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

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Description

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.
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Product details

  • Hardback | 255 pages
  • 156 x 235 x 22.86mm | 245g
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 1304 Equations, 33 in-text boxes; 26 Illustrations, black and white
  • 1420078747
  • 9781420078749
  • 1,170,465

Table of contents

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 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


Poincare Group: Inhomogeneous Lorentz Group


Homogeneous (Later Restricted) Lorentz Group


Poincare 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.
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Review quote

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
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About Ken J. Barnes

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