The Dirac Operator over Abelian Finite Groups

TL;DR

This paper constructs a Dirac operator on finite abelian groups, providing a mathematical foundation for lattice fermions that addresses the fermion doubling problem and maintains chiral symmetry.

Contribution

It introduces a novel algebraic framework generalizing Clifford algebras to define Dirac operators on abelian finite groups, linking to fermion doubling solutions.

Findings

Provides a Dirac operator construction on finite abelian groups

Establishes algebraic structures akin to square-roots of translation operators

Connects to approaches solving fermion doubling and preserving chiral symmetry

Abstract

In this paper we show how to construct a Dirac operator on a lattice in complete analogy with the continuum. In fact we consider a more general problem, that is, the Dirac operator over an abelian finite group (for which a lattice is a particular example). Our results appear to be in direct connexion with the so called fermion doubling problem. In order to find this Dirac operator we need to introduce an algebraic structure (that generalizes the Clifford algebras) where we have quantities that work as square-root of the translation operator. Quantities like these square-roots have been used recently in order to provide an approach to fermions on the lattice that is free from doubling and has chiral invariance in the massless limit, and our studies seem to give a mathematical basis to it.