Gauge Theory and Variational Principles

By: DAVID BLEECKER

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Paperback | 10 December 2005

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181 Pages

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20.9 x 13.6 x 1.27

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This text provides a framework for describing and organizing the basic forces of nature and the interactions of subatomic particles. A detailed and self-contained mathematical account of gauge theory, it is geared toward beginning graduate students and advanced undergraduates in mathematics and physics. This well-organized treatment supplements its rigor with intuitive ideas.
Starting with an examination of principal fiber bundles and connections, the text explores curvature; particle fields, Lagrangians, and gauge invariance; Lagrange's equation for particle fields; and the inhomogeneous field equation. Additional topics include free Dirac electron fields; interactions; calculus on frame bundle; and unification of gauge fields and gravitation. The text concludes with references, a selected bibliography, an index of notation, and a general index.

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Non-Fiction Science Physics Classical Mathematics Wave Mechanics including Vibration & Acoustics Mathematics Geometry Differential & Riemannian Geometry Science in General Quantum Physics & Quantum Mechanics & Quantum Field Theory

Preface	p. ix
Preliminaries	p. 1
Multilinear Algebra and Forms	p. 2
Manifolds and Tensor Analysis	p. 6
Lie Groups and Lie Algebras	p. 18
Principal Fiber Bundles and Connections	p. 23
Principal Fiber Bundles	p. 26
Connections	p. 29
Curvature and g-Valued Differential Forms	p. 34
Graded Lie Algebra of g-Valued Forms	p. 35
Curvature	p. 37
Particle Fields, Lagrangians, Gauge Invariance	p. 42
Particle Fields	p. 43
Gauge Transformations	p. 46
Lagrangians and Gauge Invariance	p. 50
Lagrange's Equation for Particle Fields	p. 54
The Principle of Least Action	p. 55
Some Machinery	p. 56
Lagrange's Equation	p. 60
The Inhomogeneous Field Equation	p. 64
The Current	p. 65
Inhomogeneous Field Equation	p. 68
Free Dirac Electron Fields	p. 71
Covering the Lorentz Group	p. 73
The Levi-Cevita Connection	p. 77
Spin Structures and the Lagrangian	p. 81
Dirac's Equation	p. 84
Interactions	p. 89
Bundle Splicing	p. 90
The (Nonfree) Dirac Electron Field	p. 95
The Nucleon in a Yang-Mills Potential	p. 99
Calculus on Frame Bundle	p. 106
Tensor Fields on L(M)	p. 107
Pseudo-Riemannian Geometry	p. 109
Metric Variations	p. 116
Unification of Gauge Fields and Gravitation	p. 120
Gradients of Metric-Dependent Functionals	p. 122
Conservation Laws from External Symmetry	p. 127
The Einstein-Yang-Mills Action Principle	p. 133
Additional Topics	p. 141
Geodesics and Forces on Classical Particles	p. 141
Utiyama's Theorem	p. 147
Spontaneous Symmetry Breaking	p. 154
Characteristic Classes, Monopoles, and Instantons	p. 161
References	p. 171
Selected Bibliography	p. 173
Index of Notation	p. 175
Index	p. 177
Errata	p. 181
Table of Contents provided by Ingram. All Rights Reserved.

Gauge Theory and Variational Principles

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