Group Theory in a Nutshell for Physicists

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

I 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

II 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

III 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

IV 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

V 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) 312

V.3 The Lie Algebra of SU(3) and Its Root Vectors

V.4 Group Theory Guides Us into the Microscopic World

VI 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 364

VI.4 The Killing-Cartan Classification of Lie Algebras

VI.5 Dynkin Diagrams

VII 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

VIII Part VIII: The Expanding Universe

VIII.1 Contraction and Extension

VIII.2 The Conformal Algebra

VIII.3 The Expanding Universe from Group Theory

IX 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

About the Author

A. Zee is professor of physics at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. His books include Quantum Field Theory in a Nutshell, Einstein Gravity in a Nutshell, and Fearful Symmetry: The Search for Beauty in Modern Physics (all Princeton).