On the uniqueness of Barrett's solution to the fermion doubling problem in Noncommutative Geometry

TL;DR

This paper proves that Barrett's solution to the fermion doubling problem in Noncommutative Geometry is essentially unique under certain invariance conditions, and suggests a simplified alternative approach.

Contribution

It demonstrates the uniqueness of Barrett's solution under invariance assumptions and proposes a simplified modification to the fermionic action as an alternative.

Findings

Barrett's solution is unique up to trivial modifications.

A simple fermionic action modification can replace the explicit projection.

The solution requires the KO-dimension of the triple to be zero.

Abstract

A solution of the so-called fermion doubling problem in Connes' Noncommutative Standard Model has been given by Barrett in 2006 in the form of Majorana-Weyl conditions on the fermionic field. These conditions define a ${\cal U}_{J,\chi}$-invariant subspace of the correct physical dimension, where ${\cal U}_{J,\chi}$ is the group of Krein unitaries commuting with the chirality and real structure. They require the KO-dimension of the total triple to be $0$. In this paper we show that this solution is, up to some trivial modifications, and under some mild assumptions on the finite triple, the only one with this invariance property. We also observe that a simple modification of the fermionic action can act as a substitute for the explicit projection on the physical subspace.