The Standard Model as an extension of the noncommutative algebra of forms

TL;DR

This paper extends Connes' noncommutative geometry framework for the Standard Model by incorporating differential graded structures, enabling derivation of key axioms for the full model and a Lorentzian version.

Contribution

It introduces a differential graded algebra extension approach that derives Standard Model axioms and is compatible with the full model and Lorentzian geometry.

Findings

Axioms derived from algebra associativity in the differential graded setting.

Compatibility with the full Standard Model achieved.

Lorentzian version of the noncommutative geometry formulated.

Abstract

The Standard Model of particle physics can be deduced from a small number of axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra extension in the sense of Eilenberg. By doing so, they could deduce three axioms of the NCG Standard Model (i.e. order zero, order one and massless photon) from the single requirement that the extended algebra be associative. However, their approach was only applied to the finite algebra and fails the full model. By taking into account the differential graded structure of the algebra of noncommutative differential forms, we obtain a formulation where the same three axioms are deduced from the associativity of the extended differential graded algebra, but which is now also compatible with the full Standard Model. Finally, we present a Lorentzian version of the noncommutative geometry of the Standard Model and we show that the three axioms still hold if the four-dimensional manifold has a Lorentzian metric.