Path-integral solution of the one-dimensional Dirac quantum cellular automaton

TL;DR

This paper introduces an analytical solution to a path integral formulation of a one-dimensional Dirac quantum cellular automaton, connecting quantum automata with quantum walks and providing explicit solutions using Jacobi polynomials.

Contribution

It presents the first analytical solution of a discrete path integral for a Dirac quantum cellular automaton in one dimension, linking automata, quantum walks, and path integrals.

Findings

Analytical solution expressed in terms of Jacobi polynomials.

Establishes isomorphism between quantum automata and quantum walks.

Provides a new discrete path integral formulation for the Dirac automaton.

Abstract

Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.