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Valter Moretti Fundamental Mathematical Structures of Quantum Theory Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation Fundamental Mathematical Structures of Quantum Theory Valter Moretti Fundamental Mathematical Structures of Quantum Theory Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation 123 ValterMoretti UniversityofTrentoandTrentoInstitute forFundamentalPhysicsandApplications Trento,Italy ISBN978-3-030-18345-5 ISBN978-3-030-18346-2 (eBook) https://doi.org/10.1007/978-3-030-18346-2 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents 1 GeneralPhenomenologyoftheQuantumWorldandElementary Formalism .................................................................... 1 1.1 ThePhysicsofQuantumSystems...................................... 1 1.1.1 WhenIsaPhysicalSystemaQuantumSystem?.............. 1 1.1.2 BasicPropertiesofQuantumSystems......................... 2 1.2 ElementaryQuantumFormalism:TheFinite-DimensionalCase..... 4 1.2.1 TimeEvolution ................................................. 8 1.3 A First Look at the Infinite-DimensionalCase, CCRs andQuantizationProcedures........................................... 10 1.3.1 TheL2(R,dx)Model.......................................... 10 1.3.2 TheL2(Rn,dnx)ModelandHeisenberg’sInequalities...... 13 1.3.3 Failure ofDirac’s Quantizationand Deformation QuantizationProcedure......................................... 14 2 HilbertSpacesandClassesofOperators.................................. 17 2.1 HilbertSpaces:ARound-Up........................................... 17 2.1.1 BasicProperties................................................. 18 2.1.2 OrthogonalityandHilbertBases............................... 19 2.1.3 TwoNotionsofHilbertOrthogonalDirectSum .............. 21 2.1.4 TensorProductofHilbertSpaces.............................. 22 2.2 Classesof(Unbounded)OperatorsonHilbertSpaces ................ 23 2.2.1 OperatorsandAbstractAlgebras .............................. 23 2.2.2 AdjointOperators............................................... 27 2.2.3 ClosedandClosableOperators ................................ 29 2.2.4 TypesofOperatorsRelevantinQuantumTheory............. 32 ∗ ⊥ 2.2.5 TheInterplayofKer,Ran, ,and .......................... 36 2.2.6 Criteriafor(Essential)Selfadjointness ........................ 37 2.2.7 PositionandMomentumOperatorsandOtherPhysical Examples........................................................ 40 v vi Contents 3 ObservablesandStatesinGeneral HilbertSpaces:SpectralTheory ........................................... 47 3.1 BasicsonSpectralTheory.............................................. 47 3.1.1 ResolventandSpectrum........................................ 47 3.1.2 SpectraofSpecialOperatorTypes............................. 52 3.2 IntegrationofProjector-ValuedMeasures ............................. 55 3.2.1 OrthogonalProjectors .......................................... 55 3.2.2 Projector-ValuedMeasures(PVMs) ........................... 58 3.2.3 PVM-IntegrationofBoundedFunctions ...................... 70 3.2.4 PVM-IntegrationofUnboundedFunctions.................... 75 3.3 SpectralDecompositionofSelfadjointOperators..................... 77 3.3.1 Spectral Theorem for Selfadjoint, Possibly Unbounded,Operators.......................................... 78 3.3.2 Some Technically Relevant Consequences oftheSpectralTheorem........................................ 88 3.3.3 JointSpectralMeasures ........................................ 92 3.3.4 MeasurableFunctionalCalculus............................... 93 3.3.5 AFirstGlanceatOne-ParameterGroupsofUnitary Operators........................................................ 95 3.4 ElementaryQuantumFormalism:ARigorousApproach............. 98 3.4.1 ElementaryFormalismfortheInfinite-DimensionalCase.... 98 3.4.2 CommutingSpectralMeasures................................. 101 3.4.3 AFirstLookattheTimeEvolutionofQuantumStates ...... 104 3.4.4 A First Look at (Continuous) Symmetries and ConservedQuantities........................................... 107 3.5 Round-UpofOperatorTopologies..................................... 109 3.6 ExistenceTheoremsofSpectralMeasures............................. 112 3.6.1 ContinuousFunctionalCalculus............................... 112 3.6.2 Existence of Spectral Measures for Bounded SelfadjointOperators........................................... 116 3.6.3 SpectralTheoremforNormalOperatorsinB(H) ............ 119 3.6.4 Existence of SpectralMeasures for Unbounded SelfadjointOperators........................................... 122 3.6.5 ExistenceofJointSpectralMeasures.......................... 124 4 FundamentalQuantumStructuresonHilbertSpaces................... 