Group Theory in a Nutshell 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 subjec...

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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)

Contents
Preface xi
A Brief Review of Linear Algebra 1
I Part I: Groups: Discrete or Continuous, Finite or Infinite
I.1 Symmetry and Groups 37
I.2 Finite Groups 55
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Contents
Preface xi
A Brief Review of Linear Algebra 1
I Part I: Groups: Discrete or Continuous, Finite or Infinite
I.1 Symmetry and Groups 37
I.2 Finite Groups 55
I.3 Rotations and the Notion of Lie Algebra 70
II Part II: Representing Group Elements by Matrices
II.1 Representation Theory 89
II.2 Schur’s Lemma and the Great Orthogonality Theorem 101
II.3 Character Is a Function of Class 114
II.4 Real, Pseudoreal, Complex Representations, and the Number of Square Roots 136
II.i1 Crystals Are Beautiful 146
II.i2 Euler’s ϕ-Function, Fermat’s Little Theorem, and Wilson’s Theorem 150
II.i3 Frobenius Groups 154
III Part III: Group Theory in a Quantum World
III.1 Quantum Mechanics and Group Theory: Parity, Bloch’s Theorem, and the Brillouin Zone 161
III.2 Group Theory and Harmonic Motion: Zero Modes 168
III.3 Symmetry in the Laws of Physics: Lagrangian and Hamiltonian 176
IV Part IV: Tensor, Covering, and Manifold
IV.1 Tensors and Representations of the Rotation Groups SO(N) 185
IV.2 Lie Algebra of SO(3) and Ladder Operators: Creation and Annihilation 203
IV.3 Angular Momentum and Clebsch-Gordan Decomposition 216
IV.4 Tensors and Representations of the Special Unitary Groups SU(N) 227
IV.5 SU(2): Double Covering and the Spinor 244
IV.6 The Electron Spin and Kramer’s Degeneracy 255
IV.7 Integration over Continuous Groups, Topology, Coset Manifold, and SO(4) 261
IV.8 Symplectic Groups and Their Algebras 277
IV.9 From the Lagrangian to Quantum Field Theory: It Is but a Skip and a Hop 283
IV.i1 Multiplying Irreducible Representations of Finite Groups: Return to the Tetrahedral Group 289
IV.i2 Crystal Field Splitting 292
IV.i3 Group Theory and Special Functions 295
IV.i4 Covering the Tetrahedron 299
V Part V: Group Theory in the Microscopic World
V.1 Isospin and the Discovery of a Vast Internal Space 303
V.2 The Eightfold Way of SU(3) 312
V.3 The Lie Algebra of SU(3) and Its Root Vectors 325
V.4 Group Theory Guides Us into the Microscopic World 337
VI Part VI: Roots, Weights, and Classification of Lie Algebras
VI.1 The Poor Man Finds His Roots 347
VI.2 Roots and Weights for Orthogonal, Unitary, and Symplectic Algebras 350
VI.3 Lie Algebras in General 364
VI.4 The Killing-Cartan Classification of Lie Algebras 376
VI.5 Dynkin Diagrams 384
VII Part VII: From Galileo to Majorana
VII.1 Spinor Representations of Orthogonal Algebras 405
VII.2 The Lorentz Group and Relativistic Physics 428
VII.3 SL(2,C) Double Covers SO(3,1): Group Theory Leads Us to the Weyl Equation 450
VII.4 From the Weyl Equation to the Dirac Equation 468
VII.5 Dirac and Majorana Spinors: Antimatter and Pseudoreality 482
VII.i1 A Hidden SO(4) Algebra in the Hydrogen Atom 491
VII.i2 The Unexpected Emergence of the Dirac Equation in Condensed Matter Physics 497
VII.i3 The Even More Unexpected Emergence of the Majorana Equation in Condensed Matter Physics 501
VIII Part VIII: The Expanding Universe
VIII.1 Contraction and Extension 507
VIII.2 The Conformal Algebra 515
VIII.3 The Expanding Universe from Group Theory 523
IX Part IX: The Gauged Universe
IX.1 The Gauged Universe 531
IX.2 Grand Unification and SU(5) 541
IX.3 From SU(5) to SO(10) 550
IX.4 The Family Mystery 560
Epilogue 565
Timeline of Some of the People Mentioned 567
Solutions to Selected Exercises 569
Bibliography 581
Index 583
Collection of Formulas 601
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