Classical Groups for Physicists

Cover......Page 1
Title page......Page 2
Preface......Page 6
Contents......Page 8
1 Introduction......Page 16
2.1 Symmetry and Atomic Quantum Numbers......Page 18
2.2 Hierarchies of Symmetry......Page 20
3.1 The Group Postulates......Page 22
3.2 Regular Matrix Groups......Page 23
3.4 Continuous Matrix Groups......Page 25
3.5 Matrix Exponential Functions......Page 29
4.1 Parameterization of the Group Elements......Page 32
4.2 Connectivity......Page 33
4.3 The Beginning of Lie Groups......Page 34
4.4 Infinitesimal Group Generators......Page 35
4.5 The Two-Dimensional Rotation Group SO(2)......Page 37
4.7 General Infinitesimal Transformations......Page 38
4.8 Infinitesimal Operators of a Lie Group......Page 40
4.9 Examples of Infinitesimal Operators......Page 43
4.10 Structure Constants of Lie Groups......Page 46
4.11 Generation of Finite Group Elements......Page 48
4.12 Finite Transformations......Page 52
5.1 Lie Algebras......Page 55
5.2 Transformation of Basis......Page 56
5.3 Homomorphisms and Isomorphisms......Page 57
5.5 Lie Algebras and Subalgebras......Page 58
5.7 Adjoint Representations of Lie Algebras......Page 59
5.8 Complex Extensions of Real Lie Algebras......Page 60
5.10 The Killing Form and Cartan\'s Criterion for Semisimple Lie\' Algebras......Page 61
5.11 Example of SO(4)......Page 63
5.12 Example of E₂......Page 64
5.14 Solvable Lie Algebras......Page 65
5.15 Nilpotent Lie Algebras......Page 66
5.16 Direct and Semidirect Sums......Page 67
5.18 The Casimir Operators......Page 68
5.21 Lie Groups and Lie Algebras......Page 70
6.2 Standard Form of the Semisimple Lie Groups......Page 72
6.3 Properties of Roots......Page 74
6.5 The Standard Form Obtained......Page 75
6.6 Further Theorems Concerning Roots......Page 77
6.8 Graphical Representation of Root Vectors......Page 80
6.9 Lie Algebras of Rank 2......Page 82
6.10 Lie Algebras of Rank 1>2......Page 85
6.11 The Exceptional Lie Algebras......Page 86
7.1 Simple Roots......Page 88
7.2 Examples of B₂ and B₃......Page 90
7.3 Dynkin Diagrams......Page 91
7.4 The Cartan Matrix......Page 92
7.5 Examples of Cart an Matrices......Page 94
7.7 Application to G₂......Page 98
7.8 C onstruction of Some Simple Lie Algebras......Page 99
8.1 Co-Weights and the Chevalley Basis......Page 102
8.2 Phases in the Chevalley Basis......Page 104
8.3 The Algebra su(3) in the Chevalley Basis......Page 105
9.1 Group Representations......Page 107
9.2 Real and Complex Representations......Page 108
9.4 Adjoint Representations......Page 109
9.5 Unitary and Nonunitary Representations......Page 110
10.1 Weights and Weight Spaces......Page 112
10.2 Theorems Concerning Weights......Page 114
10.3 The Weyl Reflection Group......Page 115
10.4 Weights and the Classification of Irreducible Representations......Page 116
10.5 Computation of the Complete Set of Weights......Page 117
10.6 Examples of Computations of Weights......Page 120
11.1 Definition......Page 124
11.3 The Weight Space for Kronecker Products......Page 125
11.4 Decomposition of the Kronecker Product......Page 126
12.1 Basic Representations......Page 128
12.2 Kronecker Powers......Page 129
12.3 Elementary Representations......Page 131
12.4 Weights of Elementary Representations......Page 133
12.5 Spin or Representations and the Groups B_n and D_n......Page 136
12.6 Labeling of Irreducible Representations......Page 138
