Classical Field Theory

Classical field theory predicts how physical fields interact with matter, and is a logical precursor to quantum field theory. This introduction focuses purely on modern classical field theory, helping graduates and researchers build an understanding of classical field theory methods before embarking on future studies in quantum field theory. It describes various classical methods for fields with negligible quantum effects, for instance electromagnetism and gravitational fields. It focuses on solutions that take advantage of classical field theory methods as opposed to applications or geometric properties. Other fields covered includes fermionic fields, scalar fields and Chern–Simons fields. Methods such as symmetries, global and local methods, Noether theorem and energy momentum tensor are also discussed, as well as important solutions of the classical equations, in particular soliton solutions. Cover Front Matter Classical Field Theory Copyright Dedication Contents Preface Acknowledgments Introduction Part I: GENERAL PROPERTIES OF FIELDS; SCALARS AND GAUGE FIELDS 1 Short Review of ClassicalMechanics 2 Symmetries, Groups, and Lie algebras; Representations 3 Examples: The Rotation Group and SU(2) 4 Review of Special Relativity: Lorentz Tensors 5 Lagrangeans and the Notion of Field; Electromagnetism as a Field Theory 6 Scalar Field Theory, Origins, and Applications 7 Nonrelativistic Examples:WaterWaves and Surface Growth 8 Classical Integrability: Continuum Limit of Discrete, Lattice, and Spin Systems 9 Poisson Brackets for Field Theory and Equations of Motion: Applications 10 Classical Perturbation Theory and Formal Solutions to the Equations of Motion 11 Representations of the Lorentz Group 12 Statistics, Symmetry, and the Spin-Statistics Theorem 13 Electromagnetism and the Maxwell Equation; Abelian Vector Fields; Proca Field 14 The Energy-Momentum Tensor 15 Motion of Charged Particles and Electromagnetic Waves; Maxwell Duality 16 The Hopfion Solution and the Hopf Map 17 Complex Scalar Field and Electric Current: Gauging a Global Symmetry 18 The Noether Theorem and Applications 19 Nonrelativistic and Relativistic Fluid Dynamics: Fluid Vortices and Knots Part II: SOLITONS AND TOPOLOGY; NON-ABELIAN THEORY 20 Kink Solutions in φ4 and Sine-Gordon, Domain Walls and Topology 21 The Skyrmion Scalar Field Solution and Topology 22 Field Theory Solitons for CondensedMatter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition 23 Radiation of a Classical Scalar Field: The Heisenberg Model 24 Derrick’s Theorem, Bogomolnyi Bound, the Abelian-Higgs System, and Symmetry Breaking 25 The Nielsen-Olesen Vortex, Topology and Applications 26 Non-Abelian Gauge Theory and the Yang–Mills Equation 27 The Dirac Monopole and Dirac Quantization 28 The ’t Hooft–Polyakov Monopole Solution and Topology 29 The BPST-’t Hooft Instanton Solution and Topology 30 General Topology and Reduction on an Ansatz 31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons 32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates Part III: OTHER SPINS OR STATISTICS; GENERAL RELATIVITY 33 Chern–Simons Terms: Emergent Gauge Fields, the Quantum Hall Effect (Integer and Fractional), Anyonic statistics 34 Chern–Simons and Self-Duality in Odd Dimensions, Its Duality to TopologicallyMassive Theory and Dualities in General 35 Particle–Vortex Duality in Three Dimensions, Particle–String Duality in Four Dimensions, and p-Form Fields in Four Dimensions 36 Fermions and Dirac Spinors 37 The Dirac Equation and Its Solutions 38 General Relativity: Metric and General Coordinate Invariance 39 The Einstein Action and the Einstein Equation 40 Perturbative Gravity: Fierz-Pauli Action, de Donder Gauge and Other Gauges, GravitationalWaves 41 Nonperturbative Gravity: The Vacuum Schwarzschild Solution 42 Deflection of Light by the Sun and Comparison with General Relativity 43 Fully Linear Gravity: Parallel Plane (pp)Waves and Gravitational ShockWave Solutions 44 Fully Linear Gravity: Parallel Plane (pp)Waves and Gravitational ShockWave Solutions 45 Time-Dependent Gravity: The Friedmann-Lemaître-Robertson-Walker (FLRW) Cosmological Solution 46 Vielbein-Spin Connection Formulation of General Relativity and Gravitational Instantons References Index An introduction to modern classical field theory, describing classical methods for fields with negligible quantum effects. This book focuses on solutions that take advantage of classical field theory methods as opposed to applications, helping students and researchers understand classical methods before embarking on studies in quantum field theory.