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  • Quantum Field Theory Demystifi ed

  • Demystified Series

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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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    TERMS OF USE

    This is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill) and its licensors reserve all rights in and to thework. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store andretrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative worksbased upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hills prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Yourright to use the work may be terminated if you fail to comply with these terms.

    THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIESAS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THEWORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY,EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITEDTO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and itslicensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or itslicensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of orinability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shallapply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

    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

    For more information about this title, click here

  • 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