Fundamental Mathematical Structures of Quantum Theory

345 Pages·2019·4.251 MB·English

by  Valter Moretti

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
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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