A Lorentzian version of the non-commutative geometry of the standard model of particle physics

TL;DR

This paper develops a Lorentzian non-commutative geometric framework for the standard model, addressing fermion doubling and enabling realistic neutrino mass modeling.

Contribution

It introduces a Lorentzian signature in non-commutative geometry, resolving fermion doubling and incorporating right-handed neutrinos with the see-saw mechanism.

Findings

Elimination of fermion doubling in Lorentzian signature

Formulation of the fermionic spectral action with right-handed neutrinos

Compatibility with neutrino mass generation mechanisms

Abstract

A formulation of the non-commutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes' internal space geometry so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the see-saw mechanism of neutrino mass generation.