Discrete spacetime and relativistic quantum particles

Abstract

We study a single quantum particle in discrete spacetime evolving in a causal way. We see that in the continuum limit any massless particle with a two dimensional internal degree of freedom obeys the Weyl equation, provided that we perform a simple relabeling of the coordinate axes or demand rotational symmetry in the continuum limit. It is surprising that this occurs regardless of the specific details of the evolution: it would be natural to assume that discrete evolutions giving rise to relativistic dynamics in the continuum limit would be very special cases. We also see that the same is not true for particles with larger internal degrees of freedom, by looking at an example with a three dimensional internal degree of freedom that is not relativistic in the continuum limit. In the process we give a formula for the Hamiltonian arising from the continuum limit of massless and massive particles in discrete spacetime.