Symmetry of equations of quantum mechanics

478 Pages·1994·2.646 MB·English

by  Fushchich, Nikitin.

SYMMETRIES OF EQUATIONS
OF QUANTUM MECHANICS
TABLEOFCONTENTS
ChapterI.LOCALSYMMETRYOFBASISEQUATIONSOF
RELATIVISTICQUANTUMTHEORY
1.LocalSymmetryoftheKlein-Gordon-FockEquation
1.1.Introduction ............................................ 1
1.2.TheIAoftheKGFEquation ............................... 3
1.3.Symmetryofthed’AlembertEquation........................ 5
1.4.LorentzTransformations................................... 6
1.5.ThePoincaréGroup ...................................... 9
1.6.TheConformalTransformations............................ 12
1.7.TheDiscreteSymmetryTransformations ..................... 14
2.LocalSymmetryoftheDiracEquation
2.1.TheDiracEquation ...................................... 16
2.2.VariousFormulationsoftheDiracequation ................... 17
2.3.AlgebraoftheDiracMatrices ............................. 19
2.4.SOsandIAs ........................................... 19
2.5.TheIAoftheDiracEquationintheClassM ................. 20
1
2.6.TheOperatorsofMassandSpin ........................... 22
2.7.ManifestlyHermitianFormofPoincaréGroupGenerators ....... 23
2.8.SymmetriesoftheMasslessDiracEquation .................. 24
2.9.LorentzandConformalTransformationsofSolutionsoftheDirac
Equation ............................................... 25
2.10.P-,T-,andC-Transformations ............................. 27
3.Maxwell’sEquations
3.1.Introduction .......................................... 28
3.2.VariousFormulationsofMaxwell’sEquations ................ 30
i
3.3.TheEquationfortheVector-Potential ....................... 32
3.4.TheIAofMaxwell’sEquationsintheClassM ............... 33
1
3.5.LorentzandConformalTransformations ..................... 34
3.6.SymmetryUndertheP-,T-,andC-Transformations ........... 38
3.7.RepresentationsoftheConformalAlgebraCorrespondingtoaField
withArbitraryDiscreteSpin ............................... 39
3.8.CovariantRepresentationsoftheAlgebrasAP(1,3)andAC(1,3) .. 40
3.9.ConformalTransformationsforAnySpin .................... 43
ChapterII.REPRESENTATIONSOFTHEPOINCARÉALGEBRA
ANDWAVEEQUATIONSFORARBITRARYSPIN
4.IRofthePoincaréAlgebra
4.1.Introduction ........................................... 44
4.2.CasimirOperators ....................................... 45
4.3.BasisofanIR ......................................... 46
4.4.TheExplicitFormoftheLubanski-PauliVector ............... 48
4.5.IRoftheAlgebraA(c,n) ................................. 50
1
4.6.ExplicitRealizationsofthePoincaréAlgebra ................. 53
4.7.ConnectionswiththeCanonicalRealizationsof
Shirokov-Foldy-Lomont-Moses ............................. 55
4.8.CovariantRepresentations ................................ 58
5.RepresentationsoftheDiscreteSymmetryTransformations
5.1.Introduction ........................................... 60
5.2.NonequivalentMultiplicatorsoftheGroupG ................ 62
8
5.3.TheGeneralFormoftheDiscreteSymmetryOperators ......... 64
5.4.TheOperatorsP,T,andCforRepresentationofClassI ......... 67
5.5.RepresentationsofClassII ................................ 70
5.6.RepresentationsofClassesIII-IV .......................... 71
5.7.RepresentationsofClassV ............................... 73
5.8.ConcludingRemarks .................................... 75
6.Poincaré-InvariantEquationsofFirstOrder
6.1.Introduction ........................................... 75
6.2.ThePoincaréInvarianceCondition ......................... 77
6.3.TheExplicitFormoftheMatricesß ....................... 78
µ
6.4.AdditionalRestristionsfortheMatricesβ ................... 79
µ
ii
6.5.TheKemmer-Duffin-Petiau(KDP)Equation .................. 81
6.6.TheDirac-Fierz-PauliEquationforaParticleofSpin3/2 ....... 82
6.7.TransitiontotheSchrödingerForm ......................... 85
7.Poincaré-InvariantEquationswithoutRedundantComponents
7.1.PreliminaryDiscussion ................................. .88
7.2.FormulationoftheProblem ............................... 89
7.3.TheExplicitFormofHamiltoniansHIandHII ................. 92
s s
7.4.DifferentialEquationsofMotionforSpinningParticles ......... 96
7.5.ConnectionwiththeShirokov-FoldyRepresentation............ 98
8.EquationsinDirac’sFormforArbitrarySpinParticles
8.1.CovariantEquationswithCoefficientsFormingtheClifford
Algebra ............................................... 100
8.2.EquationswiththeMinimalNumberofComponents .......... 101
