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

TL;DR

This paper explores the mathematical structure of the exceptional Jordan algebra and the group F_4, revealing their significance in classifying fundamental particles and gauge groups in particle physics.

Contribution

It provides a detailed analysis of the automorphism group F_4 of the exceptional Jordan algebra and its application to particle classification and Standard Model gauge groups.

Findings

F_4's intersection with gauge groups matches the Standard Model

Fundamental fermions form a basis of primitive idempotents in a Jordan subalgebra

The exceptional Jordan algebra relates to particle classification

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.