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 ? 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.

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