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

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 ${\mathcal{U}}_{J,\chi }$-invariant subspace of the correct physical dimension, where ${\mathcal{U}}_{J,\chi }$ 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.

References in corpus (6)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Why the Standard Model
• Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories
• Wick Rotation and Fermion Doubling in Noncommutative Geometry
• The Standard Model as an extension of the noncommutative algebra of forms
• Algebraic backgrounds: a framework for noncommutative Kaluza-Klein theory