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Click here for terms of use.com This page intentionally left blank www.com 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 www.com This page intentionally left blank www.com For more information about this title, click here CONTENTS Preface xv CHAPTER 1 Particle Physics and Special Relativity 1 Special Relativity 5 A Quick Overview of Particle Physics 12 Elementary Particles 14 The Higgs Mechanism 18 Grand Unification 18 Supersymmetry 19 String Theory 19 Summary 20 Quiz 20 CHAPTER 2 Lagrangian Field Theory 23 Basic Lagrangian Mechanics 23 The Action and the Equations of Motion 26 Canonical Momentum and the Hamiltonian 29 Lagrangian Field Theory 30 Symmetries and Conservation Laws 35 Conserved Currents 38 The Electromagnetic Field 39 Gauge Transformations 43 Summary 47 Quiz 47 www.com x Quantum Field Theory Demystified CHAPTER 3 An Introduction to Group Theory 49 Representation of the Group 50 Group Parameters 52 Lie Groups 52 The Rotation Group 54 Representing Rotations 55 SO(N) 58 Unitary Groups 62 Casimir Operators 67 Summary 68 Quiz 68 CHAPTER 4 Discrete Symmetries and Quantum Numbers 71 Additive and Multiplicative Quantum Numbers 71 Parity 72 Charge Conjugation 76 CP Violation 78 The CPT Theorem 80 Summary 82 Quiz 83 CHAPTER 5 The Dirac Equation 85 The Classical Dirac Field 85 Adding Quantum Theory 87 The Form of the Dirac Matrices 89 Some Tedious Properties of the Dirac Matrices 91 Adjoint Spinors and Transformation Properties 94 Slash Notation 95 Solutions of the Dirac Equation 95 Free Space Solutions 99 Boosts, Rotations, and Helicity 103 Weyl Spinors 104 Summary 107 Quiz 108 www.com Contents xi CHAPTER 6 Scalar Fields 109 Arriving at the Klein-Gordon Equation 110 Reinterpreting the Field 117 Field Quantization of Scalar Fields 117 States in Quantum Field Theory 127 Positive and Negative Frequency Decomposition 128 Number Operators 128 Normalization of the States 130 Bose-Einstein Statistics 131 Normal and Time-Ordered Products 134 The Complex Scalar Field 135 Summary 137 Quiz 137 CHAPTER 7 The Feynman Rules 139 The Interaction Picture 141 Perturbation Theory 143 Basics of the Feynman Rules 146 Calculating Amplitudes 151 Steps to Construct an Amplitude 153 Rates of Decay and Lifetimes 160 Summary 160 Quiz 160 CHAPTER 8 Quantum Electrodynamics 163 Reviewing Classical Electrodynamics Again 165 The Quantized Electromagnetic Field 168 Gauge Invariance and QED 170 Feynman Rules for QED 173 Summary 185 Quiz 185 CHAPTER 9 Spontaneous Symmetry Breaking and the Higgs Mechanism 187 Symmetry Breaking in Field Theory 189 www.com xii Quantum Field Theory Demystified Mass Terms in the Lagrangian 192 Aside on Units 195 Spontaneous Symmetry Breaking and Mass 196 Lagrangians with Multiple Particles 199 The Higgs Mechanism 202 Summary 207 Quiz 207 CHAPTER 10 Electroweak Theory 209 Right- and Left-Handed Spinors 210 A Massless Dirac Lagrangian 211 Leptonic Fields of the Electroweak Interactions 212 Charges of the Electroweak Interaction 213 Unitary Transformations and the Gauge Fields of the Theory 215 Weak Mixing or Weinberg Angle 219 Symmetry Breaking 220 Giving Mass to the Lepton Fields 222 Gauge Masses 224 Summary 231 Quiz 231 CHAPTER 11 Path Integrals 233 Gaussian Integrals 233 Basic Path Integrals 238 Summary 242 Quiz 243 CHAPTER 12 Supersymmetry 245 Basic Overview of Supersymmetry 246 Supercharge 247 Supersymmetric Quantum Mechanics 249 The Simplified Wess-Zumino Model 253 A Simple SUSY Lagrangian 254 Summary 260 Quiz 260 www.com Contents xiii Final Exam 263 Solutions to Quizzes and Final Exam 281 References 289 Index 291 www.com This page intentionally left blank www.com PREFACE Quantum field theory is the union of Einstein’s special relativity and quantum mechanics. It forms the foundation of what scientists call the standard model, which is a theoretical framework that describes all known particles and interactions with the exception of gravity. There is no time like the present to learn it—the Large Hadron Collider (LHC) being constructed in Europe will test the final pieces of the standard model (the Higgs mechanism) and look for physics beyond the standard model. In addition quantum field theory forms the theoretical underpinnings of string theory, currently the best candidate for unifying all known particles and forces into a single theoretical framework.
Quantum field theory is also one of the most difficult subjects in science. This book aims to open the door to quantum field theory to as many interested people as possible by providing a simplified presentation of the subject. This book is useful as a supplement in the classroom or as a tool for self-study, but be forewarned that the book includes the math that comes along with the subject. By design, this book is not thorough or complete, and it might even be considered by some “experts” to be shallow or filled with tedious calculations.
But this book is not written for the experts or for brilliant graduate students at the top of the class, it is written for those who find the subject difficult or impossible. Certain aspects of quantum field theory have been selected to introduce new people to the subject, or to help refresh those who have been away from physics. After completing this book, you will find that studying other quantum field theory books will be easier. You can master quantum field theory by tackling the reference list in the back of this book, which includes a list of textbooks used in the development of this one.
Frankly, while all of those books are very good and make fine references, most of them are hard to read. In fact many quantum field theory books are impossible to read. My recommendation is to work through this book first, and then tackle Quantum Field Theory in a Nutshell by Anthony Zee. Different than all other books on the subject, it’s very readable and is packed with great Copyright © 2008 by The McGraw-Hill Companies, Inc.
Click here for terms of use.com xvi Quantum Field Theory Demystified physical insight. After you’ve gone through that book, if you are looking for mastery or deep understanding you will be well equipped to tackle the other books on the list. Unfortunately, learning quantum field theory entails some background in physics and math. The bottom line is, I assume you have it.
The background I am expecting includes quantum mechanics, some basic special relativity, some exposure to electromagnetics and Maxwell’s equations, calculus, linear algebra, and differential equations. If you lack this background do some studying in these subjects and then give this book a try. Now let’s forge ahead and start learning quantum field theory. David McMahon www.com Quantum Field Theory Demystified www.com This page intentionally left blank www.