Table of Contents
Preface xi
Acknowledgments xiii
Questions xv
I When Functionals are Extremal
1 Symmetry 3
1.1 Symmetry, Invariances, and Conservation Laws 3
1.2 Meet Emmy Noether 8
2 Functionals 21
2.1 Single-Integral Functional 21
2.2 Formal Definition of a Functional 26
3 Extremals 31
3.1 The Euler-Lagrange Equation 31
3.2 Conservation Laws as Corollaries to the Euler-Lagrange Equation 43
3.3 On the Equivalence of Hamilton's Principle and Newton's Second Law 46
3.4 Where Do Functional Extremal Principles Come From? 49
3.5 Why Kinetic Minus Potential Energy? 53
3.6 Extremals with External Constraints 55
II When Functionals are Invariant
4 Invariance 71
4.1 Formal Definition of Invariance 71
4.2 The Invariance Identity 77
4.3 A More Liberal Definition of Invariance 79
5 Emmy Noether's Elegant (First) Theorem 84
5.1 Invariance + Extremal = Noether's Theorem 84
5.2 Executive Summary of Noether's Theorem 88
5.3 "Extremal" or "Stationary"? 90
5.4 An Inverse Problem: Finding Invariances 94
5.5 Adiabatic Invariance in Noether's Theorem 98
III The Invariance of Fields
6 Noether's Theorem and Fields 111
6.1 Multiple-Integral Functionals 111
6.2 Euler-Lagrange Equations for Fields 115
6.3 Canonical Momentum and the Hamiltonian Tensor for Fields 119
6.4 Equations of Continuity 122
6.5 The Invariance Identity for Fields 124
6.6 Noether's Theorem for Fields 128
6.7 Complex Fields 129
6.8 Global Gauge Transformations 133
7 Local Gauge Transformations of Fields 147
7.1 Local Gauge Invariance and Minimal Coupling 147
7.2 Electrodynamics as a Gauge Theory, Part 1: Field Tensors 153
7.3 Pure Electrodynamics, Spacetime Invariances, and Conservation Laws 159
7.4 Electrodynamics as a Gauge Theory, Part 2: Matter-Field Interactions 163
7.5 Local Gauge Invariance and Noether Currents 168
7.6 Internal Degrees of Freedom 171
7.7 Noether's Theorem and Gauged Internal Symmetries 180
8 Emmy Noether's Elegant (Second) Theorem 194
8.1 Two Noether Theorems 194
8.2 Noether's Second Theorem 199
8.3 Parametric Invariance 205
8.4 Free Fall in a Gravitational Field 211
8.5 The Gravitational Field Equations 218
8.6 The Functionals of General Relativity 226
8.7 Gauge Transformations on Spacetime 229
8.8 Noether's Resolution of an Enigma in General Relativity 231
IV Trans-Noether Invariance
9 Invariance in Phase Space 241
9.1 Phase Space 241
9.2 Hamilton's Principle in Phase Space 243
9.3 Noether's Theorem and Hamilton's Equations 245
9.4 Hamilton-Jacobi Theory 246
10 The Action as a Generator 260
10.1 Conservation of Probability and Continuous Transformations 261
10.2 The Poetry of Nature 265
Appendixes 271
A Scalars, Vectors, and Tensors 273
B Special Relativity 279
C Equations of Motion in Quantum Mechanics 286
D Conjugate Variables and Legendre Transformations 291
E The Jacobian 295
F The Covariant Derivative 299
Bibliography 305
Index 311