Lie groups, physics, and geometry : an introduction for physicists, engineers and chemists / Robert Gilmore.

Format:

Book

Author/Creator:

Gilmore, Robert, 1941-

Contributor:

Hazel M. Hussong Fund.

Language:

English

Subjects (All):

Group theory.

Lie groups.

Physical Description:

xi, 319 pages : illustrations ; 26 cm

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2008.

Summary:

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 relationship of Lie groups to many branches of mathematics and physics and illustrates these with concrete computations. Many examples of Lie groups and Lie algebras are given throughout the text, with applications of the material to physical sciences and applied mathematics. 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.

Contents:

1.1 The program of Lie 1

1.2 A result of Galois 2

1.3 Group theory background 3

1.4 Approach to solving polynomial equations 8

1.5 Solution of the quadratic equation 10

1.6 Solution of the cubic equation 11

1.7 Solution of the quartic equation 15

1.8 The quintic cannot be solved 17

2 Lie groups 24

2.1 Algebraic properties 24

2.2 Topological properties 25

2.3 Unification of algebra and topology 27

2.4 Unexpected simplification 29

3 Matrix groups 34

3.2 No constraints 35

3.3 Linear constraints 36

3.4 Bilinear and quadratic constraints 39

3.5 Multilinear constraints 42

3.6 Intersections of groups 43

3.7 Embedded groups 43

3.8 Modular groups 44

4 Lie algebras 55

4.1 Why bother? 55

4.2 How to linearize a Lie group 56

4.3 Inversion of the linearization map: EXP 57

4.4 Properties of a Lie algebra 59

4.5 Structure constants 61

4.6 Regular representation 62

4.7 Structure of a Lie algebra 63

4.8 Inner product 64

4.9 Invariant metric and measure on a Lie group 66

5 Matrix algebras 74

5.2 No constraints 74

5.3 Linear constraints 75

5.4 Bilinear and quadratic constraints 78

5.5 Multilinear constraints 80

5.6 Intersections of groups 80

5.7 Algebras of embedded groups 81

5.8 Modular groups 81

5.9 Basis vectors 81

6 Operator algebras 88

6.1 Boson operator algebras 88

6.2 Fermion operator algebras 89

6.3 First order differential operator algebras 90

7 EXPonentiation 99

7.2 The covering problem 100

7.3 The isomorphism problem and the covering group 105

7.4 The parameterization problem and BCH formulas 108

7.5 EXPonentials and physics 114

8 Structure theory for Lie algebras 129

8.1 Regular representation 129

8.2 Some standard forms for the regular representation 129

8.3 What these forms mean 133

8.4 How to make this decomposition 135

9 Structure theory for simple Lie algebras 139

9.1 Objectives of this program 139

9.2 Eigenoperator decomposition - secular equation 140

9.3 Rank 143

9.4 Invariant operators 143

9.5 Regular elements 146

9.6 Semisimple Lie algebras 147

9.7 Canonical commutation relations 151

10 Root spaces and Dynkin diagrams 159

10.1 Properties of roots 159

10.2 Root space diagrams 160

10.3 Dynkin diagrams 165

11 Real forms 172

11.2 Compact and least compact real forms 174

11.3 Cartan's procedure for constructing real forms 176

11.4 Real forms of simple matrix Lie algebras 177

11.5 Results 181

12 Riemannian symmetric spaces 189

12.2 Globally symmetric spaces 190

12.3 Rank 191

12.4 Riemannian symmetric spaces 192

12.5 Metric and measure 193

12.6 Applications and examples 194

12.7 Pseudo-Riemannian symmetric spaces 197

13 Contraction 205

13.2 Inonu-Wigner contractions 206

13.3 Simple examples of Inonu-Wigner contractions 206

13.4 The contraction U(2) to H[subscript 4] 211

14 Hydrogenic atoms 221

14.2 Two important principles of physics 222

14.3 The wave equations 223

14.4 Quantization conditions 224

14.5 Geometric symmetry SO(3) 227

14.6 Dynamical symmetry SO(4) 230

14.7 Relation with dynamics in four dimensions 233

14.8 DeSitter symmetry SO(4, 1) 235

14.9 Conformal symmetry SO(4, 2) 238

14.10 Spin angular momentum 243

14.11 Spectrum generating group 245

15 Maxwell's equations 259

15.2 Review of the inhomogeneous Lorentz group 261

15.3 Subgroups and their representations 262

15.4 Representations of the Poincare group 264

15.5 Transformation properties 270

15.6 Maxwell's equations 273

16 Lie groups and differential equations 284

16.1 The simplest case 285

16.2 First order equations 286

16.4 Additional insights 295.

Notes:

Includes bibliographical references (pages 309-312) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

9780521884006

0521884004

OCLC:

175284084