8.3.ConnectionwithEquationswithoutSuperfluousComponents ... 103
8.4.LagrangianFormulation ................................ 104
8.5.Dirac-LikeWaveEquationsasaUniversalModelofaParticle
withArbitrarySpin ...................................... 105
9.EquationsforMasslessParticles
9.1.BasicDefinitions ...................................... 108
9.2.AGroup-TheoreticalDerivationofMaxwell’sEquations ....... 109
9.3.Conformal-InvariantEquationsforFieldsofArbitrarySpin .... 110
9.4.EquationsofWeyl’sType ................................ 112
9.5.EquationsofOtherTypesforMasslessParticles .............. 115
10.RelativisticParticleofArbitrarySpininanExternalElectromagnetic
Field
10.1.ThePrincipleofMinimalInteraction ...................... 116
10.2.IntroductionofMinimalInteractionintoFirstOrderWave
Equations ............................................. 117
10.3.IntroductionofInteractionintoEquationsinDirac’sForm .... 119
10.4.AFour-ComponentEquationforSpinlessParticles .......... 121
10.5.EquationsforSystemswithVariableSpin .................. 122
10.6.IntroductionofMinimalInteractionintoEquationswithoutSuperfluous
Components ........................................... 123
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10.7.ExpansioninPowerSeriesin1/m ........................ 124
10.8.CausalityPrincipleandWaveEquationsforParticlesofArbitrary
Spin ................................................. 127
10.9.TheCausalEquationforSpin-OneParticleswithPositive
Energies .............................................. 128
ChapterIII.REPRESENTATIONSOFTHEGALILEIALGEBRAAND
GALILEI-INVARIANTWAVEEQUATIONS
11.SymmetriesoftheSchrödingerEquation
11.1.TheSchrödingerEquation .............................. 131
11.2.InvarianceAlgebraoftheSchrödingerEquation ............. 132
11.3.TheGalileiandGeneralizedGalileiAlgebras ............... 134
11.4.TheSchrödingerEquationGroup ......................... 136
11.5.TheGalileiGroup ..................................... 138
11.6.TheTransformationsPandT ............................ 139
12.RepresentationsoftheLieAlgebraoftheGalileiGroup
12.1.TheGalileiRelativityPrincipleandEquationsofQuantum
Mechanics ........................................... 140
12.2.ClassificationofIRs ................................... 141
12.3.TheExplicitFormofBasisElementsoftheAlgebraAG(1,3) ... 142
12.4.ConnectionswithOtherRealizations ...................... 144
12.5.CovariantRepresentations .............................. 146
12.6.RepresentationsoftheLieAlgebraoftheHomogeneousGalilei
Group ................................................ 148
13.Galilei-InvariantWaveEquations
13.1.Introduction ......................................... 154
13.2.Galilei-InvarianceConditions ........................... 154
13.3.AdditionalRestrictionsforMatricesß .................... 155
µ
13.4.GeneralFormofMatricesß intheBasis λ;l,m> ............ 157
µ
13.5.EquationsofMinimalDimension ......................... 159
13.6.EquationsforRepresentationswithArbitraryNilpotencyIndices 163
iv
14.Galilei-InvariantEquationsoftheSchrödingerType
14.1.UniquenessoftheSchrödingerequation ................... 165
14.2.TheExplicitFormofHamiltoniansofArbitrarySpinParticles . 167
14.3.LagrangianFormulation ................................ 170
15.GalileanParticleofArbitrarySpininanExternalElectromagneticField
15.1.IntroductionofMinimalInteractionintoFirst-OrderEquations . 171
15.2.MagneticMomentofaGalileiParticleofArbitrarySpin ...... 173
15.3.InteractionwiththeElectricField ........................ 175
15.4.Equationsfora(2s+1)-ComponentWaveFunction ........... 176
15.5.IntroductionoftheMinimalInteractionintoSchrödinger-Type
Equations ............................................ 179
15.6.AnomalousInteraction ................................ 180
ChapterIV.NONGEOMETRICSYMMETRY
16.HigherOrderSOsoftheKGFandSchrödingerEquations
16.1.TheGeneralizedApproachtoStudyingofSymmetriesofPartial
DifferentialEquations ................................... 183
16.2.SOsoftheKGFEquation ............................... 185
16.3.HiddenSymmetriesoftheKGFEquation .................. 188
16.4.HigherOrderSOsofthed’AlembertEquation .............. 190
16.5.SOsoftheSchrödingerEquation ......................... 191
16.6.HiddenSymmetriesoftheSchrödingerEquation ............ 193
16.7.SymmetriesoftheQuasi-RelativisticEvolutionEquation ..... 196
17.NongeometricSymmetriesoftheDiracEquation
17.1.TheIAoftheDiracEquationintheClassM ............... 198
1
