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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 $\varphi$, and a
$U(1)_{B-L}$ gauge boson which is Higgsed by $\varphi$. We then note a natural
extension of this construction, which describes the $SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ Pati-Salam model. Finally, we discuss a simple and
natural Jordan generalization of the exterior algebra of differential forms.