The standard model, the Pati–Salam model, and ‘Jordan geometry’

Abstract

Abstract We argue that the ordinary commutative and associative algebra of spacetime coordinates (familiar from general relativity) should perhaps be replaced, not by a noncommutative algebra (as in noncommutative geometry), but rather by a Jordan algebra (leading to a framework which we term ‘Jordan geometry’). We present the Jordan algebra (and representation) that most nearly describes the standard model of particle physics, and we explain that it actually describes a certain (phenomenologically viable) extension of the standard model: by three right-handed (sterile) neutrinos, a complex scalar field φ, and a U(1) B−L gauge boson which is Higgsed by φ. We then note a natural extension of this construction, which describes the SU(4) × SU(2)L × SU(2)R Pati–Salam model. Finally, we discuss a simple and natural Jordan generalization of the exterior algebra of differential forms.