Elementary Particle Theory, Volume 1: Quantum Mechanics

290 Pages·2018·4.879 MB·English

by  Eugene Stefanovich

EugeneStefanovich
ElementaryParticleTheory
De Gruyter Studies in
Mathematical Physics
|
Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Dmitry Gitman,São Paulo, Brazil
Alexander Lazarian, Madison, Wisconsin, USA
BorisSmirnov, Moscow, Russia
Volume 45
Eugene Stefanovich
Elementary
Particle Theory
|
Volume 1: Quantum Mechanics
MathematicsSubjectClassification2010
Primary:81-02,81P10,22E43;Secondary:47A40,81V10
Author
Dr.EugeneStefanovich
SanJose,California
USA
[email protected]
ISBN978-3-11-049088-6
e-ISBN(PDF)978-3-11-049213-2
e-ISBN(EPUB)978-3-11-049103-6
ISSN2194-3532
LibraryofCongressControlNumber:2018016481
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
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ToRegina

Contents
Listoffigures|XIII
Listoftables|XV
Postulates,statements,theorems,assertions|XVII
Conventionalnotation|XIX
Preface|XXIII
Introduction|XXIX
1 Quantumlogic|1
1.1 Whydoweneedquantummechanics?|3
1.1.1 Corpusculartheoryoflight|3
1.1.2 Wavetheoryoflight|6
1.1.3 Lightoflowintensityandotherexperiments|7
1.2 Classicallogic|8
1.2.1 Phasespaceofoneclassicalparticle|9
1.2.2 Propositionsinphasespace|9
1.2.3 Operationswithpropositions|10
1.2.4 Axiomsoflogic|11
1.2.5 Phasespacefromaxiomsofclassicallogic|13
1.2.6 Classicalobservables|14
1.3 Measurementsandprobabilities|15
1.3.1 Ensemblesandmeasurements|15
1.3.2 Statesasprobabilitymeasures|16
1.3.3 Probabilitydistributionsandstatisticalmechanics|17
1.4 Logicofquantummechanics|18
1.4.1 Partialdeterminismofquantummechanics|18
1.4.2 Axiomsofquantumlogicfromprobabilitymeasures|20
1.4.3 Compatibilityofpropositions|21
1.4.4 Logicofquantummechanics|22
1.4.5 QuantumlogicandHilbertspace|23
1.4.6 Piron’stheorem|23
1.4.7 Shouldweabandonclassicallogic?|24
1.5 PhysicsinHilbertspace|25
1.5.1 Quantumobservables|25
1.5.2 States|26
VIII | Contents
1.5.3 Completesetsofcommutingobservables|28
1.5.4 Wavefunctions|29
1.5.5 Expectationvalues|29
1.5.6 Basicrulesofclassicalandquantummechanics|30
1.6 Interpretationsofquantummechanics|31
1.6.1 Quantumnonpredictability|31
1.6.2 Collapseofwavefunction|32
1.6.3 Collapseofclassicalprobabilitydistribution|32
1.6.4 Hiddenvariables|33
1.6.5 Quantum-logicalinterpretation|34
1.6.6 Quantumrandomnessandlimitsofknowledge|35
2 Poincarégroup|37
2.1 Inertialobservers|37
2.1.1 Principleofrelativity|37
2.1.2 Inertialtransformations|38
2.2 Galileigroup|39
2.2.1 CompositionlawinGalileigroup|40
2.2.2 LiealgebraofGalileigroup|41
2.2.3 Rotationsappliedtogenerators|43
2.2.4 Spaceinversion|45
2.3 Poincarégroup|46
2.3.1 ConditionsonPoincarégenerators|46
2.3.2 LiealgebraofPoincarégroup|47
2.3.3 Boostsoftranslationgenerators|51
3 Quantummechanicsandrelativity|55
3.1 Inertialtransformationsinquantummechanics|55
3.1.1 Wigner’stheorem|55
3.1.2 Inertialtransformationsofstates|58
3.1.3 HeisenbergandSchrödingerpictures|59
3.2 UnitaryrepresentationsofPoincarégroup|60
3.2.1 Projectiverepresentationsofgroups|60
3.2.2 Generatorsofprojectiverepresentation|61
3.2.3 Commutatorsofprojectivegenerators|63
3.2.4 Cancellationofcentralcharges|65
3.2.5 Single-valuedanddouble-valuedrepresentations|68
3.2.6 Fundamentalstatementofrelativisticquantumtheory|68
3.2.7 Timeevolutioninmovingframe|70
4 Observables|71
4.1 Basicobservables|71
Contents | IX
4.1.1 Energy,momentumandangularmomentum|71
4.1.2 Operatorofvelocity|73
4.2 Casimiroperators|73
4.2.1 4-Vectors|74
4.2.2 Massoperator|74
4.2.3 Pauli–Lubanski4-vector|75
4.3 Operatorsofspinandposition|77
4.3.1 Physicalrequirements|77
4.3.2 Spinoperator|78
4.3.3 Positionoperator|79
4.3.4 Commutatorsofposition|81
4.3.5 Alternativesetofbasicoperators|83
4.3.6 Canonicalorderofoperators|84
4.3.7 Powerofoperator|85
4.3.8 Uniquenessofspinoperator|87
4.3.9 Uniquenessofpositionoperator|88
4.3.10 Boostofposition|89
5 Elementaryparticles|91
5.1 Massiveparticles|93
5.1.1 One-particleHilbertspace|93
H
5.1.2 Actionofrotationsubgroupin 0|94
5.1.3 Momentum–spinbasis|96
5.1.4 Nonuniquenessofmomentum–spinbasis|98
5.1.5 Actionoftranslationsandrotationsonbasisvectors|98
5.1.6 Actionofboostsonmomentumeigenvectors|99
5.1.7 Actionofboostsonspincomponents|100
5.1.8 Wignerangle|101
5.1.9 IrreducibilityofrepresentationUg|102
5.1.10 Methodofinducedrepresentations|102
5.2 Momentumrepresentation|103
5.2.1 Resolutionofidentity|103
5.2.2 Boosttransformation|104
5.2.3 Wavefunctioninmomentumrepresentation|106
5.3 Positionrepresentation|107
5.3.1 Basisoflocalizedfunctions|107
5.3.2 Operatorsofobservablesinpositionrepresentation|109
5.3.3 Inertialtransformationsofobservablesandstates|110
5.3.4 Timetranslationsofobservablesandstates|113
5.4 Masslessparticles|113
5.4.1 Spectraofmomentum,energyandvelocity|113
5.4.2 Representationsofsmallgroups|114