Lie Groups, Physics, and Geometry

An Introduction for Physicists, Engineers and Chemists

By: Robert Gilmore

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Hardcover | 17 January 2008

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332 Pages

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24.41 x 16.99 x 1.91

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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.

Industry Reviews

"Gilmore (physics, Drexel U.) takes an applications-base approach to Lie group theory as it relates to certain branches of applied mathematics and the physical sciences, basically distilling what he considers the most useful material of his much longer book (New York: Wiley, 1974). He begins with a discussion of Lie group theory's intellectual underpinnings in Galois theory and concludes with a chapter on the application of Lie group theory to solving differential equations, both subjects that are relatively rare in texts on Lie group theory. In between he offers chapters on matrix groups, Lie algebras, matrix algebras, operator algebras, EXPonentiation, structure theory for Lie algebras, root spaces and Dynkin diagrams, real forms, Riemannian symmetric spaces, contraction, hydrogenic atoms, and Maxwell's equations." Book News Inc.

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

Lie Groups, Physics, and Geometry

An Introduction for Physicists, Engineers and Chemists

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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