CLASSICAL FIELD THEORY - A Quick Guide - Huan Q. Bui Colby College PHYSICS & MATHEMATICS Statistics Class of 2021 February 25, 2020 Preface Greetings, Classical Field Theory, A Quick Guide to is compiled based on my inde- pendent study PH491/2: Topics in Classical Field Theory notes with professor Robert Bluhm. Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity, along with other resources, serves as the main guiding text. This text also references a number of other texts such as Quantum Field The- ory by Ryder, Quantum Field Theory by Mandl and Shaw, Gauge Theories Of The Strong, Weak, and Electromagnetic Interactions by Quigg, A First Book of Quantum Field Theory by A. Lahiri and P. B. Pal, Quantum Field Theory in a Nutshell by Zee. This text is a continuation of General Relativity and Cosmology, A Quick Guide to. Familiarity with classical mechanics, linear algebra, vector calculus, and especially general relativity is expected. There will be a quick review of general relativity where important concepts are revisited and derivations high- lighted, but familiarity with basic notions such as geodesics, Christoffel symbols, the Riemann curvature tensors, etc. is assumed. Note: As a consequence of being developed from a variety of sources, there will be some overlapping among the sections. However, the objective of each section is unique. Enjoy! Contents Preface................................2 1 Introduction to the Lagrangian and the Principle of Least Ac- tion7 1.1 A Classical-Mechanical Example..................7 2 Group Theory: a quick guide in a quick guide9 3 Introduction to Classical Field Theory 11 3.1 Relativistic Notation......................... 11 3.2 Classical Lagrangian Field Theory................. 12 3.3 Quantized Lagrangian Field Theory (primer)........... 13 3.4 Symmetries and Conservation Laws (primer)........... 15 4 Gauge Invariance 21 4.1 Introduction.............................. 21 4.2 Gauge Invariance in Classical Electrodynamics.......... 21 4.3 Phase Invariance in Quantum Mechanics.............. 24 4.4 Significance of Potentials in Quantum Theory........... 26 4.4.1 The Aharonov-Bohm Effect & The Physical Vector Potential 26 4.4.2 Path-dependent Phase Factors............... 28 4.5 Phase Invariance in Field Theory.................. 28 5 Lagrangian Field Theory in Flat Spacetime 31 5.1 Real Scalar Fields.......................... 31 5.2 Complex Scalar Fields and Electromagnetism........... 35 5.2.1 Gauge transformation of the first kind - Global Symmetry 35 5.2.2 Gauge transformation of the second kind - Local Symmetry 36 5.2.3 Motivations in the derivation of the E&M Lagrangian.. 37 5.2.4 A few remarks........................ 41 5.3 Vector Fields and Photons...................... 43 6 Symmetries and Conservation Laws in Field Theory 51 6.1 Hamiltonian formalism (primer)................... 51 3 4 CONTENTS 6.2 Overview of Noether's Theorem: A Consequence of Variational Principle................................ 51 6.3 Noether's Theorem on Symmetries and Conservation Laws.... 56 6.3.1 Space-time translations................... 60 6.3.2 Lorentz transformations................... 61 6.3.3 Internal symmetries..................... 61 7 Spontaneous Symmetry Breaking 63 7.1 Introduction.............................. 63 7.2 Continuous Global Symmetry.................... 69 7.3 An O(2) example........................... 71 7.4 Introduction to the Higgs Mechanism................ 75 8 Gravitation and Lagrangian Formulation of General Relativity 79 8.1 Review of General Relativity & Curved Spacetime........ 79 8.1.1 General Relativity...................... 79 8.1.2 Curved Spacetime...................... 79 8.2 Lagrangian Formulation....................... 80 8.3 Overview............................... 80 8.3.1 Introduction to the Lagrangian Formulation of General Relativity........................... 81 8.3.2 Variation on the metric tensor, without matter: Tµν = 0. 83 8.3.3 Variations on the metric tensor, with matter: Tµν 6= 0.. 88 8.4 Properties of Einstein Equations.................. 91 9 Diffeomorphism, Vierbein Formalism, and general spacetime symmetries 93 9.1 Overview of Diffeomorphisms and the Lie Derivative....... 93 9.2 Spacetime Symmetry......................... 97 9.2.1 Global Lorentz Transformations in Minkowski Spacetime 97 9.2.2 Diffeomorphism in Curved Spacetime........... 100 9.2.3 Local Lorentz Transformation & Vierbein Formalism... 102 9.2.4 Riemann-Cartan spacetime................. 106 10 Linearized Gravity 109 10.1 The metric.............................. 109 10.2 The Christoffel symbols and Einstein tensor............ 110 10.3 Massive Gravity........................... 123 10.3.1 A Brief History........................ 123 10.3.2 Quick review of Field Dimensions.............. 124 10.3.3 Fierz-Pauli Massive Gravity................. 126 10.3.4 Fierz-Pauli Massive Gravity with Source.......... 144 10.3.5 The St¨uckelberg Trick.................... 