Quantum Field Theory Priv.-Doz. Dr. Axel Pelster Department of Physics Lecture in Winter Term 2020/2021 Kaiserslautern, February 12, 2021 Preface The present manuscript represents the material of the lecture Quantum Field Theory, which I held as part of the Master of Science in Advanced Quantum Physics program at the Depart- ment of Physics of the Technische Universit¨atKaiserslautern during the winter term 2020/2021. Due to the Corona pandemic the 28 lectures were both live streamed and recorded. The cor- responding videos are available at the platform Panopto at the national network of the state Rhineland Palatinate. After each lecture both the corresponding hand-written notes as well as this manuscript were made available to the students. The presented material is partially inspired by former lectures of Prof. Dr. Ulrich Weiß at the Universit¨atStuttgart from I which benefitted during my own physics study. And I had the pleasure to learn from the lectures of Prof. Dr. Hagen Kleinert at the Freie Universit¨atBerlin during my postdoc time, when I supervised the corresponding exercises. The manuscript is mainly based on the two valuable books: W. Greiner and J. Reinhardt, Field Quantization, Springer, 2008 • H. Kleinert, Particles and Quantum Fields, World Scientific, 2016 • which I recommend for self-study. Additional useful information, partially including historical insights, can be found in the following selected books: J. Gleick, Genius - The Life and Science of Richard Feynman, Vintage Books, 1991 • W. Greiner, Relativistic Quantum Mechanics - Wave Equations, Springer, 2000 • W. Greiner and J. Reinhardt, Quantum Electrodynamics, Springer, 2008 • W. Greiner and J. Reinhardt, Field Quantization, Springer, 2008 • H. Kleinert, Particles and Quantum Fields, World Scientific, 2016 • M.E. Peskin and D.V. Schr¨oder, An Introduction to Quantum Field Theory, Cambridge • University Press, 1995 S. Schweber, QED and the men who made it - Feynman, Schwinger, and Tomonaga, • Princeton University Press, 1994 iii iv M. Veltman, Diagrammatica - The Path to Feynman Diagrams, Cambridge University • Press, 1994 T. Lancaster and S.J. Blundell, Quantum Field Theory for the Gifted Amateur, Oxford • University Press, 2014 With this I thank the student workers Alexander Guthmann, Joshua Krauss, and Andra Pirk- tina for typing and, partially, translating the original hand-written German lecture notes. They originated from previous lectures held at Freie Universit¨atBerlin (2000), Universtit¨atPotsdam (2010, 2011), and Technische Universit¨atKaiserslautern (2014, 2017/2018). And, finally, I am also grateful to Dr. Milan Randonji´cfor contributing to the success of the lecture in the winter term 2020/2021 by supervising the exercises. The corresponding problem sets are available at the homepage http://www-user.rhrk.uni-kl.de/~apelster/Vorlesungen/WS2021/qft.html Contents 1 Introduction 1 1.1 Standard Model . 1 1.2 Non-Relativistic Quantum Many-Body Theory . 2 1.3 Relativistic Fields and Their Quantization . 4 1.4 Quantum Electrodynamics . 4 I Non-Relativistic Quantum Many-Body Theory 7 2 Identical Particles 9 2.1 Distinguishable Particles . 9 2.2 Bosons and Fermions . 12 2.3 Non-Interacting Identical Particles . 14 3 Second Quantization 21 3.1 Harmonic Oscillator . 21 3.2 Creation and Annihilation Operators for Bosons . 25 3.3 Schr¨odingerEquation for Interacting Bosons . 26 3.4 Field Operators in Heisenberg Picture . 29 3.5 Creation and Annihilation Operators for Fermions . 31 3.6 Occupation Number Representation . 34 4 Canonical Field Quantization for Bosons 37 4.1 Action of Schr¨odinger Field . 37 4.2 Functional Derivative: Definition . 38 4.3 Functional Derivative: Application . 40 v vi CONTENTS 4.4 Euler-Lagrange Equations . 42 4.5 Hamilton Field Theory . 43 4.6 Poisson Brackets . 45 4.7 Canonical Field Quantization . 46 5 Canonical Field Quantization for Fermions 49 5.1 Grassmann Fields . 49 5.1.1 Grassmann Numbers . 49 5.1.2 Grassmann Functions . 50 5.1.3 Differentiation and Integration . 51 5.1.4 Complex Grassmann Numbers . 53 5.1.5 Grassmann Fields . 54 5.2 Lagrange Field Theory for Fermions . 54 5.3 Hamilton Field Theory for Fermions . 56 5.4 Poisson Brackets . 58 5.5 Canonical Field Quantization . 59 II Free Relativistic Fields and Their Quantization 61 6 Poincar´eGroup 63 6.1 Special Relativity . 63 6.2 Defining Representation of Lorentz Group . 67 6.3 Defining Representation of Lorentz Algebra . 69 6.4 Classification of Basis Elements . 70 6.5 Lie Theorem . 72 6.6 Rotations . 73 6.7 Boosts . 74 6.8 Scalar Field Representation . 76 6.9 Tensor/Spinor Field Representation . 78 6.10 Defining Representation of Poincar´eGroup . 79 6.11 Tensor/Spinor Representation of Poincar´eAlgebra . 81 CONTENTS vii 6.12 Casimir Operators of Poincar´eAlgebra . 82 6.13 Irreducible Representations of Poincar´eGroup . 86 6.13.1 Massive Representations . 86 6.13.2 Massless Representations . 87 6.13.3 Other Representations . 88 7 Klein-Gordon Field 89 7.1 Action and Equations of Motions . 89 7.2 Continuity Equation . 94 7.3 Canonical Field Quantization . 96 7.4 Plane Waves . 98 7.5 Fourier Operators . 100 7.6 Hamilton Operator . 101 7.7 Charge Operator . 102 7.8 Redefinition of Fourier Operators . 103 7.9 Definition of Propagator . 104 7.10 Interpretation of Propagator . 106 7.11 Calculation of Propagator . 107 7.12 Covariant Form of Propagator . 110 8 Maxwell Field 113 8.1 Maxwell Equations . 113 8.2 Local Gauge Symmetry . 114 8.3 Field Strength Tensors . 116 8.4 Four-Vector Potential . 118 8.5 Euler-Lagrange Equations . 119 8.6 Hamilton Function . 120 8.7 Canonical Field Quantization . 122 8.8 Heisenberg Equations . 124 8.9 Decomposition in Plane Waves . 125 8.10 Construction of Polarization Vectors . 127 viii CONTENTS 8.11 Properties of Polarization Vectors . 130 8.12 Fourier Operators . 132 8.13 Energy . 133 8.14 Momentum . 133 8.15 Spin Angular Momentum . 134 8.16 Definition of Maxwell Propagator . 136 8.17 Calculation of Maxwell Propagator . 137 8.18 Four-Dimensional Fourier Representation . 138 9 Dirac Field 141 9.1 Pauli Matrices . 142 9.2 Spinor Representation of Lorentz Algebra . 143 9.3 Spinor Representation of Rotations . 144 9.4 Spinor Representation of Boosts . 145 9.5 Lorentz Invariant Combinations of Weyl Spinors . 148 9.6 Dirac Action . 151 9.7 Spinor Representation of Lorentz Group . 154 9.8 Parity Transformation . 157 9.9 Neutrinos . 159 9.10 Charge conjugation . 163 9.11 Time Inversion . 166 9.12 Dirac Representation . 170 9.13 Non-Relativistic Limit . 171 9.14 Plane Waves . 176 9.14.1 Rest Frame . 176 9.14.2 Boost to Uniformly Moving Reference Frame . ..