The standard model, the exceptional Jordan algebra, and triality

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) × SU(2)L × SU(2)R × 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 (C⊗O)2 of the complex octonionic projective plane, and the existence of three generations is related to SO(8) triality.