Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories

TL;DR

This paper examines the structure of the Hilbert space in noncommutative geometry models of gauge theories, highlighting fermion doubling issues and exploring potential solutions that face physical or geometric challenges.

Contribution

It identifies fermion doubling problems in noncommutative geometry gauge models and evaluates projection methods, revealing limitations in current approaches.

Findings

Fermion doubling occurs in noncommutative geometry models.

Projection attempts either remove physical states or distort geometry.

The issue remains unresolved within existing frameworks.

Abstract

In this paper we study the structure of the Hilbert space for the recent noncommutative geometry models of gauge theories. We point out the presence of unphysical degrees of freedom similar to the ones appearing in lattice gauge theories (fermion doubling). We investigate the possibility of projecting out these states at the various levels in the construction, but we find that the results of these attempts are either physically unacceptable or geometrically unappealing.