Fundamental Theories of Physics 185 Gerard ’t Hooft The Cellular Automaton Interpretation of Quantum Mechanics Fundamental Theories of Physics Volume 185 Series Editors Henk van Beijeren, Utrecht, The Netherlands Philippe Blanchard, Bielefeld, Germany Paul Busch, York, United Kingdom Bob Coecke, Oxford, United Kingdom Dennis Dieks, Utrecht, The Netherlands Bianca Dittrich, Waterloo, Canada Detlef Dürr, München, Germany Ruth Durrer, Genève, Switzerland Roman Frigg, London, United Kingdom Christopher Fuchs, Boston, USA Giancarlo Ghirardi, Trieste, Italy Domenico J.W. Giulini, Bremen, Germany Gregg Jaeger, Boston, USA Claus Kiefer, Köln, Germany Nicolaas P. Landsman, Nijmegen, The Netherlands Christian Maes, Leuven, Belgium Mio Murao, Tokyo, Japan Hermann Nicolai, Potsdam, Germany Vesselin Petkov, Montreal, Canada Laura Ruetsche, Ann Arbor, USA Mairi Sakellariadou, London, United Kingdom Alwyn van der Merwe, Denver, USA Rainer Verch, Leipzig, Germany Reinhard Werner, Hannover, Germany Christian Wüthrich, Geneva, Switzerland Lai-Sang Young, New York City, USA The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the the- oretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also pro- vides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard. More information about this series at http://www.springer.com/series/6001 Gerard ’t Hooft The Cellular Automaton Interpretation of Quantum Mechanics Gerard ’t Hooft Institute for Theoretical Physics Utrecht University Utrecht, The Netherlands ISSN 0168-1222 ISSN 2365-6425 (electronic) Fundamental Theories of Physics ISBN 978-3-319-41284-9 ISBN 978-3-319-41285-6 (eBook) DOI 10.1007/978-3-319-41285-6 Library of Congress Control Number: 2016952241 Springer Cham Heidelberg New York Dordrecht London © The Editor(s) (if applicable) and The Author(s) 2016. The book is published open access. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Foreword When investigating theories at the tiniest conceivable scales in Nature, almost all researchers today revert to the quantum language, accepting the verdict that we shall nickname “the Copenhagen doctrine” that the only way to describe what is going on will always involve states in Hilbert space, controlled by operator equations. Returning to classical, that is, non quantum mechanical, descriptions will be forever impossible, unless one accepts some extremely contrived theoretical contraptions that may or may not reproduce the quantum mechanical phenomena observed in experiments. Dissatisfied, this author investigated how one can look at things differently. This book 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. Exam- ples 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, might be viewed as a quantum mechanical approach to analyse a system that could be classical at its core. We explain how such thoughts can conceivably be reconciled with Bell’s theorem, and how the usual objections voiced against the notion of ‘superdeterminism’ can be overcome, at least in princi- ple. Our proposal would eradicate the collapse problem and the measurement prob- lem. Even the existence of an “arrow of time” can perhaps be explained in a more elegant way than usual. Utrecht, The Netherlands Gerard ’t Hooft May 2016 v Preface This book is not in any way intended to serve as a replacement for the standard theory of quantum mechanics. A reader not yet thoroughly familiar with the basic concepts of quantum mechanics is advised first to learn this theory from one of the recommended text books [24, 25, 60], and only then pick up this book to find out that the doctrine called ‘quantum mechanics’ can be viewed as part of a marvellous mathematical machinery that places physical phenomena in a greater context, and only in the second place as a theory of Nature. This book consists of two parts. Part I deals with the many conceptual issues, without demanding excessive calculations. Part II adds to this our calculation tech- niques, occasionally returning to conceptual issues. Inevitably, the text in both parts will frequently refer to discussions in the other part, but they can be studied sepa- rately. This book is not a novel that has to be read from beginning to end, but rather a collection of descriptions and derivations, to be used as a reference. Different parts can be read in random order. Some arguments are repeated several times, but each time in a different context. Utrecht, The Netherlands Gerard ’t Hooft vii Acknowledgements The author discussed these topics with many colleagues; I often forget who said what, but it 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. Bla- sone, Eliahu Cohen, M. Duff, G. Dvali, Th. Elze, E. Fredkin, S. Giddings, S. Hawk- ing, M. Holman, H. Kleinert, R. Maimon, Th. Nieuwenhuizen, M. Porter, P. Shor, L. Susskind, R. Werner, E. Witten, W. Zurek. Utrecht, The Netherlands Gerard ’t Hooft ix Contents Part I The Cellular Automaton Interpretation as a General Doctrine 1 Motivation for This Work ....................... 3 1.1 Why an Interpretation Is Needed ................. 5 1.2 Outline of the Ideas Exposed in Part I ............... 8 1.3A19thCenturyPhilosophy.................... 12 1.4BriefHistoryoftheCellularAutomaton............. 14 1.5 Modern Thoughts About Quantum Mechanics . ........ 16 1.6Notation.............................. 17 2 Deterministic Models in Quantum Notation ............. 19 2.1 The Basic Structure of Deterministic Models . ........ 19 2.1.1 Operators: Beables, Changeables and Superimposables . 21 2.2 The Cogwheel Model ....................... 22 2.2.1 Generalizations of the Cogwheel Model: Cogwheels with N Teeth........................ 23 2.2.2 The Most General Deterministic, Time Reversible, Finite Model ........................... 25 3 Interpreting Quantum Mechanics ................... 29 3.1 The Copenhagen Doctrine . .................. 29 3.2TheEinsteinianView....................... 31 3.3 Notions Not Admitted in the CAI ................. 33 3.4 The Collapsing Wave Function and Schrödinger’s Cat ...... 35 3.5 Decoherence and Born’s Probability Axiom . ........ 37 3.6 Bell’s Theorem, Bell’s Inequalities and the CHSH Inequality . 38 3.7 The Mouse Dropping Function .................. 42 3.7.1 Ontology Conservation and Hidden Information ..... 44 3.8FreeWillandTimeInversion................... 45 4 Deterministic Quantum Mechanics .................. 49 4.1 Introduction ............................ 49 4.2TheClassicalLimitRevisited................... 52 xi xii Contents 4.3 Born’s Probability Rule ...................... 53 4.3.1TheUseofTemplates..................