Group Theory in a Nutshell for Physicists

633 Pages·2016·10.036 MB·English

by  A. Zee

G
roup Theory in a Nutshell for Physicists

G
roup Theory in a Nutshell for Physicists
A. Zee
P R I N C E T O N U N I V E R S I T Y P R E S S . P R I N C E T O N A N D O X F O R D
Copyright©2016byPrincetonUniversityPress
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Title:Grouptheoryinanutshellforphysicists/A.Zee.
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Description:Princeton,NewJersey:PrincetonUniversityPress,[2016]|
©2016|Series:Inanutshell|Includesbibliographicalreferencesand
index.
Identifiers:LCCN2015037408|ISBN9780691162690(hardcover:alk.paper)|
ISBN0691162697(hardcover:alk.paper)
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Contents
Preface xi
ABriefReviewofLinearAlgebra 1
I Part I: Groups: Discrete or Continuous, Finite or Infinite
I.1 SymmetryandGroups 37
I.2 FiniteGroups 55
I.3 RotationsandtheNotionofLieAlgebra 70
II Part II: Representing Group Elements by Matrices
II.1 RepresentationTheory 89
II.2 Schur’sLemmaandtheGreatOrthogonalityTheorem 101
II.3 CharacterIsaFunctionofClass 114
II.4 Real,Pseudoreal,ComplexRepresentations,andtheNumber
ofSquareRoots 136
II.i1 CrystalsAreBeautiful 146
II.i2 Euler’sϕ-Function,Fermat’sLittleTheorem,andWilson’sTheorem 150
II.i3 FrobeniusGroups 154
III Part III: Group Theory in a Quantum World
III.1 QuantumMechanicsandGroupTheory:Parity,Bloch’sTheorem,
andtheBrillouinZone 161
III.2 GroupTheoryandHarmonicMotion:ZeroModes 168
III.3 SymmetryintheLawsofPhysics:LagrangianandHamiltonian 176
viii | Contents
IV Part IV: Tensor, Covering, and Manifold
IV.1 TensorsandRepresentationsoftheRotationGroupsSO(N) 185
IV.2 LieAlgebraofSO(3)andLadderOperators:CreationandAnnihilation 203
IV.3 AngularMomentumandClebsch-GordanDecomposition 216
IV.4 TensorsandRepresentationsoftheSpecialUnitaryGroupsSU(N) 227
IV.5 SU(2):DoubleCoveringandtheSpinor 244
IV.6 TheElectronSpinandKramer’sDegeneracy 255
IV.7 IntegrationoverContinuousGroups,Topology,CosetManifold,
andSO(4) 261
IV.8 SymplecticGroupsandTheirAlgebras 277
IV.9 FromtheLagrangiantoQuantumFieldTheory:ItIsbutaSkip
andaHop 283
IV.i1 MultiplyingIrreducibleRepresentationsofFiniteGroups:
ReturntotheTetrahedralGroup 289
IV.i2 CrystalFieldSplitting 292
IV.i3 GroupTheoryandSpecialFunctions 295
IV.i4 CoveringtheTetrahedron 299
V Part V: Group Theory in the Microscopic World
V.1 IsospinandtheDiscoveryofaVastInternalSpace 303
V.2 TheEightfoldWayofSU(3) 312
V.3 TheLieAlgebraofSU(3)andItsRootVectors 325
V.4 GroupTheoryGuidesUsintotheMicroscopicWorld 337
VI Part VI: Roots, Weights, and Classification of Lie Algebras
VI.1 ThePoorManFindsHisRoots 347
VI.2 RootsandWeightsforOrthogonal,Unitary,andSymplecticAlgebras 350
VI.3 LieAlgebrasinGeneral 364
VI.4 TheKilling-CartanClassificationofLieAlgebras 376
VI.5 DynkinDiagrams 384
VII Part VII: From Galileo to Majorana
VII.1 SpinorRepresentationsofOrthogonalAlgebras 405
VII.2 TheLorentzGroupandRelativisticPhysics 428
VII.3 SL(2,C)DoubleCoversSO(3,1):GroupTheoryLeadsUs
totheWeylEquation 450
VII.4 FromtheWeylEquationtotheDiracEquation 468
VII.5 DiracandMajoranaSpinors:AntimatterandPseudoreality 482
VII.i1 AHiddenSO(4)AlgebraintheHydrogenAtom 491
Contents | ix
VII.i2 TheUnexpectedEmergenceoftheDiracEquationinCondensed
MatterPhysics 497
VII.i3 TheEvenMoreUnexpectedEmergenceoftheMajoranaEquation
inCondensedMatterPhysics 501
VIII Part VIII: The Expanding Universe
VIII.1 ContractionandExtension 507
VIII.2 TheConformalAlgebra 515
VIII.3 TheExpandingUniversefromGroupTheory 523
IX Part IX: The Gauged Universe
IX.1 TheGaugedUniverse 531
IX.2 GrandUnificationandSU(5) 541
IX.3 FromSU(5)toSO(10) 550
IX.4 TheFamilyMystery 560
Epilogue 565
TimelineofSomeofthePeopleMentioned 567
SolutionstoSelectedExercises 569
Bibliography 581
Index 583
CollectionofFormulas 601