1st Edition
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.
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
Result
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
Scalars
Scalar Fields
Invariant Functions
Contravariant Vectors (t →index at top)
Covariant Vectors (Co = Goes Below)
Notes
Tensors
Rotations
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)
Result
Coincidences
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
Superspace
Three Dimensional Euclidean Space (Revisited)
Covariant Derivative Operators from Right Action
Superfields
Supertransformations
The Chiral Scalar Multiplet
Superspace Methods
Covariant Definition of Component Fields
Supercharges Revisited
Invariants and Lagrangians
Superpotential
References and Problems appear at the end of each chapter.
Biography
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, 2011This 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