160 10.3.6 Nonlinear Massive Gravity................. 171 10.3.7 The Nonlinear St¨uckelberg Formalism........... 202 10.3.8 St¨uckelberg Analysis of Interacting Massive Gravity... 211 CONTENTS 5 10.4 The Λ3 theory............................ 215 10.4.1 Tuning interactions to raise the cutoff........... 215 10.4.2 The appearance of Galileons and the absence of ghosts.. 216 10.4.3 The Vainshtein radius.................... 217 10.4.4 Th Vainshtein mechanism in the Λ3 theory........ 218 10.4.5 Quantum corrections in the Λ3 theory........... 219 10.5 xACT Tutorial............................ 220 10.5.1 Importing packages...................... 220 10.5.2 xTensor Basics........................ 222 10.5.3 xPert Basics......................... 228 10.5.4 xTras Basics: Metric Variations............... 236 10.5.5 Undefining Basics: Playtime's Over!............ 241 6 CONTENTS Part 1 Introduction to the Lagrangian and the Principle of Least Action Proposition 1.0.1. All fundamental physics obeys least action principles. The action S is defined as b S = L dt: (1.1) ˆa where L is called the Lagrangian. Refer for Farlow's Partial Differential Equation, page 353, for detailed ex- planation of Lagrange's calculus of variations. I will derive the Euler-Lagrange equation(s) here, but we are not going to use it in the following subsection for the introduction to field theory for now. @L @L − @µ = 0: (1.2) @φ @(@µφ) 1.1 A Classical-Mechanical Example In this subsection we take a look at how the Lagrangian formulation of classical mechanics can give rise to Newton's second law of motion. In mechanics, the Lagrangian often takes the form: L = K − V; (1.3) 7 8PART 1. INTRODUCTION TO THE LAGRANGIAN AND THE PRINCIPLE OF LEAST ACTION where K is the kinetic energy, and V is the potential energy. Let us consider a simple example where 1 K = mx_ 2 (1.4) 2 V = V (x): (1.5) Variations on the Lagrangian gives 1 δL = δ mx_ 2 − V (x) (1.6) 2 dV = mxδ_ x_ − δx (1.7) dx dV = mx_δx_ − δx (1.8) dx d dV = m −xδx¨ + xδx_ − δx (1.9) dt dx d dV = −mxδx¨ − m xδx_ − δx: (1.10) dt dx It follows that the variations on the action gives b b dV δS = δL dt = − mx¨ + δx dt: (1.11) ˆa ˆa dx The principle of least action requires δS = 0 for all δx. Therefore it follows that dV mx¨ + = 0; (1.12) dx which is simply Newton's second law of motion in disguise. Before we move on, we should note that in order for the Lagrangian formu- lation to work in electromagnetism or in general relativity, we need to promote the Lagrangian to its relativistic version where the Lagrangian is given by b L = L d3x: (1.13) ˆa L is called the Lagrangian density, but we can colloquially refer to it as \the Lagrangian." The relativistic action hence takes the form S = L d4x; (1.14) ˆ where d4x implies integrating over all spacetime. Part 2 Group Theory: a quick guide in a quick guide 1. Consider an N-dimensional vector with complex elements. A gauge trans- formation that takes jzj2 ! jzj2 (modulus preserving) is call a U(N) gauge transformation. The letter \U" denotes \unitary." These kinds of trans- formations can be represented by a unitary matrix, which is defined as a matrix whose conjugate transpose is the same as its inverse. If z ! Uz, then jzj2 ! (Uz)y(Uz) = zyU yUz = zyz = jzj2; (2.1) hence modulus preserving. Note that the transformation eiα is unitary. It is simply a 1×1 unitary matrix. This immediately implies that complex scalars have a U(1) gauge invariance. The special group of U(N) transformations is denoted SU(N), which rep- resents those with det U = 1. 2. Consider another N-dimensional real-valued-element vector. We denote the group of orthogonal transformations O(N). Orthogonal transforma- tions are orthogonal, i.e., length-preserving. (Ox)>(Ox) = x>O>Ox = x>x = ~x2 (2.2) Once again, we denote the special group of the O(N) the SO(N) group. This group represents transformations with det O = 1. The Lorentz group is a sub-group of SO(N), as the norm is defined differently, nevertheless it is still length-preserving. We call the Lorentz group SO(3,1), signifying that one sign is different from the other three. 9 10 PART 2. GROUP THEORY: A QUICK GUIDE IN A QUICK GUIDE Part 3 Introduction to Classical Field Theory 3.1 Relativistic Notation Throughout this text, we will use the particle physics' Minkowski spacetime metric tensor: 01 0 0 0 1 B0 −1 0 0 C ηµν = B C : (3.1) @0 0 −1 0 A 0 0 0 −1 We shall assume knowledge of inner products and index-lowering/raising oper- ations (please refer to General Relativity & Cosmology: A Quick Guide to) for more details about this \indexing" business. Quick notes: the four-dimensional generalization of the gradient operator r transforms like a four-vector. If φ(x) is a scalar function, so is @φ δφ = δxµ (3.2) @xµ and so @φ ≡ @ φ ≡ φ, : (3.3) @xµ µ µ Similarly, we also have something for the contravariant four-vector: @φ ≡ @µφ ≡ φ,µ : (3.4) @xµ Lastly, we define the d'Alembertian as: @2 ≡ − @µ@ : (3.5) @t2 µ 11 12 PART 3.