17.2.SymmetriesoftheDiracEquationintheClassofIntegro-
DifferentialOperators ................................... 202
17.3.SymmetriesoftheEight-ComponentDiracEquation ......... 203
17.4.SymmetryUnderLinearandAntilinearTransformations ..... 206
17.5.HiddenSymmetriesoftheMasslessDiracEquation .......... 209
v
18.TheCompleteSetofSOsoftheDiracEquation
18.1.IntroductionandDefinitions ............................. 211
18.2.TheGeneralFormofSOsofOrdern .................... 212
18.3.AlgebraicPropertiesoftheFirst-OrderSOs ............... 213
18.4.TheCompleteSetofSOsofArbitraryOrder ................ 216
18.5.ExamplesandDiscussion .............................. 220
18.6.SOsoftheMasslessDiracEquation ...................... 221
19.SymmetriesofEquationsforArbitrarySpinParticles
19.1.SymmetriesoftheKDPEquation ........................ 223
19.2.ArbitraryOrderSOsoftheKDPEquation ................. 226
19.3.SymmetriesoftheDirac-LikeEquationsforArbitrarySpinParticl2e2s9
19.4.HiddenSymmetriesAdmittedbyAnyPoincaré-InvariantWave
Equation ............................................. 232
19.5.SymmetriesoftheLevi-LeblondEquation ................. 234
19.6.SymmetriesofGalilei-InvariantEquationsforArbitrarySpin
Particles .............................................. 236
20.NongeometricSymmetriesofMaxwell’sEquations
20.1.InvarianceUndertheAlgebraAGL(2,C) ................... 238
20.2.TheGroupofNongeometricSymmetryofMaxwell’sEquations 241
20.3.SymmetriesofMaxwell’sEquationsintheClassM ......... 243
2
20.4.SuperalgebrasofSOsofMaxwell’sEquations............... 247
20.5.SymmetriesofEquationsfortheVector-Potential ............ 249
21.SymmetriesoftheSchrödingerEquationwithaPotential
21.1.SymmetriesoftheOne-DimensionSchrödingerEquation...... 251
21.2.ThePotentialsAdmissingThird-OrderSymmetries .......... 253
21.3.Time-DependentPotentials.............................. 257
21.4.AlgebraicPropertiesofSOs ............................. 257
21.5.CompleteSetsofSOsforOne-andThree-Dimensional
Schrödingerequation .................................... 259
21.6.SOsoftheSupersymmetricOscillator ..................... 262
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22.NongeometricSymmetriesofEquationsforInteractingFields
22.1.TheDiracEquationforaParticleinanExternalField......... 263
22.2.TheSOofDiracTypeforVectorParticles ................. 267
22.3.TheDiracTypeSOsforParticlesofAnySpin ............... 268
22.4.OtherSymmetriesofEquationsforArbitrarySpinParticles .... 271
22.5.SymmetriesofaGalileiParticleofArbitrarySpinintheConstant
ElectromagneticField ................................... 272
22.6.SymmetriesofMaxwell’sEquationswithCurrentsandCharges 273
22.7.Super-andParasupersymmetries ......................... 276
22.8.SymmetriesinElasticity ................................ 278
23.ConservationLawsandConstantsofMotion
23.1.Introduction ......................................... 281
23.2.ConservationLawsfortheDiracField ................... 284
23.3.ConservationLawsfortheMasslessSpinorField ............ 285
23.4.TheProblemofDefinitionofConstantsofMotionforthe
ElectromagneticField ................................... 286
23.5.ClassicalConservationLawsfortheElectromagneticField .... 289
23.6.TheFirstOrderConstantsofMotionfortheElectromagnetic
Field ................................................. 289
23.7.TheSecondOrderConstantsofMotionfortheElectromagnetic
Field ................................................. 291
23.8.ConstantsofMotionfortheVector-Potential ................ 295
ChapterV.GENERALIZEDPOINCARÉGROUPS
24.TheGroupP(1,4)
24.1.Introduction ......................................... 297
24.2.TheAlgebraAP(1,n)................................... 298
24.3.NonequivalentRealizationsoftheTensorW ............... 299
µσ
24.4.TheBasisofanIR .................................... 302
24.5.TheExplicitFormoftheBasisElementsofthealgebraAP(1,4). 303
24.6.ConnectionwithOtherRealizations ....................... 304
25.RepresentationsoftheAlgebraAP(1,4)inthePoincaré-Basis
25.1.SubgroupStructureoftheGroupP(1,4).................... 307
25.2.Poincaré-Basis ....................................... 307
vii
25.3.ReductionP(1,4)→P(1,3)ofIRsofClassI ................ 308
25.4.ReductionP(1,4)→P(1,2) .............................. 311
25.5.ReductionofIRsfortheCasec =0 ....................... 312
1
25.6.ReductionofRepresentationsofClassIV .................. 315
25.7.ReductionP(1,n)→P(1.3) .............................. 316
