The Standard Model, The Exceptional Jordan Algebra, and Triality

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.

6 pages, 1 figure, matches version in J. Math. Phys. This paper is dedicated to the memory of Ivan Todorov