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

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.

Automated summary

In this paper, we propose a real-time lattice gauge theory (LGT) 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, which ensures unitarity and locality, with transition amplitudes vanishing outside of a lightcone on the lattice. We also present a lattice Noether's theorem for internal symmetries of this action and couple it to an electromagnetic field. Furthermore, we suggest a real-time LGT-type action for the electromagnetic field in arbitrary spacetime dimensions, deriving its classical equations of motion as lattice versions of Maxwell's equations.