On a modified electrodynamics

A modification of electrodynamics is proposed, motivated by previously unremarked paradoxes that can occur in the standard formulation. It is shown by specific examples that gauge transformations exist that radically alter the nature of a problem, even while maintaining the values of many measurable quantities. In one example, a system with energy conservation is transformed to a system where energy is not conserved. The second example possesses a ponderomotive potential in one gauge, but this important measurable quantity does not appear in the gauge-transformed system. A resolution of the paradoxes comes from noting that the change in total action arising from the interaction term in the Lagrangian density cannot always be neglected, contrary to the usual assumption. The problem arises from the information lost by employing an adiabatic cutoff of the field. This is not necessary. Its replacement by a requirement that the total action should not change with a gauge transformation amounts to a supplementary condition for gauge invariance that can be employed to preserve the physical character of the problem. It is shown that the adiabatic cutoff procedure can also be eliminated in the construction of quantum transition amplitudes, thus retaining consistency between the way in which asymptotic conditions are applied in electrodynamics and in quantum mechanics. The 'gauge-invariant electrodynamics' of Schwinger is shown to depend on an ansatz equivalent to the condition found here for maintenance of the ponderomotive potential in a gauge transformation. Among the altered viewpoints required by the modified electrodynamics, in addition to the rejection of the adiabatic cutoff, is the recognition that the electric and magnetic fields do not completely determine a physical problem, and that the electromagnetic potentials supply additional information that is required for completeness of electrodynamics.