Representations of the discrete inhomogeneous Lorentz group and Dirac wave equation on the lattice

TL;DR

This paper develops lattice-based representations of the Lorentz group and derives a Dirac wave equation for spin 1/2 particles on a discrete spacetime lattice, bridging continuous and discrete symmetries.

Contribution

It introduces a novel lattice representation of the Lorentz group and derives the Dirac equation within this discrete framework, connecting it to the continuous case.

Findings

Constructed fundamental and two-dimensional Lorentz representations on a lattice

Derived the Dirac wave equation for spin 1/2 particles on the lattice

Connected discrete inhomogeneous Lorentz group representations with the continuous case

Abstract

We propose the fundamental and two dimensional representation of the Lorentz groups on a (3+1)-dimensional hypercubic lattice, from which representations of higher dimensions can be constructed. For the unitary representation of the discrete translation group we use the kernel of the Fourier transform. From the Dirac representation of the Lorentz group (including reflections) we derive in a natural way the wave equation on the lattice for spin 1/2 particles. Finally the induced representation of the discrete inhomogeneous Lorentz group is constructed by standard methods and its connection with the continuous case is discussed.