“Beyond Quantum offers an original and fascinating contribution to the current state of the debate concerning quantum foundations. The book will be of much interest and benefit to physicists and B philosophers alike. It will be an occasion for thought and a stimulant for pursuing new work, which, as the book makes abundantly clear, is still much needed in this field, a century of debates and e controversies notwithstanding. A very welcome advance!” y Prof. Arkady Plotnitsky Purdue University, USA o n “In this volume Professor Khrennikov employs his deep knowledge of probability theory and long experience in applying it to a wide variety of problems and circumstances, to propose a theory d that goes beyond quantum mechanics. In line with the Einsteinian credo that particles are a manifestation of fields, the author transforms quantum mechanics into a classical statistical theory Q of (prequantum) random fields. Such fundamental viewpoint resolves several of the problems that beset the usual interpretations of quantum mechanics, and affords new and interesting theoretical u avenues for further enquiry. The result is a highly stimulating and original perspective of the quantum world.” a Prof. Luis de la Peña Auerbach n Instituto de Física, UNAM, Mexico t The present wave of interest in quantum foundations is caused by the tremendous u development of quantum information science and its applications to quantum computing m and quantum communication. Nowadays, this interest has increased even more because it has become clear that some of the difficulties encountered in the realization of quantum information processing (especially creation of quantum computers and designing of new quantum algorithms) are not simply technicalities but have roots at the very fundamental level. To solve such problems, quantum theory has to be reconsidered. The present book is devoted to the analysis of the probabilistic structure of quantum theory and probes the limits of classical probabilistic representation of quantum phenomena. Andrei Khrennikov is professor of applied mathematics and director of the International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science at Linnaeus University, Växjö-Kalmar, South-East Sweden. Prof. Khrrenikov’s research interests focus on foundations K of quantum physics and quantum information, foundations of probability h (in particular, studies on negative probabilities), cognitive modeling, ultrametric re BEYOND n (non-Archimedean) mathematics, dynamical systems, infinite-dimensional n analysis, quantum-like models in psychology, economics, and finances. He ik is known worldwide as the organizer of the longest series of conferences on o v QUANTUM quantum foundations in Växjö and is currently devoted to the creation of the classical random field model of quantum phenomena. Andrei Khrennikov V368 ISBN 978-981-4411-73-8 BEYOND QUANTUM PPaann SSttaannffoorrdd SSeerriieess oonn RReenneewwaabbllee EEnneerrggyy —— VVoolluummee 22 BEYOND QUANTUM Andrei Khrennikov editors Preben Maegaard Anna Krenz Wolfgang Palz The Rise of Modern Wind Energy Wind Power for the World CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140307 International Standard Book Number-13: 