A discrete model of the Dirac-K\"{a}hler equation

TL;DR

This paper develops a new discrete model of the Dirac-Kähler equation that preserves key geometric features of the continuum version, including gauge invariance, and analyzes its mathematical properties.

Contribution

It introduces a novel discrete formulation of the Dirac-Kähler equation with geometric fidelity and explores its decomposition and gauge transformation analogs.

Findings

Discrete Dirac-Kähler equation captures key geometric aspects

Decomposition into Duffin-type difference equations proven

Gauge transformation analogs for the discrete case studied

Abstract

We construct a new discrete analog of the Dirac-K\"{a}hler equation in which some key geometric aspects of the continuum counterpart are captured. We describe a discrete Dirac-K\"{a}hler equation in the intrinsic notation as a set of difference equations and prove several statements about its decomposition into difference equations of Duffin type. We study an analog of gauge transformations for the massless discrete Dirac-K\"{a}hler equations.