131 4.1 LatticesinClassicalandQuantumMechanics ........................ 131 4.1.1 ADifferentViewpointonClassicalMechanics............... 131 4.1.2 TheNotionofLattice........................................... 134 4.2 TheNon-BooleanLogicofQM........................................ 136 4.2.1 TheLatticeofQuantumElementaryObservables ............ 136 4.2.2 PartofClassicalMechanicsisHiddeninQM................. 138 4.2.3 AReasonWhyObservablesAreSelfadjointOperators ...... 143 4.3 Recovering the Hilbert Space Structure: The“Coordinatization”Problem....................................... 144 Contents vii 4.4 QuantumStatesasProbabilityMeasuresandGleason’sTheorem... 148 4.4.1 ProbabilityMeasuresonL(H)................................ 148 4.4.2 PolarDecomposition ........................................... 149 ∗ 4.4.3 TheTwo-Sided -IdealofCompactOperators................ 152 4.4.4 Trace-ClassOperators.......................................... 158 4.4.5 The Mathematical Notion of Quantum State andGleason’sTheorem......................................... 167 4.4.6 PhysicalInterpretation.......................................... 174 4.4.7 Post-measurement States: The Meaning oftheLüders-vonNeumannPostulate......................... 175 4.4.8 CompositeSystems inElementaryQM:TheUse ofTensorProducts.............................................. 178 4.5 GeneralInterplayofQuantumObservablesandQuantumStates .... 181 4.5.1 Observables,ExpectationValues,StandardDeviations....... 182 4.5.2 RelationwiththeFormalismUsedinPhysics................. 183 5 Realism,Non-Contextuality,LocalCausality,Entanglement........... 187 5.1 HiddenVariablesandno-goResults ................................... 187 5.1.1 RealisticHidden-VariableTheories............................ 188 5.1.2 TheBellandKochen–Speckerno-goTheorems.............. 188 5.1.3 AnAlternativeVersionoftheKochen–SpeckerTheorem.... 192 5.2 Realistic(Non-)ContextualTheories................................... 193 5.2.1 AnImperviousWayOut:TheNotionofContextuality....... 194 5.2.2 ThePeres–MerminMagicSquare ............................. 196 5.2.3 AState-IndependentTestonRealisticNon-Contextuality ... 198 5.3 EntanglementandtheBCHSHInequality............................. 201 5.3.1 BCHSHInequalityfromRealismandLocality............... 202 5.3.2 BCHSHInequalityandFactorizedStates ..................... 205 5.3.3 BCHSHInequalityfromRelativisticLocalCausality andRealism..................................................... 206 5.3.4 BCHSHInequalityfromRealismandNon-Contextuality.... 210 6 vonNeumannAlgebrasofObservablesandSuperselectionRules..... 213 6.1 IntroductiontovonNeumannAlgebras................................ 213 6.1.1 TheMathematicalNotionofvonNeumannAlgebra ......... 214 6.1.2 Unbounded Selfadjoint Operators Affiliated toavonNeumannAlgebra..................................... 218 6.1.3 LatticesofOrthogonalProjectorsofvonNeumann AlgebrasandFactors ........................................... 220 6.1.4 A Few Words on the Classification of Factors andvonNeumannAlgebras.................................... 223 6.1.5 Schur’sLemma ................................................. 224 6.1.6 ThevonNeumannAlgebraAssociatedtoaPVM ............ 225 6.2 vonNeumannAlgebrasofObservables................................ 228 6.2.1 ThevonNeumannAlgebraofaQuantumSystem............ 228 viii Contents 6.2.2 Complete Sets of Compatible Observables andPreparationofVectorStates............................... 229 6.3 SuperselectionRulesandOtherStructuresoftheAlgebra ofObservables .......................................................... 234 6.3.1 AbelianSuperselectionRulesandCoherentSectors.......... 234 6.3.2 GlobalGaugeGroupFormulationandNon-Abelian Superselection................................................... 238 6.3.3 Quantum States in the Presence of Abelian SuperselectionRules............................................ 241 6.3.4 TheGeneralCaseR ⊂ B(H):QuantumProbability Measures,NormalandAlgebraicStates....................... 246 6.4 CompositeSystemsandvonNeumannAlgebras:Independent Subsystems.............................................................. 248 ∗ 6.4.1 W -IndependenceandStatisticalIndependence.............. 248 6.4.2 TheSplitProperty .............................................. 251 7 QuantumSymmetries ....................................................... 253 7.1 QuantumSymmetriesAccordingtoKadisonandWigner............ 253 7.1.1 Wigner Symmetries, Kadison Symmetries andOrtho-Automorphisms..................................... 254 7.1.2 TheTheoremsofWigner,KadisonandDye .................. 257 7.1.3 ActionofSymmetriesonObservablesandPhysical Interpretation.................................................... 259 7.2 GroupsofQuantumSymmetries....................................... 