1st Edition

Group Theory for the Standard Model of Particle Physics and Beyond

By Ken J. Barnes Copyright 2010
    256 Pages 26 B/W Illustrations
    by CRC Press

    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.

    Symmetries and Conservation Laws

    Lagrangian and Hamiltonian Mechanics

    Quantum Mechanics

    Coupled Oscillators: Normal Modes

    One-Dimensional Fields: Waves

    The Final Step: Lagrange–Hamilton Quantum Field Theory

    Quantum Angular Momentum

    Index Notation

    Quantum Angular Momentum


    Matrix Representations

    Spin 1/2

    Addition of Angular Momenta

    Clebsch–Gordan Coefficients

    Matrix Representation of Direct (Outer, Kronecker) Products

    Change of Basis

    Tensors and Tensor Operators


    Scalar Fields

    Invariant Functions

    Contravariant Vectors (t →index at top)

    Covariant Vectors (Co = Goes Below)




    Vector Fields

    Tensor Operators

    Connection with Quantum Mechanics

    Specification of Rotations

    Transformation of Scalar Wave Functions

    Finite Angle Rotations

    Consistency with the Angular Momentum Commutation Rules

    Rotation of Spinor Wave Function

    Orbital Angular Momentum (x × p)

    The Spinors Revisited

    Dimensions of Projected Spaces

    Connection between the "Mixed Spinor" and the Adjoint (Regular) Representation

    Finite Angle Rotation of SO(3) Vector

    Special Relativity and the Physical Particle States

    The Dirac Equation

    The Clifford Algebra: Properties of γ Matrices

    Structure of the Clifford Algebra and Representation

    Lorentz Covariance of the Dirac Equation

    The Adjoint

    The Nonrelativistic Limit

    Poincaré Group: Inhomogeneous Lorentz Group

    Homogeneous (Later Restricted) Lorentz Group

    Poincaré Algebra

    The Casimir Operators and the States

    Internal Symmetries

    Lie Group Techniques for the Standard Model Lie Groups

    Roots and Weights

    Simple Roots

    The Cartan Matrix

    Finding All the Roots

    Fundamental Weights

    The Weyl Group

    Young Tableaux

    Raising the Indices

    The Classification Theorem (Dynkin)



    Noether’s Theorem and Gauge Theories of the First and Second Kinds

    Basic Couplings of the Electromagnetic, Weak, and Strong Interactions

    Spontaneous Symmetry Breaking and the Unification of the Electromagnetic and Weak Forces

    The Goldstone Theorem and the Consequent Emergence on Nonlinear Transforming Massless Goldstone Bosons

    The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries

    Lie Group Techniques for beyond the Standard Model Lie Groups

    The Simple Sphere

    Beyond the Standard Model

    Massive Case

    Massless Case

    Projection Operators

    Weyl Spinors and Representation

    Charge Conjugation and Majorana Spinor

    A Notational Trick

    SL(2, C) View

    Unitary Representations

    Supersymmetry: A First Look at the Simplest (N = 1) Case

    Massive Representations

    Massless Representations


    Three Dimensional Euclidean Space (Revisited)

    Covariant Derivative Operators from Right Action



    The Chiral Scalar Multiplet

    Superspace Methods

    Covariant Definition of Component Fields

    Supercharges Revisited

    Invariants and Lagrangians


    References and Problems appear at the end of each chapter.


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

    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