12.7 A Matter of Notation......Page 141
13.1 Basic Representations of the Exceptional Groups......Page 142
13.2 Labeling of Representations for the Exceptional Groups......Page 145
14.1 Scalar Products of Basic Weights......Page 148
14.2 Dimensions of Irreducible Representations......Page 150
15.1 Eigenvalues of the Quadratic Casimir Operators......Page 154
15.2 Generalized Casimir Invariants......Page 155
15.3 Invariants for Nonsemisimple Lie Groups......Page 157
15.4 Casimir Operators for SO(3) and SO(2,1)......Page 158
16.1 Topological Neighborhoods......Page 165
16.2 Topological Spaces......Page 166
16.3 Examples of Topological Spaces......Page 167
16.5 Products of Topological Spaces......Page 168
16.7 Metric Spaces......Page 169
16.9 Compact Spaces......Page 170
16.10 Homotopic Paths......Page 171
16.11 Simply Connected and Multiply Connected Spaces......Page 173
16.12 The Fundamental Group......Page 174
16.13 Universal Covering Spaces......Page 175
16.14 Topological Groups......Page 176
16.17 Topological Subgroups......Page 178
16.19 Coset Spaces and Factor Groups......Page 179
16.20 Homogeneous Spaces......Page 181
16.22 Real Simple Lie Groups and Lie Algebras......Page 182
16.23 Isomorphisms of Lie Groups and Lie Algebras......Page 188
16.24 Universal Covering Group......Page 189
17.1 The Three Parameter Lie Groups......Page 191
17.2 The Standard Form......Page 192
17.3 The Casimir Invariants......Page 193
17.4 The Elementary Representations......Page 194
17.5 Basis for the Spinor Representation......Page 195
17.6 Realization in Terms of Boson Operators......Page 196
17.7 Construction of Other Representations......Page 197
17.8 The Unitary Representations......Page 201
17.9 Matrix Elements of L₁2 and L_+-......Page 203
17.10 Finite Transformations......Page 205
17.12 Coupling Coefficients......Page 210
17;13 Specialization to SO(3)......Page 213
17.14 Coupling Coefficients for SO(2,1)......Page 216
17.15 Coupling Coefficients and Analytic Continuation......Page 218
18.1 Introduction......Page 222
18.2 A Realization of su(l,1)......Page 223
18.3 Discrete Eigenvalue Spectrum......Page 224
18.5 Three-Dimensional Isotropic Harmonic Oscillator......Page 226
18.6 The Generalized Kepler Problem......Page 227
18.7 The Two-Dimensional Kepler Problem......Page 229
18.8 The Morse Potential......Page 230
18.9 Limitations of su(l,1)......Page 231
19.1 Introduction......Page 233
19.2 Some Notation......Page 234
19.3 Tensor Operators......Page 235
19.4 Tensor Operators in SO(3)......Page 236
19.6 Coupling Coefficients......Page 237
19.7 Coupling to the Identity Representation......Page 238
19.8 The Wigner-Eckart Theorem......Page 240
19.9 Selection Rules......Page 242
19.10 Application to SO(3)......Page 243
19.11 Generalized Recoupling Coefficients......Page 245
19.12 Recoupling Coefficients for SO(3)......Page 247
19.13 Coupling Coefficients for SO(4)......Page 251
19.14 Racah\'s Factorization Lemma......Page 255
19.15 Isoscalar Factors......Page 257
19.16 Adjoint Tensor Operators......Page 258
19.17 Symmetry Properties of Coupling Coefficients......Page 260
19.18 Reciprocity and Isoscalar Factors......Page 263
19.19 Phase Conventions......Page 264
19.20 Simple Isoscalar Factors......Page 265
19.21 The Building-Up Principle......Page 266
19.22 Alternative Calculation of Isoscalar Factors......Page 276
