How to Choose a Gauge? The case of Hamiltonian Electromagnetism

TL;DR

This paper explores gauge symmetry in Hamiltonian electromagnetism, demonstrating how the formalism helps decompose electric fields and justify the Coulomb gauge through Helmholtz decomposition.

Contribution

It introduces a Hamiltonian approach to gauge fixing in electromagnetism, linking electric field decomposition with gauge choices like the Coulomb gauge.

Findings

Hamiltonian formalism pairs electric and vector potential degrees of freedom.

Decomposition of electric field guides gauge fixing, especially Coulomb gauge.

The Coulomb gauge is justified via Helmholtz decomposition of the electric field.

Abstract

We develop some ideas about gauge symmetry in the context of Maxwell's theory of electromagnetism in the Hamiltonian formalism. One great benefit of this formalism is that it pairs momentum and configurational degrees of freedom, so that a decomposition of one side into subsets can be translated into a decomposition of the other. In the case of electromagnetism, this enables us to pair degrees of freedom of the electric field with degrees of freedom of the vector potential. Another benefit is that the formalism algorithmically identifies subsets of the equations of motion that represent time-dependent symmetries. For electromagnetism, these two benefits allow us to define gauge-fixing in parallel to special decompositions of the electric field. More specifically, we apply the Helmholtz decomposition theorem to split the electric field into its Coulombic and radiative parts, and show how this gives a special role to the Coulomb gauge (i.e. div$({\bf A}) = 0$). We relate this argument to Maudlin's (2018) discussion, which advocated the Coulomb gauge.