On the initial value formulation of classical electrodynamics

Abstract

We describe a seemingly un-noticed feature of the text-book Maxwell–Lorentz system of classical electrodynamics which challenges its formulation in terms of an initial value problem. For point-charges, even after appropriate renormalization, we demonstrate that most of the generic initial data evolves to develop singularities in the electromagnetic fields along the light cones of the initial charge positions. We provide explicit formulas for the corresponding fields, demonstrate how this phenomenon renders the initial value problem ill-posed, and show how such bad initial data can be ruled out by extra conditions in addition to the Maxwell constraints. These extra conditions, however, require knowledge of the history of the solution and, as we discuss, effectively turn the Maxwell–Lorentz system into a system of delay equations much like the Fokker–Schwarzschild–Tetrode equations. For extended charges such singular light fronts persist in a smoothened form and, as we argue, yield physically doubtful solutions. Our results also apply to some extent to expectation values of field operators in quantum field theory.