A geometrical derivation of the Dirac equation

TL;DR

This paper presents a geometric derivation of the Dirac equation using a lattice model where a spin-1/2 particle moves at light speed, incorporating mass and external fields through wavefunction flips at lattice sites.

Contribution

It introduces a novel geometric lattice-based approach to derive the Dirac equation, extending from one to higher dimensions and including interactions with external fields.

Findings

Derivation of Dirac equation from lattice geometry

Extension to higher spatial dimensions

Inclusion of electromagnetic and gravitational interactions

Abstract

We give a geometrical derivation of the Dirac equation by considering a spin-1/2 particle travelling with the speed of light in a cubic spacetime lattice. The mass of the particle acts to flip the multi-component wavefunction at the lattice sites. Starting with a difference equation for the case of one spatial and one time dimensions, we generalize the approach to higher dimensions. Interactions with external electromagnetic and gravitational fields are also considered. One logical interpretation is that only at the lattice sites is the spin-1/2 particle aware of its mass and the presence of external fields.