Group Theory for the Standard Model of Particle Physics and Beyond (Series in High Energy Physics, Cosmology and Gravitation)

"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."--Publisher's description Cover Half Title Title Page Copyright Page Table of Contents Preface Acknowledgments Introduction 1 Symmetries and Conservation Laws Lagrangian and Hamiltonian Mechanics Quantum Mechanics The Oscillator Spectrum: Creation and Annihilation Operators Coupled Oscillators: Normal Modes One-Dimensional Fields: Waves The Final Step: Lagrange–Hamilton Quantum Field Theory References Problems 2 Quantum Angular Momentum Index Notation Quantum Angular Momentum Result Matrix Representations Spin 1/2 Addition of Angular Momenta Clebsch–Gordan Coefficients Notes Matrix Representation of Direct (Outer, Kronecker) Products 1/2 1/2 = 1 0 in Matrix Representation Checks Change of Basis Exercise References Problems 3 Tensors and Tensor Operators Scalars Scalar Fields Invariant Functions Contravariant Vectors (t→ Index at Top) Covariant Vectors (Co = Goes Below) Notes Tensors Notes and Properties Rotations Vector Fields Tensor Operators Scalar Operator Vector Operator Notes Connection with Quantum Mechanics Observables Rotations Scalar Fields Vector Fields 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 References Problems 4 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 Notes The Poincaré Algebra The Casimir Operators and the States References Problems 5 The Internal Symmetries References Problems 6 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 References Problems 7 Noether’s Theorem and Gauge Theories of the First and Second Kinds References Problems 8 Basic Couplings of the Electromagnetic, Weak, and Strong Interactions References Problems 9 Spontaneous Symmetry Breaking and the Unification of the Electromagnetic and Weak Forces References Problems 10 The Goldstone Theorem and the Consequent Emergence of Nonlinearly Transforming Massless Goldstone Bosons References Problems 11 The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries References Problems 12 Lie Group Techniques for beyond the Standard Model Lie Groups References Problems 13 The Simple Sphere References Problems 14 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 Notes The Chiral Scalar Multiplet Superspace Methods Covariant Definition of Component Fields Supercharges Revisited Invariants and Lagrangians Notes Superpotential References Problems Index Based on the author's well-established lecture courses, this text introduces the group theory of the symmetries of the standard model of elementary particle physics. Assuming a background in the relevant quantum field theories, it is designed for graduate students in theoretical, high energy, and particle physics.