Principles of Discrete Time Mechanics: V. The Quantisation of Maxwell's Equations

TL;DR

This paper develops a framework for quantising Maxwell's equations within discrete time mechanics, analyzing gauge choices, boundary conditions, and deriving discrete analogues of photon propagators, revealing insights into the interplay between discreteness and relativity.

Contribution

It introduces a novel approach to quantising electromagnetism in discrete time, including gauge-specific vacuum functionals and propagators, and explores implications for early universe physics.

Findings

Complete agreement with continuous theory in Coulomb gauge

Discrete time photon propagators in Landau and Feynman gauges

Insights into discreteness effects on relativity and metric analogues

Abstract

Principles of discrete time mechanics are applied to the quantisation of Maxwell's equations. Following an analysis of temporal node and link variables, we review the classical discrete time equations in the Coulomb and Lorentz gauges and conclude that electro-magneto duality does not occur in pure discrete time electromagnetism. We discuss the role of boundary conditions in our mechanics and how temporal discretisation should influence very early universe dynamics. Quantisation of the Maxwell potentials is approached via the discrete time Schwinger action principle and the Faddeev-Popov path integral. We demonstrate complete agreement in the case of the Coulomb gauge, obtaining the vacuum functional and the discrete time field commutators in that gauge. Finally, we use the Faddeev-Popov method to construct the discrete time analogues of the photon propagator in the Landau and Feynman gauges, which casts light on the break with relativity and possible discrete time analogues of the metric tensor.