Group Theory for Physicists

by Ma, Zhong-Qi

• ISBN13: 9789812771421
• ISBN10: 9812771425
• Format: Paperback
• Copyright: 2007-11-28
• Publisher: World Scientific Pub Co Inc

Summary

"This textbook explains the fundamental concepts and techniques of group theory by making use of language familiar to physicists. Application methods to physics are emphasized. New materials drawn from the teaching and research experience of the author are included. This book can be used by graduate students and young researchers in physics, especially theoretical physics. It is also suitable for some graduate students in theoretical chemistry."--BOOK JACKET.

Table of Contents

Preface	p. v
List of Symbols	p. ix
Review on Linear Algebras	p. 1
Linear Space and Basis Vector	p. 1
Linear Transformations and Linear Operators	p. 3
Similarity Transformation	p. 5
Eigenvectors and Diagonalization of a Matrix	p. 8
Inner Product of Vectors	p. 9
The Direct Product of Matrices	p. 12
Exercises	p. 13
Group and its Subsets	p. 17
Symmetry	p. 17
Group and its Multiplication Table	p. 19
Subsets in a Group	p. 27
Subgroup	p. 27
Cosets	p. 28
Conjugate Elements and the Class	p. 29
Invariant Subgroup	p. 31
Homomorphism of Two Groups	p. 33
Proper Symmetric Group of a Regular Polyhedron	p. 35
Tetrahedron, Octahedron, and Cube	p. 36
The Group Table of T	p. 38
The Group Table of O	p. 40
Regular Icosahedron and the Group Table of I	p. 40
Direct Product of Groups and Improper Point Groups	p. 44
The Direct Product of Two Groups	p. 44
Improper Point Groups	p. 44
Exercises	p. 46
Theory of Linear Representations of Groups	p. 49
Linear Representations of a Group	p. 49
Definition of a Linear Representation	p. 49
Group Algebra and the Regular Representation	p. 50
Class Operator and Class Space	p. 53
Transformation Operators for a Scalar Function	p. 54
Equivalent Representations	p. 57
Inequivalent and Irreducible Representations	p. 60
Irreducible Representations	p. 60
Schur Theorem	p. 61
Orthogonal Relation	p. 63
Completeness of Representations	p. 65
Character Tables of Finite Groups	p. 67
The Character Table of the Group T	p. 70
The Character Table of the Group O	p. 71
Self-conjugate Representation	p. 72
Subduced and Induced Representations	p. 73
Applications in Physics	p. 78
Classification of Static Wave Functions	p. 78
Clebsch-Gordan Series and Coefficients	p. 80
Wigner-Eckart Theorem	p. 81
Normal Degeneracy and Accidental Degeneracy	p. 83
An Example of Application	p. 85
Irreducible Bases in Group Algebra	p. 88
Ideal and Idempotent	p. 89
Primitive Idempotent	p. 90
Two-side Ideal	p. 92
Standard Irreducible Basis Vectors	p. 93
Exercises	p. 102
Three-Dimensional Rotation Group	p. 107
Three-dimensional Rotations	p. 107
Fundamental Concept of a Lie Group	p. 111
The Composition Functions of a Lie Group	p. 111
The Local Property of a Lie Group	p. 112
Generators and Differential Operators	p. 113
The Adjoint Representation of a Lie Group	p. 114
The Global Property of a Lie Group	p. 115
The Covering Group of SO(3)	p. 117
The Group SU(2)	p. 117
Homomorphism of SU(2) onto SO(3)	p. 118
The Group Integral	p. 120
Irreducible Representations of SU(2)	p. 124
Euler Angles	p. 124
Linear Representations of SU(2)	p. 127
Spherical Harmonics Functions	p. 131
The Lie Theorems	p. 134
Clebsch-Gordan Coefficients of SU(2)	p. 141
Direct Product of Representations	p. 141
Calculation of Clebsch-Gordan Coefficients	p. 144
Applications	p. 147
Sum of Three Angular Momentums	p. 149
Tensors and Spinors	p. 153
Vector Fields	p. 153
Tensor Fields	p. 155
Spinor Fields	p. 156
Total Angular Momentum Operator	p. 158
Irreducible Tensor Operators and Their Application	p. 160
