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

Abstract

We construct a real-time lattice-gauge-theory (LGT)-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 LGT-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.

Keywords

• quantum walks
• lattice gauge theory
• quantum automata
• quantum field theory