Group Theory for the Standard Model of Particle Physics and Beyond

by Ken J. Barnes

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 Lie 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 the 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.

Designed for graduate and advanced undergraduate students in physics, 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 students understand current knowledge about the standard model as well as the physics that potentially lies beyond the standard model.

Product Details

ISBN-13:	9781420078749
Publisher:	Taylor & Francis
Publication date:	03/10/2010
Series:	Series in High Energy Physics, Cosmology and Gravitation
Pages:	256
Product dimensions:	6.40(w) x 9.30(h) x 1.00(d)

About 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

Editorial Reviews

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

From the Publisher