Lorenz gauge and the Poisson bracket in canonical electromagnetism

TL;DR

This paper clarifies how the canonical Poisson brackets are preserved in electromagnetism under Lorenz gauge, revealing a subtle interplay between gauge conditions and relativity that ensures their validity.

Contribution

It explicates the derivation of Poisson brackets in the Lorenz gauge, showing their consistency despite gauge constraints in canonical electromagnetism.

Findings

Poisson brackets remain canonical after Lorenz gauge fixing.

Gauge conditions and relativity work together to preserve canonical structure.

The naive assumption of Poisson brackets is justified even with gauge constraints.

Abstract

In treatments of electromagnetism, it is often tacitly assumed that the vector potentials of the field and their conjugate momenta satisfy the canonical Poisson bracket relations, despite the fact that the components of the vector potential are constrained by gauge conditions. Here I explicate how this comes about by imposing Poisson bracket relations on the independent field variables remaining after the Lorenz gauge constraint is accounted for. The naively assumed Poisson brackets happens to be correct even after gauge fixing, owing to a conspiracy between the gauge conditions and the principle of relativity.