Quantum Field Theory Damon Binder 19 June 2017 1 Books and Other Resources Many books exist on quantum field theory and its applications. Here I mention just the ones that are most helpful as an introduction to the field, and which will best supplement these lecture notes. Peskin & Schroeder, An Introduction to Quantum Field Theory This is the gold standard for learning quantum field theory. In spite of its name, it contains enough material for three semesters worth of courses, and covers many advanced topics. It is well written and computationally focused. Unfortunately it introduces the path-integral rather late, instead focussing on canonical quantization. Zee, Quantum Field Theory in a Nutshell A fun introduction to the subject, with emphasis on physical insight and conceptual un- derstanding rather than on computational mastery. The author does not let the truth get in the way of a good story. Uses the path-integral throughout. Srednicki, Quantum Field Theory Another path-integral oriented book, this text aims to get to loop diagrams and the mathematical structure of QFT quickly, with little emphasis on physical applications until much later in the book. Schwartz, Quantum Field Theory and the Standard Model A recent text using the canonical approach and focussed on applications of quantum field theory to particle physics. Online Resources A lot of online resources exist on quantum field theory. Three good course notes for other introductory QFT courses are: • David Tong, http://www.damtp.cam.ac.uk/user/tong/qft.html. • Michael Luke, https://www.physics.utoronto.ca/~luke/PHY2403F/References.html. • Sidney Coleman, https://arxiv.org/abs/1110.5013. Other online resources worth checking out: • Quantum Condensed Matter Field Theory, by Ben Simons http://www.tcm.phy.cam.ac.uk/~bds10/tp3.html. • Fields, by Warren Siegal http://insti.physics.sunysb.edu/~siegel/errata.shtml. 2 Acknowledgements Most of my quantum field theory knowledge came from the textbooks and course notes listed on the previous page. These notes are heavily indebted the those resources. My main job has been to steal the good explanations from each text, and then synthesize into a coherent form. I wrote these notes for C´edricSimenel and Joseph Hope, to form the basis of their quantum field theory course. I would like to thank them for their help with this project, and also for teaching me much of what I now know about physics. Contents 1 Introduction 6 1.1 Why QFT? . .6 1.2 What is QFT good for? . .7 1.3 Notation and Conventions . .7 1.3.1 Natural Units . .7 1.3.2 Other Conventions . .9 2 Classical Field Theory 10 2.1 Examples . 11 2.2 Noether's Theorem . 13 2.2.1 Internal Symmetries . 14 2.2.2 Translations and Energy-Momentum Conservation . 15 2.3 Hamiltonian Field Theory . 15 3 The Klein-Gordon Field 17 3.1 The Harmonic Oscillator . 18 3.2 Quantizing the Klein-Gordon Field . 19 3.3 The Vacuum State . 20 3.3.1 The Casimir Effect . 21 3.3.2 The Cosmological Constant . 24 3.4 Particles . 25 3.4.1 Single Particle States . 25 3.4.2 Multiparticle States . 27 3.5 Complex Fields . 28 3.6 Non-Relativistic Fields . 29 3.6.1 The Non-Relativistic Limit . 29 3.6.2 Quantizing the Schr¨odingerField . 30 3.7 The Heisenberg Picture . 32 3.7.1 The Propagator . 33 4 Functional Methods 36 4.1 Path Integrals in Quantum Mechanics . 36 4.1.1 Correlation Functions . 38 4.1.2 The Generating Functional . 40 4.1.3 The Harmonic Oscillator . 41 4.2 Path Integrals for Free Fields . 43 4.2.1 Path Integrals for Quantum Fields . 43 4.2.2 The Klein-Gordon Field . 45 4.2.3 The Feynman Propagator . 46 4.2.4 Higher Correlation Functions . 46 4.2.5 The Inverse-Square Law . 48 3 Contents 4 4.3 Statistical Mechanics and Finite Temperatures . 50 4.3.1 The Euclidean Path-Integral . 50 4.3.2 Quantum Statistical Field Theory . 53 5 Interacting Fields 55 5.1 Perturbation Theory and Unit Analysis . 55 5.1.1 λφ4 Theory . 57 5.2 Feynman Diagrams . 57 5.3 The S-Matrix . 63 5.3.1 From Correlation Functions to the S-Matrix . 64 5.3.2 Scattering Amplitudes in λφ4 Theory . 67 5.3.3 Tree and Loop Diagrams . 68 5.4 Scattering and Decay . 69 5.4.1 Scattering . 69 5.4.2 Decay . 72 5.5 Cubic Vertices . 73 5.5.1 Meson Decay . 75 5.5.2 Meson-Meson Scattering . 76 6 Fermions 79 6.1 Lorentz Invariance . 79 6.1.1 Vectors and Tensors . 80 6.1.2 Group Representations . 80 6.1.3 Lorentz Representations . 82 6.1.4 The Spinor . 82 6.2 The Dirac Equation . 83 6.2.1 The Dirac Lagrangian . 84 6.3 Solutions of the Dirac Equation . 86 6.3.1 The Dirac Hamiltonian . 88 6.4 Quantizing the Dirac Equation . 88 6.4.1 Fermionic Statistics . 89 6.4.2 Particles . 90 6.4.3 The Dirac Propagator . 92 6.5 The Fermionic Path-Integral . 93 6.5.1 Grassmann Numbers . 93 6.5.2 Evaluating the Path-Integral . 94 6.5.3 Correlation Functions for the Dirac Field . 95 6.6 Yukawa Theory . 96 6.6.1 Feynman Rules . 97 6.6.2 Scattering of Indistinguishable Particles . 98 7 Quantum Electrodynamics 101 7.1 Gauge Invariance . 102 7.1.1 Why Gauge Invariance? . 103 7.1.2 Path-Integrals for the Electromagnetic Field . 104 7.2 Coupling Matter to Light . 107 7.2.1 Fermions . 107 7.2.2 Scalars . 108 Contents 5 7.3 Feynman Rules for QED . 108 7.3.1 Polarization of Light . 109 7.3.2 Scalar QED . 110 7.4 Electrons and Muons . 110 7.4.1 Electron-Positron Annihilation to Muons . 111 7.4.2 Electron-Muon Scattering . 116 7.4.3 Crossing Symmetry . 118 7.5 Electrons and Photons . 120 7.5.1 Compton Scattering . 121 7.5.2 Electron-Positron Annihilation to Photons . 122 8 Spontaneous Symmetry Breaking 124 8.1 Linear Sigma Models . 124 8.1.1 Discrete Symmetry Breaking . 124 8.1.2 Complex Fields . 125 8.1.3 The Linear Sigma Model . 126 8.2 Nambu-Goldstone Bosons . 127 8.2.1 Examples of Spontaneous Symmetry Breaking . 128 8.3 The Abelian Higgs Mechanism . 130 8.3.1 Propagator for a Massive Vector Boson . 131 8.3.2 Generating Fermion Masses . 132 8.4 Superfluids . 133 8.4.1 Philosophy of Effective Field Theory . 136 8.4.2 Linear Dispersion Modes . 138 8.5 Superconductors . 139 8.5.1 The Meissner Effect . 140 Chapter 1 Introduction A field is something that takes a value at every point in space and time. For instance, if you are studying a metal rod, you may consider a `temperature field’ T (x; t) which tells you the temperature at a given location x and time t. This field evolves according to the heat equation: @T @2T = −α : (1.1) @t @x2 We can likewise describe.