978-981-4411-74-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reason- able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organiza- tion that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com February12,2014 16:5 PSPBook-9inx6in 00-Beyond-Quantum-prelims Contents Preface xiii Acknowledgments xvii 1 Introduction 1 1.1 TheAuthor’sViewsonQuantumFoundations 1 1.1.1 DebatesinVa¨xjo¨ 1 1.1.1.1 Interpretationsofwavefunction 3 1.1.1.2 DebatesonBell’sinequality:Gill againstAccardi 5 1.1.1.3 Aroundloopholes:Aspect,Kwiat, Zeilinger,HessandPhilipp,Larsson andGill,Adenier,Khrennikovand Volovich,RaedtandMichielsen 6 1.1.1.4 Isitworthtryingtogobeyond quantummechanics? 9 1.1.2 TheRoleofProbability 11 1.1.3 ToHilbertSpacefromProbability 14 1.1.4 AgainstCompleteness 15 1.1.5 ‘tHooft’sDreamofPrequantumDeterminism 17 1.1.6 Einstein’sDreamofthePureFieldModel 18 1.1.7 Antiphoton 21 1.1.8 Schro¨dinger’sWaveMechanics 23 1.1.9 Bohr–Kramers–SlaterTheory 27 1.1.10OntheEvolutionofEinstein’sViews:From ClassicalElectrodynamicstoPhotonandBack 28 1.2 PrequantumClassicalStatisticalFieldTheory: Introduction 32 1.2.1 ClassicalFieldsasHiddenVariables 32 1.2.1.1 Randomfields 33 February12,2014 16:5 PSPBook-9inx6in 00-Beyond-Quantum-prelims vi Contents 1.2.2 CovarianceOperatorInterpretationofWave Function 35 1.2.3 QuantumObservablesfromQuadraticFormsof thePrequantumField 37 1.2.4 QuantumandPrequantumInterpretationsof Schro¨dingerEquation 38 1.2.5 TowardPrequantumDeterminism 40 1.2.6 RandomFieldsCorrespondingtoMixedStates 41 1.2.7 BackgroundField 42 1.2.8 CouplingbetweenSchro¨dingerandHamilton Equations 44 1.2.9 NonquadraticFunctionalsofthePrequantum FieldandViolationofBorn’sRule 45 1.2.10WaveComeback:ASolutionTooCheap? 46 1.3 WhereIsDiscreteness?DevilinDetectors? 47 1.4 OnExperimentstoTesttheEuclideanModel 49 2 ConventionalQuantumTheory:Fundamentals 51 2.1 Postulates 52 2.2 Quantization 57 2.3 InterpretationsofWaveFunction 58 2.4 Va¨xjo¨ InterpretationofQuantumMechanics 59 2.5 AShortIntroductiontoClassicalProbabilityTheory 64 2.6 QuantumConditionalProbability 70 2.7 InterferenceofProbabilitiesinQuantumMechanics 72 2.8 Two-SlitExperiment 73 2.8.1 ContextualityofData 74 2.8.2 WaveInterference 77 2.9 CorpuscularInterference 79 2.9.1 MemoryEffectsandViolationofErgodicity 79 2.9.2 ExistenceofIndivisibleTime-Quantumand DiscreteDynamics 80 2.9.2.1 Discretetimedynamics 81 2.9.2.2 Diffractionpatterninasingle-slit scattering 83 2.9.2.3 Interferenceinthetwo-slit experimentfordeterministicparticles 86 2.10InterferenceofProbabilitiesinCognitiveScience 91 February12,2014 16:5 PSPBook-9inx6in 00-Beyond-Quantum-prelims Contents vii 3 FundamentalsofPrequantumClassicalStatisticalField Theory 93 3.1 NoncompositeSystems 93 3.1.1 QuantumSystemasaClassicalRandomField 94 3.1.2 DensityOperatorIsCovarianceOperator 95 3.1.3 FunctionalRepresentationofQuantum Observables 95 3.1.4 BeyondQM 96 3.1.5 CouplingwithClassicalSignalTheory 98 3.2 CompositeSystems 100 3.2.1 OperatorRealizationofWaveFunction 101 3.2.2 BasicEquality 102 3.2.3 CouplingofClassicalandQuantum Correlations 103 3.2.4 VacuumFluctuations 106 3.2.5 SuperstrongQuantumCorrelations 107 3.3 StochasticProcessCorrespondingtoSchro¨dinger’s Evolution 108 3.3.1 Bernoulli-TypeProcess 109 3.3.2 Stochastic(Local)DynamicsintheAbsenceof Interaction 110 3.3.3 