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

Charles Sebens (California Institute of Technology, California Institute of Technology)

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Abstract: In electrostatics, we can use either potential energy or field energy to ensure conservation of energy. In full 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.  Poynting’s theorem shows that changes in electromagnetic field energy balance the work done on matter by the field, and thus that energy is conserved in electromagnetic interactions.  Pausing here, potential energy seems to have disappeared.  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.  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.

This analysis of energy in classical and quantum field theories sets the stage for discussing a puzzle of self-interaction.  In classical electrodynamics, the electron can be modeled as a spread-out distribution of energy and charge in the Dirac field.  This classical model of the electron avoids some of the problems of self-interaction facing point charge models.  However, the electron will experience self-repulsion.  This self-repulsion cannot be eliminated within classical field theory without also losing Coulomb interactions between distinct particles.  But, electron self-repulsion can be eliminated from quantum electrodynamics in the Coulomb gauge by fully normal-ordering the Coulomb term in the Hamiltonian.  After normal-ordering, the Coulomb term contains pieces describing attraction and repulsion between distinct particles and also pieces describing particle creation and annihilation, but no pieces describing self-repulsion.