The retarded van der Waals potential - revisited

TL;DR

This paper revisits the retarded van der Waals potential, deriving its asymptotic form using a full Hamiltonian and path integral methods, extending previous nonrelativistic approximations to all charge values.

Contribution

It provides a new derivation of the leading $-R^{-7}$ potential term using the full Hamiltonian and a path integral approach, without relying on the dipole approximation.

Findings

Derived the asymptotic $-R^{-7}$ potential strength for all charge values.

Used path integral representation to expand in $1/R$, not in coupling $e$.

Confirmed the validity of the potential beyond the small coupling regime.

Abstract

The retarded van-der-Waals potential, as first obtained by Casimir and Polder, is usually computed on the basis of nonrelativistic QED. The hamiltonian describes two infinitely heavy nuclei, charge $e$, separated by a distance $R$ and two spinless electrons, charge $-e$, nonrelativistically coupled to the quantized radiation field. Casimir and Polder use the dipole approximation and small coupling to the Maxwell field. We employ here the full hamiltonian and determine the asymptotic strength of the leading $-R^{-7}$ potential, which is valid for all $e$. Our computation is based on a path integral representation and expands in $1/R$, rather than in $e$.