in regions wherematter is absent where we have local flatness, and only in those regions, relative time iswell-defined.

Because of the absence of a global time concept, we have no Schrodinger equation, oreven a discrete time-step equation, that tells us how the entire universe evolves.

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Suppose we split the universe Ω into two parts, X and Ω\X . Then the edges Liin X obey a Schrodinger equation regarding their dependence on a relative time variablet (it is relative to time in Ω\X ). The Schrodinger equation is derived from Eq. (A.5),where now H and η are operators:

η = −i ∂∂L

, H = i∂

∂t. (A.12)

If the wave function is ψ(L, t) , then

(cosH)ψ(L, t) = 12

(ψ(L, t+ 1) + ψ(L, t− 1)

), (A.13)

and the action of 1/ cosh η on ψ(L, t) can be found by Fourier transforming this operator:

(cosh η)−1ψ(L, t) =

∫ ∞−∞

dy1

2 cosh(πy/2)ψ(L+ y, t) . (A.14)

So, the particle in Fig. 20 obeys the Schrodinger equation following from Eq. (A.5):

ψ(L, t+ 1) + ψ(L, t− 1) =

∫ ∞−∞

dycosµ

cosh(πy/2)ψ(L+ y, t) . (A.15)

The problem with this equation is that it involves all L values, while the polygonsforming the tessellation of the Cauchy surface, whose edge lengths are given by the Li ,will have to obey inequalities, and therefore it is not clear to us how to proceed from here.In Ref. [61] we tried to replace the edge lengths Li by the particle coordinates themselves.It turns out that they indeed have conjugated momenta that form a compact space, sothat these coordinates span some sort of lattice, but this is not a rectangular lattice, andagain the topological constraints were too difficult to be handled.

The author now suspects that, in a meaningful theory for a system of this sort, wemust require all dynamical variables to be sharply defined, so as to be able to definetheir topological winding properties. Now that would force us to search for deterministic,classical models for 2+1 dimensional gravity. In fact, the difficulty of formulating ameaningful ‘Schrodinger equation’ for a 2+1 dimensional universe, and the insight thatthis equation would (probably) have to be deterministic, was one of the first incentivesfor this author to re-investigate deterministic quantum mechanics as was done in thework reported about here: if we would consider any classical model for 2+1 dimensionalgravity with matter (which certainly can be formulated in a neat way), declaring itsclassical states to span a Hilbert space in the sense described in our work, then that couldbecome a meaningful, unambiguous quantum system[50][58][59].

Acknowledgements

The author discussed these topics with many colleagues; I often forget who said what, butit is clear that many critical remarks later turned out to be relevant and were picked up.Among them were A. Aspect, T. Banks, N. Berkovitz, M. Blasone, Th. Elze, E. Fredkin,S. Hawking, H. Kleinert, R. Maimon, Th. Nieuwenhuizen, M. Porter, P. Shor, L. Susskind,E. Witten, W. Zurek.

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