A Discrete Version of Plane Wave Solutions of the Dirac Equation in the Joyce Form

Abstract

We construct a discrete version of the plane wave solution to a discrete Dirac-Kähler 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.

2D discrete Hodge-Dirac operator on the torus

“…Recently, in [6], it was described a quite general framework of discrete calculus based on a new type of discrete geometry called script geometry. The proposed approach in this article was introduced by Dezin [7] and later developed in the author's previous papers [13,14,15,16,17,18].…”

2D Discrete Hodge–Dirac Operator on the Torus

“…In this article, we follow the approach that was initially introduced by Dezin [13] and later further developed in the author's previous papers [14][15][16][17]. We present a discretization scheme using cochains over rectangular meshes as the discrete representation of differential forms.…”

“…The similar equality holds in the case r = 1. It should be noted that the relation(17) includes not only the forms with the components r ϕ k,s and r+1 ω k,s , where the subscripts k, s would run only over the values from (14k = 1, 2, ..., N.…”