1st Edition

# Group Theory for the Standard Model of Particle Physics and Beyond

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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