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Quantum Field Theory Demystifi ed

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Quantum Field Theory Demystifi ed

David McMahon

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Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Exceptas permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any formor by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

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DOI: 10.1036/0071543821

ABOUT THE AUTHOR

David McMahon works as a researcher at Sandia National Laboratories. He has advanced degrees in physics and applied mathematics, and is the author of QuantumMechanics Demystified, Relativity Demystified, MATLAB Demystified, and several other successful books.

Copyright 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use.

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CONTENTS AT A GLANCE

CHAPTER 1 Particle Physics and Special Relativity 1

CHAPTER 2 Lagrangian Field Theory 23

CHAPTER 3 An Introduction to Group Theory 49

CHAPTER 4 Discrete Symmetries and Quantum Numbers 71

CHAPTER 5 The Dirac Equation 85

CHAPTER 6 Scalar Fields 109

CHAPTER 7 The Feynman Rules 139

CHAPTER 8 Quantum Electrodynamics 163

CHAPTER 9 Spontaneous Symmetry Breaking and the Higgs Mechanism 187

CHAPTER 10 Electroweak Theory 209

CHAPTER 11 Path Integrals 233

CHAPTER 12 Supersymmetry 245

Final Exam 263

Solutions to Quizzes and Final Exam 281

References 289

Index 291

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CONTENTS

Preface xv

CHAPTER 1 Particle Physics and Special Relativity 1Special Relativity 5A Quick Overview of Particle Physics 12Elementary Particles 14The Higgs Mechanism 18Grand Unifi cation 18Supersymmetry 19String Theory 19Summary 20Quiz 20

CHAPTER 2 Lagrangian Field Theory 23Basic Lagrangian Mechanics 23The Action and the Equations of Motion 26Canonical Momentum and the Hamiltonian 29Lagrangian Field Theory 30Symmetries and Conservation Laws 35Conserved Currents 38The Electromagnetic Field 39Gauge Transformations 43Summary 47Quiz 47

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CHAPTER 3 An Introduction to Group Theory 49Representation of the Group 50Group Parameters 52Lie Groups 52The Rotation Group 54Representing Rotations 55SO(N) 58Unitary Groups 62Casimir Operators 67Summary 68Quiz 68

CHAPTER 4 Discrete Symmetries and Quantum Numbers 71Additive and Multiplicative Quantum Numbers 71Parity 72Charge Conjugation 76CP Violation 78The CPT Theorem 80Summary 82Quiz 83

CHAPTER 5 The Dirac Equation 85The Classical Dirac Field 85Adding Quantum Theory 87The Form of the Dirac Matrices 89Some Tedious Properties of the

Dirac Matrices 91Adjoint Spinors and Transformation Properties 94Slash Notation 95Solutions of the Dirac Equation 95Free Space Solutions 99Boosts, Rotations, and Helicity 103Weyl Spinors 104Summary 107Quiz 108

x Quantum Field Theory Demystifi ed

CHAPTER 6 Scalar Fields 109Arriving at the Klein-Gordon Equation 110Reinterpreting the Field 117Field Quantization of Scalar Fields 117States in Quantum Field Theory 127Positive and Negative Frequency

Decomposition 128Number Operators 128Normalization of the States 130Bose-Einstein Statistics 131Normal and Time-Ordered Products 134The Complex Scalar Field 135Summary 137Quiz 137

CHAPTER 7 The Feynman Rules 139The Interaction Picture 141Perturbation Theory 143Basics of the Feynman Rules 146Calculating Amplitudes 151Steps to Construct an Amplitude 153Rates of Decay and Lifetimes 160Summary 160Quiz 160

CHAPTER 8 Quantum Electrodynamics 163Reviewing Classical Electrodynamics Again 165The Quantized Electromagnetic Field 168Gauge Invariance and QED 170Feynman Rules for QED 173Summary 185Quiz 185

CHAPTER 9 Spontaneous Symmetry Breaking and the Higgs Mechanism 187

Symmetry Breaking in Field Theory 189

Contents xi

Mass Terms in the Lagrangian 192Aside on Units 195Spontaneous Symmetry Breaking and Mass 196 Lagrangians with Multiple Particles 199The Higgs Mechanism 202Summary 207Quiz 207

CHAPTER 10 Electroweak Theory 209Right- and Left-Handed Spinors 210A Massless Dirac Lagrangian 211Leptonic Fields of the Electroweak Interactions 212Charges of the Electroweak Interaction 213Unitary Transformations and

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