This volume is intended as a systematic introduction to gauge field theory for advanced undergraduate and graduate students in high energy physics. The discussion is restricted to the classical (non-quantum) theory in Minkowski spacetime. Particular attention has been given to conceptual aspects of field theory, accurate definitions of basic physical notions, and thorough analysis of exact solutions to the equations of motion for interacting systems. Offers an introduction to gauge field theory for advanced undergraduate and graduate students in high energy physics. This title gives attention to conceptual aspects of field theory, accurate definitions of basic physical notions, and thorough analysis of exact solutions to the equations of motion for interacting systems. 1 Geometry of Minkowski Space 1 1.1 Spacetime 1 1.2 Affine and Metric Structures 10 1.3 Vectors, Tensors, and n-Forms 22 1.4 Lines and Surfaces 32 1.5 Poincar nvariance 38 1.6 World Lines 43 Notes 48 2 Relativistic Mechanics 51 2.1 Dynamical Law for Relativistic Particles 52 2.2 The Minkowski Force 58 2.3 Invariants of the Electromagnetic Field 65 2.4 Motion of a Charged Particle in Constant and Uniform Electromagnetic Fields 69 2.5 The Principle of Least Action. Symmetries and Conservation Laws 75 2.6 Reparametrization Invariance 90 2.7 Spinning Particle 98 2.8 Relativistic Kepler Problem 104 2.9 A Charged Particle Driven by a Magnetic Monopole 110 2.10 Collisions and Decays 113 Notes 118 3 Electromagnetic Field 123 3.1 Geometric Contents of Maxwell's Equations 124 3.2 Physical Contents of Maxwell's Equations 127 3.3 Other Forms of Maxwell's Equations 135 Notes 139 4 Solutions to Maxwell's Equations 141 4.1 Statics 141 4.2 Solutions to Maxwell's Equations: Some General Observations 152 4.3 Free Electromagnetic Field 157 4.4 The Retarded Green's Function 167 4.5 Covariant Retarded Variables 174 4.6 Electromagnetic Field Generated by a Single Charge Moving Along an Arbitrary Timelike World Line 179 4.7 Another Way of Looking at Retarded Solutions 183 4.8 Field Due to a Magnetic Monopole 187 Notes 191 5 Lagrangian Formalism in Electrodynamics 195 5.1 Action Principle. Symmetries and Conservation Laws 195 5.2 Poincar nvariance 206 5.3 Conformal Invariance 216 5.4 Duality Invariance 225 5.5 Gauge Invariance 228 5.6 Strings and Branes 235 Notes 245 6 Self-Interaction in Electrodynamics 249 6.1 Rearrangement of Degrees of Freedom 249 6.2 Radiation 258 6.3 Energy-Momentum Balance 265 6.4 The Lorentz Dirac Equation 274 6.5 Alternative Methods of Deriving the Equation of Motion for a Dressed Charged Particle 278 Notes 283 7 Lagrangian Formalism for Gauge Theories 285 7.1 The Yang Mills Wong Theory 285 7.2 The Standard Model 294 7.3 Lattice Formulation of Gauge Theories 298 Notes 305 8 Solutions to the Yang Mills Equations 307 8.1 The Yang Mills Field Generated by a Single Quark 309 8.2 Ansatz 317 8.3 The Yang Mills Field Generated by Two Quarks 320 8.4 The Yang Mills Field Generated by N Quarks 326 8.5 Stability 331 8.6 Vortices and Monopoles 334 8.7 Two Phases of the Subnuclear Realm 343 Notes 348 9 Self-Interaction in Gauge Theories 353 9.1 Rearrangement of the Yang Mills Wong Theory 353 9.2 Self-Consistency 358 9.3 Paradoxes 360 Notes 365 10 Generalizations 367 10.1 Rigid Particle 367 10.2 Different Dimensions 372 10.2.1 Two Dimensions 374 10.2.2 Six Dimensions 376 10.3 Is the Dimension D = 3 Indeed Distinguished? 383 10.4 Nonlinear Electrodynamics 385 10.5 Nonlocal Interactions 393 10.6 Action at a Distance 401 Notes 407 Mathematical Appendices 411 A. Differential Forms 411 B. Lie Groups and Lie Algebras 416 C. The Gamma Matrices and Dirac Spinors 423 D. Conformal Transformations 427 E. Grassmannian Variables 434 F. Distributions 437 Notes 446 References 449 Index 469