The standard model, the Pati-Salam model, and "Jordan geometry"

TL;DR

This paper proposes replacing the traditional algebra of spacetime coordinates with a Jordan algebra, offering a new geometric framework that naturally extends the standard model and the Pati-Salam model in particle physics.

Contribution

It introduces a Jordan algebra-based framework called 'Jordan geometry' that models the standard model and its extension, providing a novel algebraic approach to particle physics theories.

Findings

Jordan algebra describes an extension of the standard model with sterile neutrinos and a $U(1)_{B-L}$ gauge boson.

The framework naturally extends to the Pati-Salam model.

A Jordan generalization of differential forms is proposed.

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.