26.RepresentationsoftheAlgebraAP(1,4)intheG(1,3)-andE(4)-Basises
26.1.TheG(1,3)-Basis...................................... 319
26.2.RepresentationswithPnP 〉0 ............................. 321
n
26.3.RepresentationsofClassesII-IV .......................... 324
26.4.CovariantRepresentations .............................. 325
26.5.TheE(4)-Basis ....................................... 327
26.6.RepresentationsofthePoincaréAlgebraintheG(1,2)-Basis ... 328
27.WaveEquationsInvariantUnderGeneralizedPoincaréGroups
27.1.PreliminaryNotes ..................................... 330
27.2.GeneralizedDiracEquations ............................ 331
27.3.TheGeneralizedKemmer-Duffin-PetiauEquations........... 334
27.4.CovariantSystemsofEquations .......................... 335
Chapter6.EXACTSOLUTIONSOFLINEARANDNONLINEAR
EQUATIONSOFMOTION
28.ExactSolutionsofRelativisticWaveEquationsforParticlesofArbitrary
Spin
28.1.Introduction ......................................... 338
28.2.FreeMotionofParticles ................................ 339
28.3.RelativisticParticleofArbitrarySpininHomogeneousMagnetic
Field ................................................. 341
28.4.AParticleofArbitrarySpinintheFieldofthePlane
ElectromagneticWave ................................... 345
29.RelativisticParticlesofArbitrarySpinintheCoulombField
29.1.SeparationofVariablesinaCentralField .................. 347
29.2.SolutionofEquationsforRadialFunctions ................. 349
29.3.EnergyLevelsofaRelativisticParticleofArbitrarySpininthe
CoulombField ......................................... 351
viii
30.ExactSolutionsofGalilei-InvariantWaveEquations
30.1.PreliminaryNotes .................................... 354
30.2.NonrelativisticParticleintheConstantandHomogeneous
MagneticField ........................................ 355
30.3.NonrelativisticParticleofArbitrarySpininCrossedElectricand
MagneticFields ....................................... 357
30.4.NonrelativisticParticleofArbitrarySpinintheCoulombField . 359
31.NonlinearEquationsInvariantUnderthePoincaréandGalileiGroups
31.1.Introduction ......................................... 362
31.2.SymmetryAnalysisandExactSolutionsoftheScalarNonlinear
WaveEquation ......................................... 362
31.3.SymmetriesandExactSolutionsoftheNonlinearDiracEquation 365
34.4.EquationsofSchrödingertypeInvariantUndertheGalileiGroup 368
34.5.SymmetriesofNonlinearEquationsofElectrodynamics ....... 371
31.6.GalileiRelativityPrincipleandtheNonlinearHeatEquation ... 374
31.7.ConditionalSymmetryandExactSolutionsoftheBoussinesq
Equation .............................................. 377
31.8.ExactSolutionsofLinearandNonlinearSchrödingerequation . 381
Chapter7.TWO-PARTICLEEQUATIONS
32.Two-ParticleEquationsInvariantUndertheGalileiGroup
32.1.PreliminaryNotes .................................... 384
32.2.EquationsforSpinlessParticles .......................... 385
32.3.EquationsforSystemsofParticlesofArbitrarySpin ......... 388
32.4.Two-ParticleEquationsofFirstOrder ..................... 390
32.5.EquationsforInteractingParticlesofArbitrarySpin .......... 392
33.Quasi-RelativisticandPoincaré-InvariantTwo-ParticleEquations
33.1.PreliminaryNotes .................................... 395
33.2.TheBreitEquation .................................... 396
33.3.TransformationtotheQuasidiagonalForm ................. 397
33.4.TheBreitEquationforParticlesofEqualMasses ............ 399
33.5.Two-ParticleEquationsInvariantUndertheGroupP(1,6) ..... 401
33.6.AdditionalConstantsofMotionforTwo-andThree-Particle
Equations ............................................. 403
ix
34.ExactlySolvableModelsofTwo-ParticleSystems
34.1.TheNonrelativisticModel .............................. 405
34.2.TheRelativisticTwo-ParticleModel ...................... 405
34.3.SolutionsofTwo-ParticleEquations ...................... 407
34.4.DiscussingofSpectraoftheTwo-ParticleModels ........... 409
Appendix1.LieAlgebras,SuperalgebrasandParasuperalgebras ........ 414
Appendix2.GeneralizedKillingTensors ............................ 416
Appendix3.MatrixElementsofScalarOperatorsintheBasisofSpherical
Spinors .................................................. 420
References ..................................................... 424
AdditionalListofReferences ...................................... 453
ListofAbbreviations ............................................. 457
Index .......................................................... 458
x