Octonions, exceptional Jordan algebra and the role of the group F₄ in particle physics

2018/05/31 by Ivan Todorov, Svetla Drenska · 3 citations
Physics and Astronomy · Mathematics · #hep-th #math-ph #math.MP

paper · pdf · doi:10.1007/s00006-018-0899-y

published as Adv. Appl. Clifford Alg. 28 (2018) 82 · Final version - new section 5 and references added, typos corrected; 51 pages, 2 figures

arxiv created 2018/09/27 · arxiv updated 2018/09/28

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 F₄ 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 F₄ 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 JSpin₉ of J.

Cited by

• Higgs and Yukawa Structure in a Clifford Algebra Model with Three Generations and S₃ Family Symmetry
• Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry
• Octonions, complex structures and Standard Model fermions

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• Exceptional quantum algebra for the standard model of particle physics
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• Exceptional quantum geometry and particle physics