Group theory for the standard model of particle physics and beyond

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.

Author Notes

Ken J. Barnes is a Professor Emeritus in the School of Physics and Astronomy at the University of Southampton.

Preface	p. ix
Acknowledgments	p. xi
Introduction	p. xiii
1 Symmetries and Conservation Laws	p. 1
Lagrangian and Hamiltonian Mechanics	p. 2
Quantum Mechanics	p. 6
The Oscillator Spectrum: Creation and Annihilation Operators	p. 8
Coupled Oscillators: Normal Modes	p. 10
One-Dimensional Fields: Waves	p. 13
The Final Step: Lagrange-Hamilton Quantum Field Theory	p. 16
References	p. 20
Problems	p. 20
2 Quantum Angular Momentum	p. 23
Index Notation	p. 23
Quantum Angular Momentum	p. 25
Result	p. 27
Matrix Representations	p. 28
Spin ½	p. 28
Addition of Angular Momenta	p. 30
Clebsch-Gordan Coefficients	p. 32
Notes	p. 33
Matrix Representation of Direct (Outer, Kronecker) Products	p. 34
½ ⊗ ½ = 1 ⊕ 0 in Matrix Representation	p. 35
Checks	p. 36
Change of Basis	p. 37
Exercise	p. 38
References	p. 38
Problems	p. 38
3 Tensors and Tensor Operators	p. 41
Scalars	p. 41
Scalar Fields	p. 42
Invariant functions	p. 42
Contravariant Vectors (t → Index at Top)	p. 43
Covariant Vectors (Co = Goes Below)	p. 44
Notes	p. 44
Tensors	p. 45
Notes and Properties	p. 45
Rotations	p. 47
Vector Fields	p. 48
Tensor Operators	p. 49
Scalar Operator	p. 49
Vector Operator	p. 49
Notes	p. 50
Connection with Quantum Mechanics	p. 51
Observables	p. 51
Rotations	p. 52
Scalar Fields	p. 52
Vector Fields	p. 53
Specification of Rotations	p. 55
Transformation of Scalar Wave Functions	p. 56
Finite Angle Rotations	p. 57
Consistency with the Angular Momentum Commutation Rules	p. 58
Rotation of Spinor Wave Function	p. 58
Orbital Angular Momentum (x × p)	p. 60
The Spinors Revisited	p. 65
Dimensions of Projected Spaces	p. 67
Connection between the "Mixed Spinor" and the Adjoint (Regular) Representation	p. 67
Finite Angle Rotation of SO(3) Vector	p. 68
References	p. 69
Problems	p. 69
4 Special Relativity and the Physical Particle States	p. 71
The Dirac Equation	p. 71
The Clifford Algebra: Properties of ¿ Matrices	p. 72
Structure of the Clifford Algebra and Representation	p. 74
Lorentz Covariance of the Dirac Equation	p. 76
The Adjoint	p. 78
The Nonrelativistic Limit	p. 79
Poincaré Group: Inhomogeneous Lorentz Group	p. 80
Homogeneous (Later Restricted) Lorentz Group	p. 82
Notes	p. 84
The Poincaré Algebra	p. 88
The Casimir Operators and the States	p. 89
References	p. 93
Problems	p. 93
5 The Internal Symmetries	p. 95
References	p. 105
Problems	p. 105
6 Lie Group Techniques for the Standard Model Lie Groups	p. 107
Roots and Weights	p. 108
Simple Roots	p. 111
The Cartan Matrix	p. 113
Finding All the Roots	p. 113
Fundamental Weights	p. 115
The Weyl Group	p. 116
Young Tableaux	p. 117
Raising the Indices	p. 117
The Classification Theorem (Dynkin)	p. 119
Result	p. 119
Coincidences	p. 119
References	p. 120
Problems	p. 120
7 Noether's Theorem and Gauge Theories of the First and Second Kinds	p. 125
References	p. 129
Problems	p. 129
8 Basic Couplings of the Electromagnetic, Weak, and Strong Interactions	p. 131
References	p. 136
Problems	p. 136
9 Spontaneous Symmetry Breaking and the Unification of the Electromagnetic and Weak Forces	p. 139
References	p. 144
Problems	p. 145
10 The Goldstone Theorem and the Consequent Emergence of Nonlinearly Transforming Massless Goldstone Bosons	p. 147
References	p. 151
Problems	p. 151
11 The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries	p. 153
References	p. 155
Problems	p. 155
12 Lie Group Techniques for beyond the Standard Model Lie Groups	p. 157
References	p. 159
Problems	p. 160
13 The Simple Sphere	p. 161
References	p. 181
Problems	p. 182
14 Beyond the Standard Model	p. 185
Massive Case	p. 188
Massless Case	p. 188
Projection Operators	p. 189
Weyl Spinors and Representation	p. 190
Charge Conjugation and Majorana Spinor	p. 192
A Notational Trick	p. 194
SL(2, C) View	p. 194
Unitary Representations	p. 195
Supersymmetry: A First Look at the Simplest (N = 1) Case	p. 196
Massive Representations	p. 197
Massless Representations	p. 199
Superspace	p. 200
Three-Dimensional Euclidean Space (Revisited)	p. 200
Covariant Derivative Operators from Right Action	p. 207
Superfields	p. 209
Supertransformations	p. 211
Notes	p. 211
The Chiral Scalar Multiplet	p. 212
Superspace Methods	p. 213
Covariant Definition of Component Fields	p. 214
Supercharges Revisited	p. 214
Invariants and Lagrangians	p. 217
Notes	p. 220
Superpotential	p. 221
References	p. 225
Problems	p. 225
Index	p. 22

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents