Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists Book

Preface     xi
Introduction     1
The program of Lie     1
A result of Galois     2
Group theory background     3
Approach to solving polynomial equations     8
Solution of the quadratic equation     10
Solution of the cubic equation     11
Solution of the quartic equation     15
The quintic cannot be solved     17
Example     18
Conclusion     21
Problems     22
Lie groups     24
Algebraic properties     24
Topological properties     25
Unification of algebra and topology     27
Unexpected simplification     29
Conclusion     29
Problems     30
Matrix groups     34
Preliminaries     34
No constraints     35
Linear constraints     36
Bilinear and quadratic constraints     39
Multilinear constraints     42
Intersections of groups     43
Embedded groups     43
Modular groups     44
Conclusion     46
Problems     47
Lie algebras     55
Why bother?     55
How to linearize a Lie group     56
Inversion of the linearization map: EXP     57
Properties of a Lie algebra     59
Structure constants     61
Regular representation     62
Structure of a Lie algebra     63
Inner product     64
Invariant metric and measure on a Lie group     66
Conclusion     69
Problems     69
Matrix algebras     74
Preliminaries     74
No constraints     74
Linear constraints     75
Bilinear and quadratic constraints     78
Multilinear constraints     80
Intersections of groups     80
Algebras of embedded groups     81
Modular groups     81
Basis vectors     81
Conclusion     83
Problems     83
Operator algebras     88
Boson operator algebras     88
Fermion operator algebras     89
First order differential operator algebras     90
Conclusion     93
Problems     93
EXPonentiation     99
Preliminaries      99
The covering problem     100
The isomorphism problem and the covering group     105
The parameterization problem and BCH formulas     108
EXPonentials and physics     114
Conclusion     119
Problems     120
Structure theory for Lie algebras     129
Regular representation     129
Some standard forms for the regular representation     129
What these forms mean     133
How to make this decomposition     135
An example     136
Conclusion     136
Problems     137
Structure theory for simple Lie algebras     139
Objectives of this program     139
Eigenoperator decomposition - secular equation     140
Rank     143
Invariant operators     143
Regular elements     146
Semisimple Lie algebras     147
Canonical commutation relations     151
Conclusion     153
Problems     154
Root spaces and Dynkin diagrams     159
Properties of roots     159
Root space diagrams     160
Dynkin diagrams     165
Conclusion      168
Problems     168
Real forms     172
Preliminaries     172
Compact and least compact real forms     174
Cartan's procedure for constructing real forms     176
Real forms of simple matrix Lie algebras     177
Results     181
Conclusion     182
Problems     183
Riemannian symmetric spaces     189
Brief review     189
Globally symmetric spaces     190
Rank     191
Riemannian symmetric spaces     192
Metric and measure     193
Applications and examples     194
Pseudo-Riemannian symmetric spaces     197
Conclusion     198
Problems     198
Contraction     205
Preliminaries     205
Inonu-Wigner contractions     206
Simple examples of Inonu-Wigner contractions     206
The contraction U(2) to H[subscript 4]     211
Conclusion     216
Problems     217
Hydrogenic atoms     221
Introduction     221
Two important principles of physics     222
The wave equations      223
Quantization conditions     224
Geometric symmetry SO(3)     227
Dynamical symmetry SO(4)     230
Relation with dynamics in four dimensions     233
DeSitter symmetry SO(4, 1)     235
Conformal symmetry SO(4, 2)     238
Spin angular momentum     243
Spectrum generating group     245
Conclusion     249
Problems     250
Maxwell's equations     259
Introduction     259
Review of the inhomogeneous Lorentz group     261
Subgroups and their representations     262
Representations of the Poincare group     264
Transformation properties     270
Maxwell's equations     273
Conclusion     275
Problems     275
Lie groups and differential equations     284
The simplest case     285
First order equations     286
An example     290
Additional insights     295
Conclusion     302
Problems     303
Bibliography     309
Index     313