Determine whether the total vacuum electromagnetic field is exactly free

Determine a priori whether the total vacuum electromagnetic field generated in a background-field configuration is exactly free or non-free, particularly when the background field is constant or a plane wave, in order to resolve which renormalization prescription applies.

Background

The paper analyzes tadpole renormalization in strong-field quantum electrodynamics and shows that idealized free background fields require special care. When the total vacuum electromagnetic field satisfies the free Maxwell equations, manipulating the wave operator before renormalization can make the renormalization relation ambiguous and can eliminate the counterterm contributions needed for a finite result.

The authors prescribe treating the background as generated by a nonzero four-current during renormalization and taking the free-field limit only afterward. They explicitly note that, outside special cases such as constant and plane-wave fields, it may not be possible to know in advance whether the total vacuum field is exactly free.