Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata

TL;DR

This paper introduces a simple quantum cellular automaton model for Dirac particles on a lattice, preserving key symmetries and connecting to continuum Weyl equations, with implications for lattice fermion theories.

Contribution

It presents a novel, simple quantum cellular automaton that models Dirac particles on a lattice, maintaining unitarity and chiral symmetry, and links automata to continuum Weyl equations.

Findings

Automaton preserves unitarity and chiral symmetry.

In the continuum limit, automata lead to Weyl equations.

Connection established between automata and lattice fermion theories.

Abstract

A discretized time evolution of the wave function for a Dirac particle on a cubic lattice is represented by a very simple quantum cellular automaton. In each evolution step the updated value of the wave function at a given site depends only on the values at the nearest sites, the evolution is unitary and preserves chiral symmetry. Moreover, it is shown that the relationship between Dirac particles and cellular automata operating on two component objects on a lattice is indeed very close. 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 its connection with path integrals and theories of fermions on a lattice is outlined.