Seminarium KMMF "Teoria Dwoistości"

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 of Hermitian operators (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 phi, and a U(1)_{B-L} gauge boson which is Higgsed by phi. We then note a natural extension of this construction, which describes the Pati-Salam model of unification. Finally, we discuss a simple and natural Jordan generalization of the exterior algebra of differential forms.