Vacuum fluctuation and retardation effects on long-range potentials

The Casimir-Polder $R^{-7\mathrm{}}$ retarded potential between two neutral (polarizable) atoms at a separation $R$, which replaces the nonrelativistic $R^{-6\mathrm{}}$ van der Waals potential for $R$ sufficiently large, has been derived both by standard techniques and by assuming the atoms to be within a large volume with conducting walls and considering vacuum-fluctuation effects. The latter technique is here used to provide simple derivations of the $R^{-5\mathrm{}}$ potential between a charged point particle and a polarizable system—an addition to the usual $R^{-4\mathrm{}}$ polarization potential—and the $R^{-3\mathrm{}}$ two-photon contribution to the potential between two charged point particles. It also suggests a new, $R^{-2\mathrm{}}$ potential—an addition to the usual $R^{-1\mathrm{}}$ image potential—between a charged point particle and a conducting wall.