Introduction to the Classical Theory of Particles and Fields (eBook)

Preface 5
Contents 10
1 Geometry of Minkowski Space 13
1.1 Spacetime 13
1.2 Affine and Metric Structures 22
1.3 Vectors, Tensors, and n-Forms 34
1.4 Lines and Surfaces 44
1.5 Poincare Invariance 50
1.6 World Lines 55
Notes 60
2 Relativistic Mechanics 63
2.1 Dynamical Law for Relativistic Particles 64
2.2 The Minkowski Force 70
2.3 Invariants of the Electromagnetic Field 77
2.4 Motion of a Charged Particle in Constant and Uniform Electromagnetic Fields 81
2.5 The Principle of Least Action. Symmetries and Conservation Laws 87
2.6 Reparametrization Invariance 102
2.7 Spinning Particle 110
2.8 Relativistic Kepler Problem 116
2.9 A Charged Particle Driven by a Magnetic Monopole 122
2.10 Collisions and Decays 125
Notes 130
3 Electromagnetic Field 134
3.1 Geometric Contents of Maxwell’s Equations 135
3.2 Physical Contents of Maxwell’s Equations 138
3.3 Other Forms of Maxwell’s Equations 146
Notes 150
4 Solutions to Maxwell’s Equations 152
4.1 Statics 152
4.2 Solutions to Maxwell’s Equations: Some General Observations 163
4.3 Free Electromagnetic Field 168
4.4 The Retarded Green’s Function 178
4.5 Covariant Retarded Variables 185
4.6 Electromagnetic Field Generated by a Single Charge Moving Along an Arbitrary Timelike World Line 190
4.7 Another Way of Looking at Retarded Solutions 194
4.8 Field Due to a Magnetic Monopole 198
Notes 202
5 Lagrangian Formalism in Electrodynamics 205
5.1 Action Principle. Symmetries and Conservation Laws 205
5.2 Poincare Invariance 216
5.3 Conformal Invariance 226
5.4 Duality Invariance 235
5.5 Gauge Invariance 238
5.6 Strings and Branes 245
Notes 255
6 Self-Interaction in Electrodynamics 259
6.1 Rearrangement of Degrees of Freedom 259
6.2 Radiation 268
6.3 Energy-Momentum Balance 275
6.4 The Lorentz–Dirac Equation 284
6.5 Alternative Methods of Deriving the Equation of Motion for a Dressed Charged Particle 288
Notes 293
7 Lagrangian Formalism for Gauge Theories 295
7.1 The Yang–Mills–Wong Theory 295
7.2 The Standard Model 304
7.3 Lattice Formulation of Gauge Theories 308
Notes 315
8 Solutions to the Yang–Mills Equations 317
8.1 The Yang–Mills Field Generated by a Single Quark 319
8.2 Ansatz 327
8.3 The Yang–Mills Field Generated by Two Quarks 330
8.4 The Yang–Mills Field Generated by N Quarks 336
8.5 Stability 341
8.6 Vortices and Monopoles 344
8.7 Two Phases of the Subnuclear Realm 353
Notes 358
9 Self-Interaction in Gauge Theories 362
9.1 Rearrangement of the Yang–Mills–Wong Theory 362
9.2 Self-Consistency 367
9.3 Paradoxes 369
Notes 374
10 Generalizations 376
10.1 Rigid Particle 376
10.2 Different Dimensions 381
10.3 Is the Dimension D = 3 Indeed Distinguished? 392
10.4 Nonlinear Electrodynamics 394
10.5 Nonlocal Interactions 402
10.6 Action at a Distance 410
Notes 416
Mathematical Appendices 419
A. Differential Forms 419
B. Lie Groups and Lie Algebras 424
C. The Gamma Matrices and Dirac Spinors 431
D. Conformal Transformations 435
E. Grassmannian Variables 442
F. Distributions 445
Notes 454
References 456
Index 476