Group theory has long been an important computational tool for physicists, but, with the advent of the Standard Model, it has become a powerful conceptual tool as well. This book introduces physicists to many of the fascinating mathematical aspects of group theory, and mathematicians to its physics applications. Designed for advanced undergraduate and graduate students, this book gives a comprehensive overview of the main aspects of both finite and continuous group theory, with an emphasis on applications to fundamental physics. Finite groups are extensively discussed, highlighting their irreducible representations and invariants. Lie algebras, and to a lesser extent Kac-Moody algebras, are treated in detail, including Dynkin diagrams. Special emphasis is given to their representations and embeddings. The group theory underlying the Standard Model is discussed, along with its importance in model building. Applications of group theory to the classification of elementary particles are treated in detail.
Author(s): Pierre Ramond
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 322
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
1 Preface: the pursuit of symmetries......Page 13
2 Finite groups: an introduction......Page 16
2.1 Group axioms......Page 17
2.2 Finite groups of low order......Page 18
2.3 Permutations......Page 31
2.4.1 Conjugation......Page 34
2.4.2 Simple groups......Page 37
2.4.3 Sylow’s criteria......Page 39
2.4.4 Semi-direct product......Page 40
2.4.5 Young Tableaux......Page 43
3.1 Introduction......Page 45
3.2 Schur's lemmas......Page 47
3.3 The A4 character table......Page 53
3.4 Kronecker products......Page 56
3.5 Real and complex representations......Page 58
3.6 Embeddings......Page 60
3.7 Zn character table......Page 64
3.8 Dn character table......Page 65
3.9 Q2n character table......Page 68
3.10 Some semi-direct products......Page 70
3.11 Induced representations......Page 73
3.12 Invariants......Page 76
3.13 Coverings......Page 79
4.1 Finite Hilbert spaces......Page 81
4.2 Fermi oscillators......Page 82
4.3 Infinite Hilbert spaces......Page 84
5.1 Introduction......Page 90
5.2 Some representations......Page 94
5.3 From Lie algebras to Lie groups......Page 98
5.4 SU (2) right arrow SU (1,1)......Page 101
5.5.1 The isotropic harmonic oscillator......Page 105
5.5.2 The Bohr atom......Page 107
5.5.3 Isotopic spin......Page 111
6.1 SU (3) algebra......Page 114
6.2 Alpha-Basis......Page 118
6.3 Omega-Basis......Page 119
6.4 Alpha'-Basis......Page 120
6.5 The triplet representation......Page 122
6.6 The Chevalley basis......Page 124
6.7.1 The isotropic harmonic oscillator redux......Page 126
6.7.2 The Elliott model......Page 127
6.7.3 The Sakata model......Page 129
6.7.4 The Eightfold Way......Page 130
7 Classification of compact simple Lie algebras......Page 135
7.1 Classification......Page 136
7.2 Simple roots......Page 141
7.3 Rank-two algebras......Page 143
7.4 Dynkin diagrams......Page 146
7.5 Orthonormal bases......Page 152
8.1 Representation basics......Page 155
8.2 A3 fundamentals......Page 156
8.3 The Weyl group......Page 161
8.4 Orthogonal Lie algebras......Page 163
8.5 Spinor representations......Page 165
8.5.1 SO (2n) spinors......Page 166
8.5.2 SO (2n + 1) spinors......Page 168
8.5.3 Clifford algebra construction......Page 171
8.6 Casimir invariants and Dynkin indices......Page 176
8.7 Embeddings......Page 180
8.8 Oscillator representations......Page 190
8.9 Verma modules......Page 192
8.9.1 Weyl dimension formula......Page 199
8.9.2 Verma basis......Page 200
9 Finite groups: the road to simplicity......Page 202
9.1 Matrices over Galois fields......Page 204
9.1.1 PSL2(7)......Page 209
9.1.2 A doubly transitive group......Page 210
9.2 Chevalley groups......Page 213
9.3 A fleeting glimpse at the sporadic groups......Page 217
10.1 Serre presentation......Page 220
10.2 Affine Kac–Moody algebras......Page 222
10.3 Super algebras......Page 228
11 The groups of the Standard Model......Page 233
11.1 Space-time symmetries......Page 234
11.1.1 The Lorentz and Poincaré groups......Page 235
11.1.2 The conformal group......Page 243
11.2 Beyond space-time symmetries......Page 247
11.2.1 Color and the quark model......Page 251
11.3 Invariant Lagrangians......Page 252
11.4 Non-Abelian gauge theories......Page 255
11.5 The Standard Model......Page 256
11.6 Grand Unification......Page 258
11.7.1 Finite SU (2) and SO (3) subgroups......Page 261
11.7.2 Finite SU (3) subgroups......Page 264
12.1 Hurwitz algebras......Page 266
12.2 Matrices over Hurwitz algebras......Page 269
12.3 The Magic Square......Page 271
Appendix 1 Properties of some finite groups......Page 277
Appendix 2 Properties of selected Lie algebras......Page 289
References......Page 319
Index......Page 320