19.23 Coupled Tensor Operators......Page 278
19.24 Coupled Tensor Operators for SO(3)......Page 280
20.1 Introduction......Page 283
20.2 Second Quantization and the Harmonic Oscillator......Page 284
20.3 The Groups U(3) and SU(3)......Page 285
20.4 Rotational Symmetry......Page 286
20.5 Some SU(3) Tensor Operators......Page 287
20.6 Reduced Matrix Elements......Page 290
20.7 The Quadratic Casimir Operator......Page 292
20.8 Ladder Operators in SU(3)......Page 293
20.10 Commutation Relations......Page 294
20.11 A Larger Group for the Oscillator......Page 297
20.12 Subgroups of Sp(6,R)......Page 298
20.14 A Dynamical Group for the Oscillator......Page 301
20.15 Group Contractions and the Dynamical Group......Page 303
20.17 Tensor Operators for the SO(2, 1) X SO(3) Subgroup......Page 305
20.18 Matrix Elements of Multiple Operators......Page 307
21.1 Introduction......Page 312
21.2 SO(4) and Hydrogen Energy Levels......Page 315
21.4 Reduced Matrix Elements of A......Page 317
21.5 Ladder Operators in SO(4)......Page 319
21.6 Boson Operators and SO(4)......Page 321
21.7 Dynamical Group of the Hydrogen Atom......Page 322
21.8 The Casimir Operators......Page 326
21.9 The SO(4,1) Subgroup......Page 327
21.10 Further Subgroups of SO (4,2)......Page 328
21.11 SO(4,2) Bases and Hydrogenic Atoms......Page 329
21.12 A Coordinate Realization of SO(4,2)......Page 334
21.13 A Physical Realization of SO(4,2)......Page 335
21.14 Tilted States of the Hydrogen Atom......Page 336
21.15 A Dilatation-Operator Realization of SO¹(2, 1)X S0²(2, 1)......Page 338
21.16 The Electric Dipole Operator......Page 340
21.17 Galilean Boosts......Page 344
21.18 Lorentzian Boosts......Page 346
21.19 Infinite-Component Wave Equations......Page 347
21.20 Example of Hydrogen......Page 352
21.21 A Finite-Dimensional Realization of SO(4,2)......Page 355
21.22 Reformulation of the Dirac Theory of the Electron......Page 358
21.23 The Hydrogen Atom with Spin......Page 359
21.24 The Conformal Group and SO(4,2)......Page 360
21.25 Concluding Remarks......Page 363
22.1 Introduction......Page 364
22.2 States of a Fermion Shell......Page 365
22.3 The Supergroup......Page 367
22.4 Two Important Subgroups......Page 368
22.5 A Unitary Subgroup......Page 369
22.6 Tensor Operators and Annihilation and Creation Operators......Page 370
22.7 A Coupled Tensor Operator......Page 371
22.8 Further Subgroups......Page 372
22.9 Classification for the j = 7/2 Shell......Page 382
22.10 Seniority......Page 374
22.11 The Quasi-Spin Formalism......Page 375
22.12 Quasi-Spin Classification of States......Page 376
22.13 Quasi-Spin for Annihilation and Creation Operators......Page 377
22.14 Symmetry Classification of Operators......Page 378
22.15 Interaction of Particles in a Central Field......Page 381
A.1 Introduction......Page 387
A.2 S-Functions......Page 388
A.3 Outer S-Function Multiplication......Page 390
A.4 S-Function Division......Page 392
A.6 Characters of Groups as S-Functions......Page 393
A.7 Reduction of the Number of Parts of an S-Function......Page 394
A.8 Branching Rules......Page 395
A.9 Kronecker Products for Continuous Groups......Page 396
A.10 Outer Plethysm of S-Functions......Page 397
A.11 Inner Plethysm of S-Functions......Page 400
A.12 Machine Calculation of S-Function Properties......Page 401
References......Page 402
Author Index......Page 422
Subject Index......Page 428