A New Construction for Spinor Wave Equations

TL;DR

This paper extends a discrete wave propagation framework to spinor fields in 1+1 dimensions, deriving the Dirac equation in different representations through elementary process symmetries.

Contribution

It introduces a novel discrete construction method for spinor wave equations, generalizing previous scalar wave approaches to include spinor fields.

Findings

Dirac equation in Majorana-Weyl representation derived

Standard Dirac equation obtained via symmetry relations

Framework applicable to inhomogeneous media and arbitrary dimensions

Abstract

The construction of discrete scalar wave propagation equations in arbitrary inhomogeneous media was recently achieved by using elementary dynamical processes realizing a discrete counterpart of the Huygens principle. In this paper, we generalize this approach to spinor wave propagation. Although the construction can be formulated on a discrete lattice of any dimension, for simplicity we focus on spinors living in 1+1 space-time dimensions. The Dirac equation in the Majorana-Weyl representation is directly recovered by incorporating appropriate symmetries of the elementary processes. The Dirac equation in the standard representation is also obtained by using its relationship with the Majorana-Weyl representation.