Solutions for the Klein-Gordon and Dirac equations on the lattice based on Chebyshev polynomials

TL;DR

This paper develops a Chebyshev polynomial-based method within a discrete Clifford calculus framework to solve lattice Klein-Gordon and Dirac equations, offering insights into lattice fermion doubling.

Contribution

It introduces a novel multivector calculus scheme using Chebyshev polynomials for discretizing Klein-Gordon and Dirac equations on the lattice.

Findings

Solutions for Dirac equations derived from Klein-Gordon solutions

Framework simplifies lattice fermion analysis

Implications for fermion doubling problem discussed

Abstract

The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein-Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein-Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.