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

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.