  QED perturbation theory has been conjectured to break down in sufficiently
strong backgrounds, obstructing the analysis of strong-field physics. We show
that the breakdown occurs even in classical electrodynamics, at lower field
strengths than previously considered, and that it may be cured by resummation.
As a consequence, an analogous resummation is required in QED. A detailed
investigation shows, for a range of observables, that unitarity removes
diagrams previously believed to be responsible for the breakdown of QED
perturbation theory.
