Quantum Theory as an Emergent Phenomenon: The Statistical Mechanics of Matrix Models as the Precursor of Quantum Field Theory

238 Pages·2004·1.07 MB·English

by  Stephen L. Adler

QUANTUM THEORY AS AN
EMERGENT PHENOMENON
TheStatisticalMechanicsofMatrixModels
asthePrecursorofQuantumFieldTheory
Quantum mechanics is our most successful physical theory. However, it raises
conceptual issues that have perplexed physicists and philosophers of science for
decades.Thisbookdevelopsanewapproach,basedontheproposalthatquantum
theoryisnotacomplete,finaltheory,butisinfactanemergentphenomenonaris-
ing from a deeper level of dynamics. The dynamics at this deeper level is taken
tobeanextensionofclassicaldynamicstonon-commutingmatrixvariables,with
cyclicpermutationinsideatraceusedasthebasiccalculationaltool.Withplausible
assumptions, quantum theory is shown to emerge as the statistical thermodynam-
ics of this underlying theory, with the canonical commutation–anticommutation
relationsderivedfromageneralizedequipartitiontheorem.Brownianmotioncor-
rectionstothisthermodynamicsarearguedtoleadtostatevectorreductionandto
theprobabilisticinterpretationofquantumtheory,makingcontactwithrecentphe-
nomenologicalproposalsforstochasticmodificationstoSchro¨dingerdynamics.
STEPHEN L. ADLER received his Ph.D. degree in theoretical physics from
Princeton. He has been a Professor in the School of Natural Sciences at the In-
stitute for Advanced Study since 1969, and from 1979 to 2003 held the State of
NewJerseyAlbertEinsteinProfessorshipthere.
Dr.Adler’sresearchhasincludedseminalpapersincurrentalgebras,sumrules,
perturbation theory anomalies, and high energy neutrino processes. Dr. Adler has
also done important work on neutral current phenomenology, strong field elec-
tromagneticprocesses,accelerationmethodsforMonteCarloalgorithms,induced
gravity, non-Abelian monopoles, and models for quark confinement. For nearly
twentyyearshehasbeenstudyingembeddingsofstandardquantummechanicsin
largermathematicalframeworks,withresultsdescribedinthisvolume.
Dr.AdlerisamemberoftheNationalAcademyofSciences,andisaFellowof
theAmericanPhysicalSociety,theAmericanAcademyofArtsandSciences,and
the American Association for the Advancement of Science. He received the J. J.
Sakurai Prize in particle phenomenology, awarded by the American Physical So-
ciety,in1988,andtheDiracPrizeandMedalawardedbytheInternationalCenter
forTheoreticalPhysicsinTrieste,in1998.

QUANTUM THEORY AS AN EMERGENT
PHENOMENON
The Statistical Mechanics of Matrix Models
as the Precursor of Quantum Field Theory
STEPHEN L. ADLER
InstituteforAdvancedStudy,Princeton
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521831949
© S. L. Adler 2004
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2004
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ToSarahBrett-Smith,withloveandadmiration

Contents
Acknowledgements pagex
Introductionandoverview 1
1 Thequantummeasurementproblem 2
2 Reinterpretationsofquantummechanicalfoundations 6
3 Motivationsforbelievingthatquantummechanics
isincomplete 9
4 Anoverviewofthisbook 13
5 Briefhistoricalremarksontracedynamics 18
1 Tracedynamics:theclassicalLagrangianandHamiltonian
dynamicsofmatrixmodels 21
1.1 Bosonicandfermionicmatricesandthecyclic
traceidentities 21
1.2 Derivativeofatracewithrespecttoanoperator 24
1.3 LagrangianandHamiltoniandynamicsofmatrixmodels 27
1.4 ThegeneralizedPoissonbracket,itsproperties,
andapplications 29
1.5 TracedynamicscontrastedwithunitaryHeisenberg
picturedynamics 32
2 Additionalgenericconservedquantities 39
2.1 Thetrace“fermionnumber”N 39
˜
2.2 TheconservedoperatorC 42
2.3 Conservedquantitiesforcontinuumspacetimetheories 52
2.4 Anillustrativeexample:aDiracfermioncoupled
toascalarKlein–Gordonfield 58
2.5 Symmetriesofconservedquantitiesunder p ↔q 62
F F
3 Tracedynamicsmodelswithglobalsupersymmetry 64
3.1 TheWess–Zuminomodel 64
3.2 ThesupersymmetricYang–Millsmodel 67
vii
viii Contents
3.3 ThematrixmodelforMtheory 70
3.4 Superspaceconsiderationsandremarks 72
4 Statisticalmechanicsofmatrixmodels 75
4.1 TheLiouvilletheorem 76
4.2 Thecanonicalensemble 81
4.3 Themicrocanonicalensemble 88
4.4 Gaugefixinginthepartitionfunction 93
4.5 ReductionoftheHilbertspacemoduloi 100
eff
4.6 Globalunitaryfixing 106
5 Theemergenceofquantumfielddynamics 117
5.1 ThegeneralWardidentity 119
5.2 Variationofthesourceterms 124
5.3 Approximations/assumptionsleadingtotheemergence
ofquantumtheory 128
5.4 Restrictionsontheunderlyingtheoryimpliedbyfurther
Wardidentities 139
5.5 DerivationoftheSchro¨dingerequation 147
5.6 EvasionoftheKochen–SpeckertheoremandBell
inequalityarguments 151
6 BrownianmotioncorrectionstoSchro¨dingerdynamicsandthe
emergenceoftheprobabilityinterpretation 156
6.1 Scenariosleadingtothelocalizationandtheenergy-driven
stochasticSchro¨dingerequations 157
6.2 ProofofreductionwithBornruleprobabilities 170
6.3 Phenomenologyofstochasticreduction–reduction
rateformulas 174
6.4 Phenomenologyofenergy-drivenreduction 175
6.5 Phenomenologyofreductionbycontinuousspontaneous
localization 185
7 Discussionandoutlook 190
Appendices 193
AppendixA:Modificationsinrealandquaternionic
Hilbertspace 194
AppendixB:AlgebraicproofoftheJacobiidentityforthe
generalizedPoissonbracket 194
AppendixC:Symplecticstructuresintracedynamics 198
AppendixD:Gammamatrixidentitiesforsupersymmetric
tracedynamicsmodels 201
AppendixE:Tracedynamicsmodelswithoperator
gaugeinvariance 204
Contents ix
AppendixF:PropertiesofWightmanfunctionsneededfor
reconstructionoflocalquantumfieldtheory 206
AppendixG:BRSTinvariancetransformationforglobal
unitaryfixing 208
References 212
Index 220