Close examination of the ground-state Casimir–Polder interaction: time-ordered versus covariant formalism and radiative corrections

The purpose of this paper is twofold. First, we compare, in detail, the derivation of the Casimir–Polder interaction using time-ordered perturbation theory, to the matching of the scattering amplitude using quantum electrodynamics. In the first case, a total of twelve time-ordered diagrams need to be considered, while in the second case, one encounters only two Feynman diagrams, namely, the ladder and crossed-ladder contributions. For ground-state interactions, we match the contribution of six of the time-ordered diagrams against the corresponding Feynman diagrams, showing the consistency of the two approaches. Second, we also examine the leading radiative correction to the long-range interaction, which is of relative order $O\left({\alpha }^3\right)$. In doing so, we uncover logarithmic terms, in both the interatomic distance as well as the fine-structure constant, in higher-order corrections to the Casimir–Polder interaction. (paper)