StochasticNonlocalDynamics 113 3.4 CorrelationsoftheComponentsofthePrequantum Field 115 3.5 PCSFTFormalismforClassicalElectromagnetic Field-1 116 3.6 DiscussionofaPossibleExperimentalVerificationof PCSFT 120 3.7 PhotonicField 121 3.8 CorrelationbetweenPolarizationVectorsof EntangledPhotons 122 3.9 FunctionalsofPrequantumFieldsCorrespondingto OperatorsofPhotonPolarization 125 3.10ClassicalRepresentationofHeisenberg’sUncertainty Relation 126 3.10.1TheRoleoftheRandomBackground 127 3.10.2CouplingofDispersions 129 February12,2014 16:5 PSPBook-9inx6in 00-Beyond-Quantum-prelims viii Contents 3.11TowardViolationofBorn’sRule:Descriptionofa SimpleExperiment 132 3.11.1DeviationfromBorn’sRuleforFourth-Order NonlinearitiesinDetection 133 3.11.2ViolationofBorn’sRuleforGaussianStates 135 3.11.3CouplingwithMeasurementTheoryfor PrequantumRandomSignals 137 3.12WhyGaussian? 137 3.12.1PulseProcesses 138 3.12.2WavePulses 140 3.12.3WhereAretheSourcesofPrequantumPulses? 141 3.13OnCorrespondencebetweenQuantumObservables andClassicalVariables 143 3.13.1IndependenceofComponentsofPrequantum RandomFieldsCorrespondingtoFactorizable QuantumStates 145 4 PrequantumDynamicsfromHamiltonianEquationsonthe Infinite-DimensionalPhaseSpace 149 4.1 HamiltonianMechanics 151 4.1.1 QuadraticHamiltonFunction 151 4.1.2 SymplecticallyInvariantQuadraticForms 152 4.1.3 DynamicsforSymplecticallyInvariant QuadraticHamiltonFunctions 153 4.1.4 ComplexRepresentationofHamiltonian Dynamics 154 4.2 SymplecticRepresentationofSchro¨dingerDynamics 156 4.3 ClassicalandQuantumStatisticalModels 158 4.3.1 ClassicalStatisticalModel 158 4.3.2 QuantumStatisticalModel 159 4.4 MeasuresonHilbertSpaces 159 4.5 LiftingofPointwiseDynamicstoSpacesofVariables andMeasures 161 4.6 DispersionPreservingDynamics 162 4.7 DynamicsintheSpaceofPhysicalVariables 163 4.7.1 ArbitraryQuadraticVariables 163 4.7.2 J-invariantVariables 164 4.7.3 ComplexStructure 164 February12,2014 16:5 PSPBook-9inx6in 00-Beyond-Quantum-prelims Contents ix 4.8 ProbabilisticDynamics 165 4.8.1 ArbitraryMeasure 165 4.8.2 J-invariantMeasures 165 4.8.3 ComplexStructure 166 4.8.4 DynamicsintheSpaceofStatisticalStates 167 4.8.5 ComplexCovarianceOperator 168 4.8.6 DynamicsofComplexCovarianceOperator: TheCaseofanArbitraryMeasure 170 4.9 DetailedAnalysisofDispersionPreservingDynamics 171 4.9.1 PhysicalDiscussiononDispersionPreserving PrequantumDynamics 173 5 QuantumMechanicsasApproximationofStatistical MechanicsofClassicalFields 175 5.1 TheTaylorApproximationofAveragesforFunctions ofRandomVariables 177 5.2 QuantumModel:Finite-DimensionalCase 180 5.2.1 RealQuantumMechanics 180 5.2.2 ComplexQuantumMechanics 180 5.3 Prequantum(cid:2)→QuantumCorrespondence: Finite-DimensionalCase 181 5.3.1 One-DimensionalCase 181 5.3.2 MultidimensionalCase 185 5.3.3 PrequantumPhaseSpaceinthe Two-DimensionalCase 187 5.3.4 PrequantumPhaseSpaceinMultidimensional Case 192 5.4 PrequantumPhaseSpace:Infinite-DimensionalCase 194 5.5 GaussianMeasuresCorrespondingtoPureQuantum States 198 5.6 IllustrationofthePrequantum(cid:2)→QuantumCoupling intheCaseofQubitMechanics 198 5.6.1 ClassicalRepresentationforSpinOperators 198 5.6.2 PrequantumGaussianMeasures CorrespondingtoPureStates 200 5.7 PrequantumClassicalStatisticalFieldTheory 202 5.7.1 InterpretationofMass 205