Irreducible Tensor Operators	p. 160
Wigner-Eckart Theorem	p. 163
Selection Rule and Relative Intensity of Radiation	p. 164
Lande Factor and Zeeman Effects	p. 166
An Isolated Quantum n-body System	p. 169
Separation of the Motion of Center-of-Mass	p. 169
Quantum Two-body System	p. 171
Quantum Three-body System	p. 172
Quantum n-body System	p. 176
Exercises	p. 180
Symmetry of Crystals	p. 185
Symmetric Group of Crystals	p. 185
Crystallographic Point Groups	p. 187
Elements in a Crystallographic Point Group	p. 187
Proper Crystallographic Point Groups	p. 189
Improper Crystallographic Point Group	p. 193
Crystal Systems and Bravais Lattice	p. 195
Restrictions on Vectors of Crystal Lattice	p. 195
Triclinic Crystal System	p. 198
Monoclinic Crystal System	p. 198
Orthorhombic Crystal System	p. 199
Trigonal and Hexagonal Crystal System	p. 200
Tetragonal Crystal System	p. 204
Cubic Crystal System	p. 205
Space Group	p. 208
Symmetric Elements	p. 208
Symbols of a Space Group	p. 211
Method for Determining the Space Groups	p. 213
Example for the Space Groups in Type A	p. 215
Example for the Space Groups in Type B	p. 216
Analysis of the Symmetry of a Crystal	p. 218
Linear Representations of Space Groups	p. 220
Irreducible Representations of T	p. 220
Star of Wave Vectors and Group of Wave Vectors	p. 222
Representation Matrices of Elements in S	p. 224
Irreducible Representations of S(k[subscript 1])	p. 225
The Bloch Theorem	p. 227
Energy Band in a Crystal	p. 228
Exercises	p. 229
Permutation Groups	p. 231
Multiplication of Permutations	p. 231
Permutations	p. 231
Cycles	p. 233
Classes in a Permutation Group	p. 234
Alternating Subgroups	p. 236
Transposition of Two Neighbored Objects	p. 237
Young Patterns, Young Tableaux, and Young Operators	p. 237
Young Patterns	p. 237
Young Tableaux	p. 238
Young Operators	p. 240
Fundamental Property of Young Operators	p. 242
Products of Young Operators	p. 244
Irreducible Representations of S[subscript n]	p. 246
Primitive Idempotents in the Group Algebra of S[subscript n]	p. 246
Orthogonal Primitive Idempotents of S[subscript n]	p. 249
Calculation of Representation Matrices for S[subscript n]	p. 253
Calculation of Characters by Graphic Method	p. 257
The Permutation Group S[subscript 3]	p. 259
Inner Product of Irreducible Representations of S[subscript n]	p. 261
Real Orthogonal Representation of S[subscript n]	p. 262
Outer Product of Irreducible Representations of S[subscript n]	p. 268
Representations of S[subscript n+m] and Its Subgroup S[subscript n multiply sign in circle] S[subscript m]	p. 268
Littlewood-Richardson Rule	p. 271
Exercises	p. 273
Lie Groups and Lie Algebras	p. 277
Lie Algebras and its Structure Constants	p. 277
The Global Property of a Lie Group	p. 277
The Local Property of a Lie Group	p. 278
The Lie Algebra	p. 281
The Killing Form and the Cartan Criteria	p. 283
The Regular Form of a Semisimple Lie Algebra	p. 285
The Inner Product in a Semisimple Lie Algebra	p. 285
The Cartan Subalgebra	p. 286
Regular Commutative Relations of Generators	p. 287
The Inner Product of Roots	p. 289
Positive Roots and Simple Roots	p. 292
Classification of Simple Lie Algebras	p. 295
Angle between Two Simple Roots	p. 295
Dynkin Diagrams	p. 296
The Cartan Matrix	p. 302
Classical Simple Lie Algebras	p. 302
The SU(N) Group and its Lie Algebra	p. 302
The SO(N) Group and its Lie Algebra	p. 307
The USp(2l) Group and its Lie Algebra	p. 309
Representations of a Simple Lie Algebra	p. 313
Representations and Weights	p. 313
Weight Chain and Weyl Reflections	p. 316
Mathematical Property of Representations	p. 319
Fundamental Dominant Weights	p. 320
