On the Role of Division, Jordan and Related Algebras in Particle Physics

This monograph surveys the role of some associative and non-associative algebras, remarkable by their ubiquitous appearance in contemporary theoretical physics, particularly in particle physics. It concerns the interplay between division algebras, specifically quaternions and octonions, between Jordan and related algebras on the one hand, and unified theories of the basic interactions on the other. Selected applications of these algebraic structures are discussed: quaternion analyticity of Yang–Mills instantons, octonionic aspects of exceptional broken gauge, supergravity theories, division algebras in anyonic phenomena and in theories of extended objects in critical dimensions. The topics presented deal primarily with original contributions by the authors.

• Introduction
• Quaternions:
  • Algebraic Structures
  • Jordan Formulation, H-Hilbert Spaces and Groups
  • Vector Products, Parallelisms and Quaternionic Manifolds
  • Quaternionic Function Theory
  • Arithmetics of Quaternions
  • Selected Physical Applications
  • Historical Notes
• Octonions:
  • Algebraic Structures
  • Octonionic Hilbert Spaces, Exceptional Groups and Algebras
  • Vector Products, Parallelisms on S⁷ and Octonionic Manifolds
  • Octonionic Function Theory
  • Arithmetics of Octonions
  • Some Physical Applications
  • Historical Notes
• Division, Jordan Algebras and Extended Objects:
  • Dyson's 3-Fold Way: Time Reversal and Berry Phases
  • Essential Hopf Fibrations and D≥3 Anyonic Phenomena
  • The Super-Poincaré Group and Super Extended Objects
• References
• Index