A single-particle framework for unitary lattice gauge theory in discrete time

TL;DR

This paper develops a unitary, local lattice gauge theory framework for a single-particle spin-1/2 system in (1+1) dimensions using quantum walks, including coupling to electromagnetic fields and deriving lattice Maxwell's equations.

Contribution

It introduces a novel single-particle lattice gauge theory framework based on quantum walks, with a lattice Noether's theorem and a method to couple to electromagnetic fields.

Findings

Constructed a unitary, local lattice action for a spin-1/2 particle.

Derived lattice Maxwell's equations from the proposed framework.

Provided a lattice Noether's theorem for internal symmetries.

Abstract

We construct a real-time lattice-gauge-theory-type action for a spin-1/2 matter field of a single particle on a (1+1)-dimensional spacetime lattice. The framework is based on a discrete-time quantum walk, and is hence inherently unitary and strictly local, i.e., transition amplitudes exactly vanish outside of a lightcone on the lattice. We then provide a lattice Noether's theorem for internal symmetries of this action. We further couple this action to an electromagnetic field by a minimal substitution on the lattice. Finally, we suggest a real-time lattice-gauge-theory-type action for the electromagnetic field in arbitrary spacetime dimensions, and derive its classical equations of motion, which are lattice versions of Maxwell's equations.