The Standard Model, The Exceptional Jordan Algebra, and Triality

TL;DR

This paper explores the deep mathematical connections between the exceptional Jordan algebra, triality, and the standard model of particle physics, proposing a geometric interpretation involving complex octonions and unification theories.

Contribution

It reveals a novel relationship between the exceptional Jordan algebra's complexification and the standard model, linking fermion generations to $SO(8)$ triality and geometric structures.

Findings

Single fermion generation described by tangent space of complex octonionic projective plane

Three fermion generations related to $SO(8)$ triality

Potential geometric interpretation of standard model unification

Abstract

Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing relationship between the complexification of this algebra and the standard model of particle physics, its minimal left-right-symmetric $SU(3)\times SU(2)_{L}\times SU(2)_{R}\times U(1)$ extension, and $Spin(10)$ unification. This suggests a geometric interpretation, where a single generation of standard model fermions is described by the tangent space $(\mathbb{C}\otimes\mathbb{O})^{2}$ of the complex octonionic projective plane, and the existence of three generations is related to $SO(8)$ triality.