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More About This Textbook
Overview
Based on the author's well-established courses, Group Theory for the Standard Model of Particle Physics and Beyond explores the use of symmetries through descriptions of the techniques of Lie groups and it algebras. The text develops the models, theoretical framework, and mathematical tools to understand these symmetries.
After linking symmetries with conservation laws, the book works through me mathematics of angular momentum and extends operators and functions of classical mechanics to quantum mechanics. It then covers the mathematical framework for special relativity and the internal symmetries of the standard model of elementary particle physics. In the chapter on Noether's theorem, the author explains how Lagrangian formalism provides a natural framework for the quantum mechanical interpretation of symmetry principles. He then examines electromagnetic, weak, and strong interactions; spontaneous symmetry breaking; the elusive Higgs boson; and supersymmetry. He also introduces new techniques based on extending space-time into dimensions described by anticommuting coordinates.
Features
Explores the modern framework of special relativity, including Lorentz covariance and the Poincaré group
Describes Lie group techniques for the standard model and beyond
Provides a stepping-stone to Einstein's work on general relativity
Goes beyond the standard model to explore supertransformations, superspace methods, and more
Includes many worked examples and problems in each chapter
This text provides succinct yet complete coverage of the group theory of the symmetries of the standard model of elementary particle physics. It will help readers understand current knowledge about the standard model as well as the physics that potentially lies beyond the standard model.
Editorial Reviews
From the Publisher
The book is clearly written … In addition to references, there are copious problems at the end of each chapter which add to the value of the book … This readable text will be of value to theoreticians entering the area of quantum field theory and also to more seasoned researchers in other areas of physics who wish to remind themselves of the basic group theoretical underpinning of that most fundamental of all physical theories.
—Allan I. Solomon, Contemporary Physics, 52, 2011
This book provides a lucid and readable account of group theory relevant to gauge theories and is a welcome addition to the available texts in the area. … The presentation of difficult topics is clear and suitable for a reader new to the subject, while enough material is included to make this book useful as a reference for more experienced researchers. … The material is a pleasure to read and enlightening. … Overall, this book is well written and presents this important topic in an excellent and clear way. … readers with a more theoretical background will find this book an essential read. In conclusion, every student and researcher in high energy physics should read this excellent book.
—Robert Appleby, Reviews, Volume 11, Issue 2, 2010
Product Details
Related Subjects
Meet the Author
Ken J. Barnes is a Professor Emeritus in the School of Physics and Astronomy at the University of Southampton.
Table of Contents
Preface ix
Acknowledgments xi
Introduction xiii
1 Symmetries and Conservation Laws 1
Lagrangian and Hamiltonian Mechanics 2
Quantum Mechanics 6
The Oscillator Spectrum: Creation and Annihilation Operators 8
Coupled Oscillators: Normal Modes 10
One-Dimensional Fields: Waves 13
The Final Step: Lagrange-Hamilton Quantum Field Theory 16
References 20
Problems 20
2 Quantum Angular Momentum 23
Index Notation 23
Quantum Angular Momentum 25
Result 27
Matrix Representations 28
Spin ½ 28
Addition of Angular Momenta 30
Clebsch-Gordan Coefficients 32
Notes 33
Matrix Representation of Direct (Outer, Kronecker) Products 34
½ ⊗ ½ = 1 ⊕ 0 in Matrix Representation 35
Checks 36
Change of Basis 37
Exercise 38
References 38
Problems 38
3 Tensors and Tensor Operators 41
Scalars 41
Scalar Fields 42
Invariant functions 42
Contravariant Vectors (t → Index at Top) 43
Covariant Vectors (Co = Goes Below) 44
Notes 44
Tensors 45
Notes and Properties 45
Rotations 47
Vector Fields 48
Tensor Operators 49
Scalar Operator 49
Vector Operator 49
Notes 50
Connection with Quantum Mechanics 51
Observables 51
Rotations 52
Scalar Fields 52
Vector Fields 53
Specification of Rotations 55
Transformation of Scalar Wave Functions 56
Finite Angle Rotations 57
Consistency with the Angular Momentum Commutation Rules 58
Rotation of Spinor Wave Function 58
Orbital Angular Momentum (x × p) 60
The Spinors Revisited 65
Dimensions of Projected Spaces 67
Connection between the "Mixed Spinor" and the Adjoint (Regular) Representation 67
Finite Angle Rotation of SO(3) Vector 68
References 69
Problems 69
4 Special Relativity and the Physical Particle States 71
The Dirac Equation 71
The Clifford Algebra: Properties of γ Matrices 72
Structure of the Clifford Algebra and Representation 74
Lorentz Covariance of the Dirac Equation 76
The Adjoint 78
The Nonrelativistic Limit 79
Poincaré Group: Inhomogeneous Lorentz Group 80
Homogeneous (Later Restricted) Lorentz Group 82
Notes 84
The Poincaré Algebra 88
The Casimir Operators and the States 89
References 93
Problems 93
5 The Internal Symmetries 95
References 105
Problems 105
6 Lie Group Techniques for the Standard Model Lie Groups 107
Roots and Weights 108
Simple Roots 111
The Cartan Matrix 113
Finding All the Roots 113
Fundamental Weights 115
The Weyl Group 116
Young Tableaux 117
Raising the Indices 117
The Classification Theorem (Dynkin) 119
Result 119
Coincidences 119
References 120
Problems 120
7 Noether's Theorem and Gauge Theories of the First and Second Kinds 125
References 129
Problems 129
8 Basic Couplings of the Electromagnetic, Weak, and Strong Interactions 131
References 136
Problems 136
9 Spontaneous Symmetry Breaking and the Unification of the Electromagnetic and Weak Forces 139
References 144
Problems 145
10 The Goldstone Theorem and the Consequent Emergence of Nonlinearly Transforming Massless Goldstone Bosons 147
References 151
Problems 151
11 The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries 153
References 155
Problems 155
12 Lie Group Techniques for beyond the Standard Model Lie Groups 157
References 159
Problems 160
13 The Simple Sphere 161
References 181
Problems 182
14 Beyond the Standard Model 185
Massive Case 188
Massless Case 188
Projection Operators 189
Weyl Spinors and Representation 190
Charge Conjugation and Majorana Spinor 192
A Notational Trick 194
SL(2, C) View 194
Unitary Representations 195
Supersymmetry: A First Look at the Simplest (N = 1) Case 196
Massive Representations 197
Massless Representations 199
Superspace 200
Three-Dimensional Euclidean Space (Revisited) 200
Covariant Derivative Operators from Right Action 207
Superfields 209
Supertransformations 211
Notes 211
The Chiral Scalar Multiplet 212
Superspace Methods 213
Covariant Definition of Component Fields 214
Supercharges Revisited 214
Invariants and Lagrangians 217
Notes 220
Superpotential 221
References 225
Problems 225
Index 22