Spacetime Quantum Mechanics and the Quantum Mechan

arXiv:gr-qc/9304006v26Apr1993UCSBTH92-21SPACETIMEQUANTUMMECHANICSANDTHEQUANTUMMECHANICSOFSPACETIME∗JamesB.HartleDepartmentofPhysics,UniversityofCaliforniaSantaBarbara,CA93106-9530USATABLEOFCONTENTSI.IntroductionII.TheQuantumMechanicsofClosedSystemsII.1.QuantumMechanicsandCosmologyII.2.ProbabilitiesinGeneralandProbabilitiesinQuantumMechanicsII.3.ProbabilitiesforaTimeSequenceofMeasurementsII.4.Post-EverettQuantumMechanicsII.5.TheOriginsofDecoherenceinOurUniverseII.6.TheCopenhagenApproximationII.7.QuasiclassicalDomainsIII.DecoherenceinGeneral,DecoherenceinParticular,andtheEmergenceofClassicalBehaviorIII.1.AMoreGeneralFormulationoftheQuantumMechanicsofClosedSystemsIII.1.1.Fine-GrainedandCoarse-GrainedHistoriesIII.1.2.TheDecoherenceFunctionalIII.1.3.Prediction,Retrodiction,andStatesIII.1.4.TheDecoherenceFunctionalinPathIntegralFormIII.2.TheEmschModelIII.3.LinearOscillatorModelsIII.3.1.SpecificationIII.3.2.TheInfluencePhaseandDecoherenceIII.4.TheEmergenceofaQuasiclassicalDomainIV.GeneralizedQuantumMechanicsIV.1.ThreeElementsIV.2.HamiltonianQuantumMechanicsasaGeneralizedQuantumMechanicsIV.3.Sum-Over-HistoriesQuantumMechanicsforTheorieswithaTimeIV.4.DifferencesandEquivalencesbetweenHamiltonianandSum-Over-HistoriesQuantumMechanicsforTheorieswithaTimeIV.5.ClassicalPhysicsandtheClassicalLimitofQuantumMechanicsIV.6.GeneralizationsofHamiltonianQuantumMechanicsIV.7.ATime-NeutralFormulationofQuantumMechanics∗Lecturesgivenatthe1992LesHouches´Ecoled’´et´e,GravitationetQuantifications,July9–17,1992.V.TheSpacetimeApproachtoNon-RelativisticQuantumMechanicsV.1.AGeneralizedSum-Over-HistoriesQuantumMechanicsforNon-RelativisticSystemsV.2.EvaluatingPathIntegralsV.2.1ProductFormulaeV.2.2Phase-SpacePathIntegralsV.3.ExamplesofCoarseGrainingsV.3.1.AlternativesatDefiniteMomentsofTimeV.3.2.AlternativesDefinedbyaSpacetimeRegionV.3.3.ASimpleExampleofaDecoherentSpacetimeCoarse-GrainingV.4.CoarseGrainingsbyFunctionalsofthePathsV.4.1.GeneralCoarseGrainingsV.4.2.CoarseGrainingsDefiningMomentumV.5.TheRelationbetweentheHamiltonianandGeneralizedSum-Over-HistoriesFormulationsofNon-RelativisticQuantumMechanicsVI.AbelianGaugeTheoriesVI.1.GaugeandReparametrizationInvarianceVI.2.CoarseGrainingsoftheElectromagneticFieldVI.3.SpecificExamplesVI.4.ConstraintsVI.5.ADMandDiracQuantizationVII.ModelswithaSingleReparametrizationInvarianceVII.1.ReparametrizationInvarianceinGeneralVII.2.ConstraintsandPathIntegralsVII.3.ParametrizedNon-RelativisticQuantumMechanicsVII.4.TheRelativisticWorldLine—FormulationwithaPreferredTimeVII.5.TheRelativisticWorldLine—FormulationwithoutaPreferredTimeVII.5.1.Fine-GrainedHistories,CoarseGrainings,andDecoherenceFunctionalVII.5.2.ExplicitExamplesVII.5.3.ConnectionwithFieldTheoryVII.5.4.NoEquivalentHamiltonianFormulationVII.5.5.TheProbabilityoftheConstraintVII.6.RelationtoDiracQuantizationVIII.GeneralRelativityVIII.1.GeneralRelativityandQuantumGravityVIII.2.Fine-GrainedHistoriesofMetricsandFieldsandtheirSimplicialApproximationVIII.3.CoarseGrainingsofSpacetimeVIII.4.TheDecoherenceFunctionalforGeneralRelativityVIII.4.1.Actions,Invariance,ConstraintsVIII.4.2.TheClassOperatorsVIII.4.3.AdjoiningInitialandFinalConditionsVIII.5.Discussion—TheProblemofTimeVIII.6.Discussion—ConstraintsVIII.7.SimplicialModelsVIII.8.InitialandFinalConditionsinCosmology2IX.SemiclassicalPredictionsIX.1.TheSemiclassicalRegimeIX.2.TheSemiclassicalApproximationtotheQuantumMechanicsofaNon-RelativisticParticleIX.3.TheSemiclassicalApproximationfortheRelativisticParticleIX.4.TheApproximationofQuantumFieldTheoryinSemiclassicalSpacetimeIX.5.RulesforSemiclassicalPredictionandtheEmergenceofTimeX.SummationAcknowledgmentsNotationandConventionsReferences3I.IntroductionTheselecturesarenotaboutthequantizationofanyparticulartheoryofgravitation.Rathertheyareabouthowtoformulatequantummechanicsgenerallyenoughsothatitcananswerquestionsinanyquantumtheoryofspacetime.Theyarenotconcernedwithanyparticulartheoryofthedynamicsofgravitybutratherwiththequantumframeworkforpredictioninsuchtheoriesgenerally.Itisreasonabletoaskwhyanelementarycourseoflecturesonquantummechanicsshouldbeneededinaschoolonthequantizationofgravity.Wehavestandardcoursesinquantummechanicsthataretaughtineverygraduateschool.Whyaren’tthesesufficient?Theyarenotsufficientbecausetheformulationsofquantummechanicsusuallytaughtinthesecoursesisinsufficientlygeneralforconstructingaquantumtheoryofgravitysuitableforapplicationtoallthedomainsinwhichwewouldliketoapplyit.Thereareatleasttwocountsonwhichtheusualformulationsofquantummechanicsarenotgeneralenough:Theydonotdiscussthequantummechanicsofclosedsystemssuchastheuniverseasawhole,andtheydonotaddressthe“problemoftime”inquantumgravity.TheS-matrixisoneimportantquestiontowhichquantumgravityshouldsupplyananswer.Wecannotexpecttotestitsmatrix-elementsthatinvolveexternal,Planck-energygravitonsanytimeinthenearfuture.However,wemighthopethat,sincegravitycouplesuniversallytoallformsofmatter,wemightseeimprintsofPlanckscalephysicsintestablescatteringexperimentsatmoreaccessibleenergieswithmorefamiliarconstituents.