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

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. 1. Introduction The hypothesis of a discrete space and time has attracted the attention of physicists for different reasons [1] [3]: i) as a mathematical tool in order to get rid of the infinities in quantum field theories with the help of a cut-off in momentum space or, what is equivalent, a lattice for the space-time coordinates [7]. This model is similar to s...