261 7.2.1 Unitary(-Projective)Representationsof Groups ofQuantumSymmetries........................................ 262 7.2.2 RepresentationsComprisingAnti-UnitaryOperators......... 264 7.2.3 Unitary-ProjectiveRepresentationsofLie Groups andBargmann’sTheorem...................................... 265 7.2.4 InequivalentUnitary-ProjectiveRepresentations andSuperselectionRules....................................... 269 7.2.5 Continuous Unitary-Projective and Unitary RepresentationsofR ........................................... 271 7.2.6 StronglyContinuousOne-ParameterUnitaryGroups: Stone’sTheorem................................................ 274 7.2.7 TimeEvolution,HeisenbergPictureandQuantum NoetherTheorem............................................... 280 7.3 MoreonStronglyContinuousUnitaryRepresentationsofLie Groups................................................................... 285 7.3.1 StronglyContinuousUnitaryRepresentations ................ 286 7.3.2 FromtheGårdingSpacetoNelson’sTheorem................ 288 7.3.3 Pauli’sTheorem................................................. 296 8 TheAlgebraicFormulation................................................. 299 8.1 PhysicalMotivations.................................................... 299 8.2 ObservablesandStatesintheAlgebraicFormalism .................. 301 Contents ix ∗ 8.2.1 TheC -AlgebraCase........................................... 301 ∗ 8.2.2 The -AlgebraCase............................................. 303 8.2.3 ConsistencyofaProbabilisticInterpretation.................. 304 8.3 TheGNSConstructionsandTheirConsequences..................... 306 ∗ 8.3.1 TheGNSReconstructionTheorem:TheC -Algebra Case ............................................................. 307 ∗ 8.3.2 TheGNSReconstructionTheorem:The -AlgebraCase..... 309 8.3.3 NormalStates................................................... 312 8.3.4 TheGelfand-NajmarkTheorem................................ 313 8.3.5 Pure States, Irreducible Representations and SuperselectionRules............................................ 314 ∗ 8.4 Examples:WeylC -Algebras.......................................... 317 8.5 SymmetriesandAlgebraicFormulation ............................... 321 8.5.1 SymmetriesandSpontaneousSymmetryBreaking........... 321 8.5.2 GroupsofSymmetriesintheAlgebraicApproach............ 323 References......................................................................... 327 Index............................................................................... 333 Introduction This book faithfully reflects and perfects the 63-h MSc course, Mathematical Physics:QuantumandQuantum-RelativisticTheories,ItaughtatTrentoUniversity intheacademicyear2017–2018.(Thatcourseisasweepingexpansionofthemini- courseheld atthe “XXIV InternationalFall WorkshoponGeometryandPhysics” inZaragozainSeptember2015.)Theoverallintentionistopresentthemachinery neededto formalizeanddevelopphysics’ideasaboutquantumtheoriesin Hilbert spaces both rigorously and in a concise and self-contained way. Notably, the last ∗ chaptereyestheC -algebraformulationandprovesthebasicrelevantpropositions of that theory.Chapter 5 addresses issues related to the philosophicalfoundations ofquantumtheories,suchasrealism,non-contextuality,andlocality. As a matter of fact, the reader is introduced to the beautiful web of mutual connectionsexistingbetweenlogic,lattice theory,probability,andspectraltheory, includingthebasictheoryofvonNeumannalgebrasthatunderpinsthemathematics of quantum theories. This book should appeal to a dual readership: on one hand mathematicians who wish to acquire the tools that unlock the physical aspects of quantum theories and on the other physicists eager to solidify their understand- ing of the mathematical scaffolding of quantum theories. Several examples and solvedexercisesaccompanythemathematicalstatements—mostofwhichcarefully demonstrated—and physical motivations are provided for every mathematical notion. That said, I must point out that this is not a manualon (higher) quantum mechanics. There are many (very good) books that treat standard or advanced material such as the Schrödinger equation using the proper machinery of PDEs, forwhichreasonthosetopicsarenotfoundhere. Some of the present contents appear in [Mor18], other parts are completely new, for instance Chap. 5, the last section of Chap. 6, and some material in Chap. 8. Despite a gooddegreeof ideologicaloverlap,[Mor18] is morecomplete mathematically, but its 950+ pages do not make it suitable for a single Master course. Most of the proofs here are in fact novel, because they were developed autonomously to reflect the relative conciseness of the lectures to which this text isacompanion. Thebookisorganizedasfollows. xi