The Casimir Operator of Order 2	p. 321
Main Data of Simple Lie Algebras	p. 322
Lie Algebra A[subscript l] and Lie Group SU(l + 1)	p. 323
Lie Algebra B[subscript l] and Lie Group SO(2l + 1)	p. 324
Lie Algebra C[subscript l] and Lie Group USp(2l)	p. 325
Lie Algebra D[subscript l] and Lie Group SO(2l)	p. 325
Lie Algebra G[subscript 2]	p. 327
Lie Algebra F[subscript 4]	p. 327
Lie Algebra E[subscript 6]	p. 328
Lie Algebra E[subscript 7]	p. 329
Lie Algebra E[subscript 8]	p. 330
Block Weight Diagrams	p. 331
Chevalley Bases	p. 331
Orthonormal Basis States	p. 333
Method of Block Weight Diagram	p. 335
Some Representations of A[subscript 2]	p. 337
Some Representations of C[subscript 3]	p. 338
Planar Weight Diagrams	p. 342
Clebsch-Gordan Coefficients	p. 343
Representations in the CG Series	p. 344
Method of Dominant Weight Diagram	p. 345
Reductions of Direct Product Representations in A[subscript 2]	p. 347
Exercises	p. 350
Unitary Groups	p. 353
Irreducible Representations of SU(N)	p. 353
Reduction of a Tensor Space	p. 354
Basis Tensors in the Tensor Subspace	p. 356
Chevalley Bases of Generators in SU(N)	p. 362
Inequivalent and Irreducible Representations	p. 363
Dimensions of Representations of SU(N)	p. 364
Subduced Representations with Respect to Subgroups	p. 366
Orthonormal Irreducible Basis Tensors	p. 367
Orthonormal Basis Tensors in T[subscript mu superscript lambda]	p. 368
Orthonormal Basis Tensors in S[subscript n]	p. 373
Direct Product of Tensor Representations	p. 373
Outer Product of Tensors	p. 373
Covariant and Contravariant Tensors	p. 377
Traceless Mixed Tensors	p. 379
Adjoint Representation of SU(N)	p. 382
SU(3) Symmetry and Wave Functions of Hadrons	p. 383
Quantum Numbers of Quarks	p. 384
Planar Weight Diagrams	p. 386
Mass Formulas	p. 391
Wave Functions of Mesons	p. 393
Wave Functions of Baryons	p. 395
Exercises	p. 397
Real Orthogonal Groups	p. 399
Tensor Representations of SO(N)	p. 399
Tensors of SO(N)	p. 399
Irreducible Basis Tensors of SO(2l + 1)	p. 403
Irreducible Basis Tensors of SO(2l)	p. 408
Dimensions of Irreducible Tensor Representations	p. 411
Adjoint Representation of SO(N)	p. 414
Tensor Representations of O(N)	p. 415
[Gamma] Matrix Groups	p. 416
Property of [Gamma] Matrix Groups	p. 417
The Case N = 2l	p. 418
The Case N = 2l + 1	p. 421
Spinor Representations of SO(N)	p. 422
Covering Groups of SO(N)	p. 422
Fundamental Spinors of SO(N)	p. 425
Direct Products of Spinor Representations	p. 426
Spinor Representations of Higher Ranks	p. 427
Dimensions of the Spinor Representations	p. 430
Rotational Symmetry in N-Dimensional Space	p. 432
Orbital Angular Momentum Operators	p. 432
Spherical Harmonic Functions	p. 432
Schrodinger Equation for a Two-body System	p. 434
Schrodinger Equation for a Three-body System	p. 435
Dirac Equation in (N + 1)-dimensional Space-time	p. 438
The SO(4) Group and the Lorentz Group	p. 444
Irreducible Representations of SO(4)	p. 445
Single-valued Representations of O(4)	p. 448
The Lorentz Group	p. 450
Irreducible Representations of L[subscript p]	p. 451
The Covering Group of L[subscript p]	p. 454
Classes of L[subscript p]	p. 456
Irreducible Representations of L[subscript h]	p. 457
Exercises	p. 459
The Symplectic Groups	p. 461
Irreducible Representations of USp(2l)	p. 461
Decomposition of the Tensor Space of USp(2l)	p. 461
Orthonormal Irreducible Basis Tensors	p. 463
Dimensions of Irreducible Representations	p. 468
Physical Application	p. 471
Exercises	p. 472
Identities on Combinatorics	p. 473
Covariant and Contravariant Tensors	p. 475
The Space Groups	p. 477
Bibliography	p. 481
Index	p. 487
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