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Introduction to Classical Field Theory Utah State University DigitalCommons@USU All Complete Monographs 12-16-2020 Introduction to Classical Field Theory Charles G. Torre Department of Physics, Utah State University, [email protected] Follow this and additional works at: https://digitalcommons.usu.edu/lib_mono Part of the Applied Mathematics Commons, Cosmology, Relativity, and Gravity Commons, Elementary Particles and Fields and String Theory Commons, and the Geometry and Topology Commons Recommended Citation Torre, Charles G., "Introduction to Classical Field Theory" (2020). All Complete Monographs. 3. https://digitalcommons.usu.edu/lib_mono/3 This Book is brought to you for free and open access by DigitalCommons@USU. It has been accepted for inclusion in All Complete Monographs by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Introduction to Classical Field Theory C. G. Torre Department of Physics Utah State University Version 1.3 December 2020 2 About this text This is a quick and informal introduction to the basic ideas and mathematical methods of classical relativistic field theory. Scalar fields, spinor fields, gauge fields, and gravitational fields are treated. The material is based upon lecture notes for a course I teach from time to time at Utah State University on Classical Field Theory. The following is version 1.3 of the text. It is roughly the same as version 1.2. The update to 1.3 includes: • numerous small improvements in exposition; • fixes for a number of typographical errors and various bugs; • a few new homework problems. I am grateful to the students who participated in the 2020 Pandemic Special Edition of the course for helping me to improve the text. The students were: Lyle Arnett, Eli Atkin, Alex Chanson, Guillermo Frausto, Tyler Hansen, Kevin Rhine, and Ben Shaw. c 2016, 2020 C. G. Torre 3 4 Contents Preface 9 1 What is a classical field theory? 13 1.1 Example: waves in an elastic medium . 13 1.2 Example: Newtonian gravitational field . 14 1.3 Example: Maxwell's equations . 14 2 Klein-Gordon field 17 2.1 The Klein-Gordon equation . 17 2.2 Solving the KG equation . 18 2.3 A small digression: one particle wave functions . 20 2.4 Variational principle . 21 2.5 Getting used to δ. Functional derivatives. 26 2.6 The Lagrangian . 27 2.7 The Euler-Lagrange equations . 28 2.8 Jet Space . 31 2.9 Miscellaneous generalizations . 33 2.9.1 External \sources" . 33 2.9.2 Self-interacting field . 34 2.9.3 KG in arbitrary coordinates . 35 2.9.4 KG on any spacetime . 39 2.10 PROBLEMS . 41 3 Symmetries and conservation laws 43 3.1 Conserved currents . 43 3.2 Conservation of energy . 45 3.3 Conservation of momentum . 47 3.4 Energy-momentum tensor . 48 5 6 CONTENTS 3.5 Conservation of angular momentum . 50 3.6 Variational symmetries . 51 3.7 Infinitesimal symmetries . 53 3.8 Divergence symmetries . 55 3.9 A first look at Noether's theorem . 57 3.10 Time translation symmetry and conservation of energy . 58 3.11 Space translation symmetry and conservation of momentum . 59 3.12 Conservation of energy-momentum revisited . 61 3.13 Angular momentum revisited . 62 3.14 Spacetime symmetries in general . 64 3.15 Internal symmetries . 65 3.16 The charged KG field and its internal symmetry . 66 3.17 More generally. 69 3.18 SU(2) symmetry . 70 3.19 A general version of Noether's theorem . 74 3.20 \Trivial" conservation laws . 75 3.21 Conservation laws in terms of differential forms . 78 3.22 PROBLEMS . 80 4 The Hamiltonian formulation 83 4.1 Review of the Hamiltonian formulation of mechanics . 83 4.2 Hamiltonian formulation of the scalar field . 89 4.3 Poisson brackets . 94 4.4 Symmetries and conservation laws . 95 4.5 PROBLEMS . 98 5 Electromagnetic field theory 101 5.1 Review of Maxwell's equations . 101 5.2 Electromagnetic Lagrangian . 103 5.3 Gauge symmetry . 107 5.4 Noether's second theorem in electromagnetic theory . 110 5.5 Noether's second theorem . 112 5.6 The canonical energy-momentum tensor . 115 5.7 Improved Maxwell energy-momentum tensor . 116 5.8 The Hamiltonian formulation of Electromagnetism. 118 5.8.1 Phase space . 119 5.8.2 Equations of motion . 121 CONTENTS 7 5.8.3 The electromagnetic Hamiltonian and gauge transformations . 122 5.9 PROBLEMS . 125 6 Scalar Electrodynamics 129 6.1 Electromagnetic field with scalar sources . 129 6.2 Minimal coupling: the gauge covariant derivative . 131 6.3 Global and Local Symmetries . 133 6.4 A lower-degree conservation law . 138 6.5 Scalar electrodynamics and fiber bundles . 142 6.6 PROBLEMS . 146 7 Spontaneous symmetry breaking 147 7.1 Symmetry of laws versus symmetry of states . 147 7.2 The \Mexican hat" potential . 150 7.3 Dynamics near equilibrium and Goldstone's theorem . 154 7.4 The Abelian Higgs model . 158 7.5 PROBLEMS . 161 8 The Dirac field 163 8.1 The Dirac equation . 163 8.2 Representations of the Poincar´egroup . 