# A discrete version of plane wave solutions of the Dirac equation in the Joyce form

## TL;DR

This paper develops a discrete geometric scheme for plane wave solutions of the Dirac-Kähler equation in Joyce form, analyzing conditions for their validity in a discretized setting.

## Contribution

It introduces a novel discrete discretization scheme combining forward and backward differences for the Dirac-Kähler equation in Joyce form.

## Key findings

- A geometric discretization scheme for the Dirac-Kähler equation is proposed.
- Conditions for discrete plane wave solutions to satisfy the discrete Joyce equation are identified.
- The scheme bridges continuous and discrete formulations of Dirac equations.

## Abstract

We construct a discrete version of the plane wave solution to a discrete Dirac-K\"{a}hler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under which a discrete plane wave solution satisfies a discrete Joyce equation are discussed.