Group Theory in a Nutshell for Physicists (In a Nutshell, 17)

✍ Scribed by Anthony Zee

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 633
• Edition: Illustrated
• Category: Library

✦ Synopsis

A concise, modern textbook on group theory written especially for physicists

Although group theory is a mathematical subject, it is indispensable to many areas of modern theoretical physics, from atomic physics to condensed matter physics, particle physics to string theory. In particular, it is essential for an understanding of the fundamental forces. Yet until now, what has been missing is a modern, accessible, and self-contained textbook on the subject written especially for physicists.

Group Theory in a Nutshell for Physicists fills this gap, providing a user-friendly and classroom-tested text that focuses on those aspects of group theory physicists most need to know. From the basic intuitive notion of a group, A. Zee takes readers all the way up to how theories based on gauge groups could unify three of the four fundamental forces. He also includes a concise review of the linear algebra needed for group theory, making the book ideal for self-study.

• Provides physicists with a modern and accessible introduction to group theory
• Covers applications to various areas of physics, including field theory, particle physics, relativity, and much more
• Topics include finite group and character tables; real, pseudoreal, and complex representations; Weyl, Dirac, and Majorana equations; the expanding universe and group theory; grand unification; and much more
• The essential textbook for students and an invaluable resource for researchers
• Features a brief, self-contained treatment of linear algebra
• An online illustration package is available to professors
• Solutions manual (available only to professors)

✦ Table of Contents

Cover
Title
Copyright
Dedication
Contents
Preface
A Brief Review of Linear Algebra
Part I: Groups: Discrete or Continuous, Finite or Infinite
I.1 Symmetry and Groups
I.2 Finite Groups
I.3 Rotations and the Notion of Lie Algebra
Part II: Representing Group Elements by Matrices
II.1 Representation Theory
II.2 Schur’s Lemma and the Great Orthogonality Theorem
II.3 Character Is a Function of Class
II.4 Real, Pseudoreal, Complex Representations, and the Number of Square Roots
II.i1 Crystals Are Beautiful
II.i2 Euler’s ϕ-Function, Fermat’s Little Theorem, and Wilson’s Theorem
II.i3 Frobenius Groups
Part III: Group Theory in a Quantum World
III.1 Quantum Mechanics and Group Theory: Parity, Bloch’s Theorem, and the Brillouin Zone
III.2 Group Theory and Harmonic Motion: Zero Modes
III.3 Symmetry in the Laws of Physics: Lagrangian and Hamiltonian
Part IV: Tensor, Covering, and Manifold
IV.1 Tensors and Representations of the Rotation Groups SO(N)
IV.2 Lie Algebra of SO(3) and Ladder Operators: Creation and Annihilation
IV.3 Angular Momentum and Clebsch-Gordan Decomposition
IV.4 Tensors and Representations of the Special Unitary Groups SU(N)
IV.5 SU(2): Double Covering and the Spinor
IV.6 The Electron Spin and Kramer’s Degeneracy
IV.7 Integration over Continuous Groups, Topology, Coset Manifold, and SO(4)
IV.8 Symplectic Groups and Their Algebras
IV.9 From the Lagrangian to Quantum Field Theory: It Is but a Skip and a Hop
IV.i1 Multiplying Irreducible Representations of Finite Groups: Return to the Tetrahedral Group
IV.i2 Crystal Field Splitting
IV.i3 Group Theory and Special Functions
IV.i4 Covering the Tetrahedron
Part V: Group Theory in the Microscopic World
V.1 Isospin and the Discovery of a Vast Internal Space
V.2 The Eightfold Way of SU(3)
V.3 The Lie Algebra of SU(3) and Its Root Vectors
V.4 Group Theory Guides Us into the Microscopic World
Part VI: Roots, Weights, and Classification of Lie Algebras
VI.1 The Poor Man Finds His Roots
VI.2 Roots and Weights for Orthogonal, Unitary, and Symplectic Algebras
VI.3 Lie Algebras in General
VI.4 The Killing-Cartan Classification of Lie Algebras
VI.5 Dynkin Diagrams
Part VII: From Galileo to Majorana
VII.1 Spinor Representations of Orthogonal Algebras
VII.2 The Lorentz Group and Relativistic Physics
VII.3 SL(2,C) Double Covers SO(3,1): Group Theory Leads Us to the Weyl Equation
VII.4 From the Weyl Equation to the Dirac Equation
VII.5 Dirac and Majorana Spinors: Antimatter and Pseudoreality
VII.i1 A Hidden SO(4) Algebra in the Hydrogen Atom
VII.i2 The Unexpected Emergence of the Dirac Equation in Condensed Matter Physics
VII.i3 The Even More Unexpected Emergence of the Majorana Equation in Condensed Matter Physics
Part VIII: The Expanding Universe
VIII.1 Contraction and Extension
VIII.2 The Conformal Algebra
VIII.3 The Expanding Universe from Group Theory
Part IX: The Gauged Universe
IX.1 The Gauged Universe
IX.2 Grand Unification and SU(5)
IX.3 From SU(5) to SO(10)
IX.4 The Family Mystery
Epilogue
Timeline of Some of the People Mentioned
Solutions to Selected Exercises
Bibliography
Index
Collection of Formulas