When dealing with bound states (e.g., atoms or nuclei) many calculations are most efficiently carried out using old-fashioned perturbation theory. It is difficult, however, to see how gauge invariance manifests at the level of the amplitude (this is related to the non-conservation of energy in OFPT).
In this talk I will explain how a partial-fraction trick can reorganize amplitudes into a form that is manifestly gauge invariant. Remarkably, when applied to the Compton tensor, the resulting amplitude decomposes into four separate gauge invariant pieces that are naturally organized around a heavy-mass expansion. I will comment on potential implications for atomic, nuclear, and hadronic physics applications involving QED.