Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata

Very simply unitary cellular automata on a cubic lattice are introduced to model a discretized time evolution of the wave functions for spinning particles. In each evolution step the undated value of the wave function at a given site depends only on the values at the nearest sites. The discretized evolution is also unitary and preserves chiral symmetry. The case of the spin-1/2 particle is studied in detail and it is shown that every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and is shown to reproduce the retarded propagator in the continuum limit. Generalizations to include massive particles (Dirac theory), spin-1 particles (Maxwell theory), and higher-spin particles are also described