The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics

TL;DR

This paper explores how potential energy in electromagnetic systems appears, disappears, and reappears depending on the context, matter representation, and quantization, highlighting nuanced energy accounting in classical and quantum electrodynamics.

Contribution

It clarifies the conditions under which electromagnetic potential energy is present or absent in classical and quantum frameworks, especially regarding matter fields like the Dirac field.

Findings

Potential energy disappears when using standard electromagnetic field energy density.

Potential energy reappears when matter is modeled by the Dirac field.

In quantum electrodynamics, potential energy becomes an interaction term in the Hamiltonian.

Abstract

In electrostatics, we can use either potential energy or field energy to ensure conservation of energy. In electrodynamics, the former option is unavailable. To ensure conservation of energy, we must attribute energy to the electromagnetic field and, in particular, to electromagnetic radiation. If we adopt the standard energy density for the electromagnetic field, then potential energy seems to disappear. However, a closer look at electrodynamics shows that this conclusion actually depends on the kind of matter being considered. Although we cannot get by without attributing energy to the electromagnetic field, matter may still have electromagnetic potential energy. Indeed, if we take the matter to be represented by the Dirac field (in a classical precursor to quantum electrodynamics), then it will possess potential energy (as can be seen by examining the symmetric energy-momentum tensor of the Dirac field). Thus, potential energy reappears. Upon field quantization, the potential energy of the Dirac field becomes an interaction term in the Hamiltonian operator of quantum electrodynamics.