Lie groups, physics, and geometry: an introduction

Robert Gilmore0521884004, 9780521884006, 9780511379277

Describing many of the most important aspects of Lie group theory, this book presents the subject in a ‘hands on’ way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 13
1.1 The program of Lie……Page 15
1.2 A result of Galois……Page 16
1.3 Group theory background……Page 17
1.4 Approach to solving polynomial equations……Page 22
1.5 Solution of the quadratic equation……Page 24
1.6 Solution of the cubic equation……Page 25
1.7 Solution of the quartic equation……Page 29
1.8 The quintic cannot be solved……Page 31
1.9 Example……Page 32
1.10 Conclusion……Page 35
1.11 Problems……Page 36
2.1 Algebraic properties……Page 38
2.2 Topological properties……Page 39
2.3 Unification of algebra and topology……Page 41
2.5 Conclusion……Page 43
2.6 Problems……Page 44
3.1 Preliminaries……Page 48
3.2 No constraints……Page 49
3.3 Linear constraints……Page 50
3.4 Bilinear and quadratic constraints……Page 53
3.5 Multilinear constraints……Page 56
3.7 Embedded groups……Page 57
3.8 Modular groups……Page 58
3.9 Conclusion……Page 60
3.10 Problems……Page 61
4.1 Why bother?……Page 69
4.2 How to linearize a Lie group……Page 70
4.3 Inversion of the linearization map: EXP……Page 71
4.4 Properties of a Lie algebra……Page 73
4.5 Structure constants……Page 75
4.6 Regular representation……Page 76
4.7 Structure of a Lie algebra……Page 77
4.8 Inner product……Page 78
4.9 Invariant metric and measure on a Lie group……Page 80
4.11 Problems……Page 83
5.2 No constraints……Page 88
5.3 Linear constraints……Page 89
5.4 Bilinear and quadratic constraints……Page 92
5.6 Intersections of groups……Page 94
5.9 Basis vectors……Page 95
5.11 Problems……Page 97
6.1 Boson operator algebras……Page 102
6.2 Fermion operator algebras……Page 103
6.3 First order differential operator algebras……Page 104
6.5 Problems……Page 107
7.1 Preliminaries……Page 113
7.2 The covering problem……Page 114
7.3 The isomorphism problem and the covering group……Page 119
7.4 The parameterization problem and BCH formulas……Page 122
7.5.1 Dynamics……Page 128
7.5.2 Equilibrium thermodynamics……Page 130
7.6 Conclusion……Page 133
7.7 Problems……Page 134
8.2 Some standard forms for the regular representation……Page 143
8.3 What these forms mean……Page 147
8.4 How to make this decomposition……Page 149
8.6 Conclusion……Page 150
8.7 Problems……Page 151
9.1 Objectives of this program……Page 153
9.2 Eigenoperator decomposition — secular equation……Page 154
9.4 Invariant operators……Page 157
9.5 Regular elements……Page 160
9.6 Semisimple Lie algebras……Page 161
9.6.2 Properties of roots……Page 162
9.6.3 Structure constants……Page 163
9.6.4 Root reflections……Page 164
9.7 Canonical commutation relations……Page 165
9.8 Conclusion……Page 167
9.9 Problems……Page 168
10.1 Properties of roots……Page 173
10.2 Root space diagrams……Page 174
10.3 Dynkin diagrams……Page 179
10.5 Problems……Page 182
11.1 Preliminaries……Page 186
11.2 Compact and least compact real forms……Page 188
11.3 Cartan’s procedure for constructing real forms……Page 190
11.4 Real forms of simple matrix Lie algebras……Page 191
11.4.2 Subfield restriction……Page 192
11.4.3 Field embeddings……Page 194
11.5 Results……Page 195
11.6 Conclusion……Page 196
11.7 Problems……Page 197
12.1 Brief review……Page 203
12.2 Globally symmetric spaces……Page 204
12.3 Rank……Page 205
12.4 Riemannian symmetric spaces……Page 206
12.5 Metric and measure……Page 207
12.6 Applications and examples……Page 208
12.7 Pseudo-Riemannian symmetric spaces……Page 211
12.9 Problems……Page 212
13.1 Preliminaries……Page 219
13.3.1 The contraction SO(3) … ISO(2)……Page 220
13.3.2 The contraction SO(4) … ISO(3)……Page 222
13.3.3 The contraction SO(4,1) … ISO(3,1)……Page 224
13.4.1 Contraction of the algebra……Page 225
13.4.2 Contraction of the Casimir operators……Page 226
13.4.4 Contraction of representations……Page 227
13.4.6 Contraction of matrix elements……Page 228
13.4.8 Contraction of special functions……Page 229
13.5 Conclusion……Page 230
13.6 Problems……Page 231
14.1 Introduction……Page 235
14.2 Two important principles of physics……Page 236
14.3 The wave equations……Page 237
14.4 Quantization conditions……Page 238
14.5 Geometric symmetry SO(3)……Page 241
14.6 Dynamical symmetry SO(4)……Page 244
14.7 Relation with dynamics in four dimensions……Page 247
14.8 DeSitter symmetry SO(4,1)……Page 249
14.9.1 Schwinger representation……Page 252
14.9.2 Dynamical mappings……Page 254
14.9.3 Lie algebra of physical operators……Page 256
14.10 Spin angular momentum……Page 257
14.11 Spectrum generating group……Page 259
14.11.1 Bound states……Page 260
14.11.2 Scattering states……Page 261
14.11.3 Quantum defect……Page 262
14.12 Conclusion……Page 263
14.13 Problems……Page 264
15.1 Introduction……Page 273
15.2.2 Inhomogeneous Lorentz group……Page 275
15.3.1 Translations { I,a }……Page 276
15.3.3 Representations of SO(3,1)……Page 277
15.4.1 Manifestly covariant representations……Page 278
15.4.2 Unitary irreducible representations……Page 280
15.5 Transformation properties……Page 284
15.6 Maxwell’s equations……Page 287
15.8 Problems……Page 289
16 Lie groups and differential equations……Page 298
16.1 The simplest case……Page 299
16.2 First order equations……Page 300
16.2.3 Determining equation……Page 301
16.2.4 New coordinates……Page 302
16.3 An example……Page 304
16.4.2 Higher degree equations……Page 309
16.4.4 Second order equations……Page 310
16.4.5 Reduction of order……Page 312
16.4.7 Partial differential equations: Laplace’s equation……Page 313
16.4.8 Partial differential equations: heat equation……Page 314
16.4.9 Closing remarks……Page 315
16.5 Conclusion……Page 316
16.6 Problems……Page 317
Bibliography……Page 323
Index……Page 327