Finite Pseudo-Riemannian Spectral Triples and The Standard Model

TL;DR

This paper extends the spectral triple framework to pseudo-Riemannian geometries over finite algebras, applying it to the Standard Model and revealing new geometric interpretations of physical symmetries.

Contribution

It introduces a pseudo-Riemannian generalization of spectral triples for finite geometries and connects this to the Standard Model, offering novel geometric insights.

Findings

Finite pseudo-Riemannian spectral triples can model the Standard Model.

The lepton number symmetry is interpreted as a shadow of a pseudo-Riemannian structure.

The formalism provides a new geometric perspective on physical symmetries.

Abstract

Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard Model spectral triple in this formalism and interpret the physical symmetry preserving the lepton number as a shadow of a finite pseudo-Riemannian structure.