166 8.3 The spinor representation . 169 8.4 Dirac Lagrangian . 174 8.5 Poincar´esymmetry . 175 8.6 Energy . 176 8.7 Anti-commuting fields . 178 8.8 Coupling to the electromagnetic field . 180 8.9 PROBLEMS . 182 9 Non-Abelian gauge theory 185 9.1 SU(2) doublet of KG fields, revisited . 186 9.2 Local SU(2) symmetry . 189 9.3 Infinitesimal gauge transformations . 192 9.4 Geometrical interpretation: parallel propagation . 194 9.5 Geometrical interpretation: curvature . 196 9.6 Lagrangian for the YM field . 199 9.7 The source-free Yang-Mills equations . 200 8 CONTENTS 9.8 Yang-Mills with sources . 202 9.9 Noether theorems . 203 9.10 Non-Abelian gauge theory in general . 205 9.11 Chern-Simons Theory . 207 9.12 PROBLEMS . 213 10 Gravitational field theory 217 10.1 Spacetime geometry . 217 10.2 The Geodesic Hypothesis . 221 10.3 The Principle of General Covariance . 223 10.4 The Einstein-Hilbert Action . 226 10.5 Vacuum spacetimes . 229 10.6 Diffeomorphism symmetry and the contracted Bianchi identity 230 10.7 Coupling to matter - scalar fields . 233 10.8 The contracted Bianchi identity revisited . 236 10.9 PROBLEMS . 239 11 Goodbye 241 11.1 Suggestions for Further Reading . 241 Preface This document was created to support a course in classical field theory which gets taught from time to time here at Utah State University. In this course, hopefully, you acquire information and skills that can be used in a variety of places in theoretical physics, principally in quantum field theory, particle physics, electromagnetic theory, fluid mechanics and general relativity. As you may know, it is usually in courses on such subjects that the techniques, tools, and results we shall develop here are introduced { if only in bits and pieces as needed. As explained below, it seems better to give a unified, systematic development of the tools of classical field theory in one place. If you want to be a theoretical/mathematical physicist you must see this stuff at least once. If you are just interested in getting a deeper look at some fundamental/foundational physics and applied mathematics ideas, this is a good place to do it. The traditional physics curriculum supports a number of classical1 field theories. In particular, there is (i) the \Newtonian theory of gravity", based upon the Poisson equation for the gravitational potential and Newton's laws, and (ii) electromagnetic theory, based upon Maxwell's equations and the Lorentz force law. Both of these field theories appear in introductory physics courses as well as in upper level courses. Einstein provided us with another important classical field theory { a relativistic gravitational theory { via his general theory of relativity. This subject takes some investment in geo- metrical technology to adequately explain. It does, however, also typically get a course of its own. These courses (Newtonian gravity, electrodynam- ics, general relativity) are traditionally used to cover a lot of the concepts 1Here, and in all that follows, the term \classical" is to mean \not quantum", e.g., as in \the classical limit". Sometimes people use \classical" to also mean non-relativistic; we shall definitely not being doing that here. Indeed, every field theory we shall consider is \relativistic". 9 10 Preface and methodology of classical field theory. The other field theories that are important (e.g., Dirac, Yang-Mills, Klein-Gordon) typically arise, physically speaking, not as classical field theories but as quantum field theories, and it is usually in a course in quantum field theory that these other field theories are described. So, in a typical physics curriculum, it is through such courses that a student normally gets exposed to the tools and results of classical field theory. This book reflects an alternative approach to learning classical field theory, which I will now try to justify. The traditional organization of material just described, while natural in some ways, overlooks the fact that field theories have many fundamental fea- tures in common { features which are most easily understood in the classical limit { and one can get a really good feel for what is going on in \the big pic- ture" by exploring these features in a general, systematic way. Indeed, many of the basic structures appearing in classical field theory (Lagrangians, field equations, symmetries, conservation laws, gauge transformations, etc.) are of interest in their own right, and one should try to master them in general, not just in the context of a particular example (e.g., electromagnetism), and not just in passing as one studies the quantum field. Moreover, once one has mastered a sufficiently large set of such field theoretic tools, one is in a good position to discern how this or that field theory differs from its cousins in some key structural way.
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