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

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