Discrete Differential Geometry and Lattice Field Theory

TL;DR

This paper develops a discrete differential calculus for lattice field theory, preserving Lorentz invariance explicitly, and applies it to fundamental equations like Maxwell, Klein-Gordon, and Dirac.

Contribution

It introduces a difference calculus analogous to differential geometry, ensuring Lorentz invariance in lattice field theories, which improves upon previous methods.

Findings

Lorentz invariance is preserved in lattice formulations

The approach applies to Maxwell, Klein-Gordon, and Dirac equations

Provides a new framework for discrete differential geometry in physics

Abstract

We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our approach has the advantage with respect to other attempts [Ref. 2-6] that the Lorentz invariance is automatically preserved as it can be seen explicitely in the Maxwell, Klein-Gordon and Dirac equations on the lattice.