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Group Theory

Book Description

Group theory has long been an important computational tool for physicists, but, with the advent of the Standard Model, it has become a powerful conceptual tool as well. This book introduces physicists to many of the fascinating mathematical aspects of group theory, and mathematicians to its physics applications. Designed for advanced undergraduate and graduate students, this book gives a comprehensive overview of the main aspects of both finite and continuous group theory, with an emphasis on applications to fundamental physics. Finite groups are extensively discussed, highlighting their irreducible representations and invariants. Lie algebras, and to a lesser extent Kac-Moody algebras, are treated in detail, including Dynkin diagrams. Special emphasis is given to their representations and embeddings. The group theory underlying the Standard Model is discussed, along with its importance in model building. Applications of group theory to the classification of elementary particles are treated in detail.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Contents
  7. 1. Preface: the pursuit of symmetries
  8. 2. Finite groups: an introduction
    1. 2.1 Group axioms
    2. 2.2 Finite groups of low order
    3. 2.3 Permutations
    4. 2.4 Basic concepts
      1. 2.4.1 Conjugation
      2. 2.4.2 Simple groups
      3. 2.4.3 Sylow’s criteria
      4. 2.4.4 Semi-direct product
      5. 2.4.5 Young Tableaux
  9. 3. Finite groups: representations
    1. 3.1 Introduction
    2. 3.2 Schur’s lemmas
    3. 3.3 The A4 character table
    4. 3.4 Kronecker products
    5. 3.5 Real and complex representations
    6. 3.6 Embeddings
    7. 3.7 Zn character table
    8. 3.8 Dn character table
    9. 3.9 Q2n character table
    10. 3.10 Some semi-direct products
    11. 3.11 Induced representations
    12. 3.12 Invariants
    13. 3.13 Coverings
  10. 4. Hilbert spaces
    1. 4.1 Finite Hilbert spaces
    2. 4.2 Fermi oscillators
    3. 4.3 Infinite Hilbert spaces
  11. 5. SU(2)
    1. 5.1 Introduction
    2. 5.2 Some representations
    3. 5.3 From Lie algebras to Lie groups
    4. 5.4 SU(2) → SU (1, 1)
    5. 5.5 Selected SU(2) applications
      1. 5.5.1 The isotropic harmonic oscillator
      2. 5.5.2 The Bohr atom
      3. 5.5.3 Isotopic spin
  12. 6. SU(3)
    1. 6.1 SU(3) algebra
    2. 6.2 α-Basis
    3. 6.3 ω-Basis
    4. 6.4 α' -Basis
    5. 6.5 The triplet representation
    6. 6.6 The Chevalley basis
    7. 6.7 SU(3) in physics
      1. 6.7.1 The isotropic harmonic oscillator redux
      2. 6.7.2 The Elliott model
      3. 6.7.3 The Sakata model
      4. 6.7.4 The Eightfold Way
  13. 7. Classification of compact simple Lie algebras
    1. 7.1 Classification
    2. 7.2 Simple roots
    3. 7.3 Rank-two algebras
    4. 7.4 Dynkin diagrams
    5. 7.5 Orthonormal bases
  14. 8. Lie algebras: representation theory
    1. 8.1 Representation basics
    2. 8.2 A3 fundamentals
    3. 8.3 The Weyl group
    4. 8.4 Orthogonal Lie algebras
    5. 8.5 Spinor representations
      1. 8.5.1 SO(2n) spinors
      2. 8.5.2 SO(2n + 1) spinors
      3. 8.5.3 Clifford algebra construction
    6. 8.6 Casimir invariants and Dynkin indices
    7. 8.7 Embeddings
    8. 8.8 Oscillator representations
    9. 8.9 Verma modules
      1. 8.9.1 Weyl dimension formula
      2. 8.9.2 Verma basis
  15. 9. Finite groups: the road to simplicity
    1. 9.1 Matrices over Galois fields
      1. 9.1.1 PSL2(7)
      2. 9.1.2 A doubly transitive group
    2. 9.2 Chevalley groups
    3. 9.3 A fleeting glimpse at the sporadic groups
  16. 10. Beyond Lie algebras
    1. 10.1 Serre presentation
    2. 10.2 Affine Kac–Moody algebras
    3. 10.3 Super algebras
  17. 11. The groups of the Standard Model
    1. 11.1 Space-time symmetries
      1. 11.1.1 The Lorentz and Poincaré groups
      2. 11.1.2 The conformal group
    2. 11.2 Beyond space-time symmetries
      1. 11.2.1 Color and the quark model
    3. 11.3 Invariant Lagrangians
    4. 11.4 Non-Abelian gauge theories
    5. 11.5 The Standard Model
    6. 11.6 Grand Unification
    7. 11.7 Possible family symmetries
      1. 11.7.1 Finite SU (2) and SO(3) subgroups
      2. 11.7.2 Finite SU (3) subgroups
  18. 12. Exceptional structures
    1. 12.1 Hurwitz algebras
    2. 12.2 Matrices over Hurwitz algebras
    3. 12.3 The Magic Square
  19. Appendix 1: Properties of some finite groups
  20. Appendix 2: Properties of selected Lie algebras
  21. References
  22. Index