A Discrete Dirac–Kähler Equation Using a Geometric Discretisation Scheme

Abstract

Discrete models of the Dirac-Kähler equation and the Dirac equation in the Hestenes form are discussed. A discrete version of the plane wave solutions to a discrete analogue of the Hestenes equation is established.

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

“…A discrete analogue of the Hodge star operator is one of the main distinctive features of the present discretisation scheme as compared to [17]. In [17], to define a discrete counterpart of the Hodge star the combinatorial double complex construction which is too awkward and some unclear geometrically is used. Now we define the operation * : K r (4) → K 4−r (4) for an arbitrary basis element s k = s k0 ⊗ s k1 ⊗ s k2 ⊗ s k3 by the rule…”

“…Note that replacing ∪ by ∧ in (2.16) gives the usual definition of the Hodge star on Λ r (M ). Hence (2.16) defines a discrete analogue of * which is more natural than the one defined in [17] (2.20)…”

“…As a new result, we announce here the construction of a discrete plane wave solution based on both forward and backward difference operators. This suggests that such a discrete model is more close to the continuum counterpart as the comparison with [17,18]. The clear intrinsically defined difference representations of the main equations and an explicit formula for the general solution are an adventure of our discretisation method.…”

“…This means that the geometric properties and the algebraic relationships between the differential, the exterior product, and the Hodge star operator are expected to be captured in the case of their discrete counterparts. This work is a continuation of that studied in the papers [15][16][17][18][19]. In this paper, we are mainly interested in a discrete model in which both forward and backward differences operators are used.…”

“…In [18], the same problem was considered by using a discretisation scheme based on a combinatorial double complex construction. Such a discretisation method was also applied in the paper [17] to construct a similar solution for a discrete Dirac-Kähler equation in the Hestenes form. However, the double complex construction generates difference operators of the forward type only and it is not enough to obtain an exact geometric counterpart of the Clifford algebra C (1,3).…”