Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics

Abstract

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F_4 of J and its maximal Borel - de Siebenthal subgroups are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in F_4 is demonstrated to coincide with the gauge group of the Standard Model of particle physics. The first generation's fundamental fermions form a basis of primitive idempotents in the euclidean extension of the Jordan subalgebra JSpin_9 of J.

arXiv commentFinal version - new section 5 and references added, typos corrected; 51 pages, 2 figures

References in corpus (6)

• Division Algebras and Quantum Theory
• Exceptional quantum geometry and particle physics
• Grand Unification in the Spectral Pati-Salam Model
• Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry
• The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6
• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems

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