van der Waals Force, Dispersion Theory, and Singularities on Second Riemann Sheets

Previous work on the long-range electromagnetic forces between neutral particles is extended to obtain a more detailed understanding of the atomic or molecular van der Waals forces within the framework of covariant dispersion theory. It is shown that the modulation of the two-photon exchange potential $V_{2\gamma }$ from the London form $V_{2\gamma }\sim R^{-6\mathrm{}}$ to the Casimir-Polder form $V_{2\gamma }\sim R^{-7\mathrm{}}$ has an interesting dispersion-theoretic interpretation: It arises as a consequence of singularities in the momentum transfer $t$ of the scattering amplitude $F(s, t)$ in the physical region of $t$ but on an unphysical Riemann sheet. The connection of the present approach with that of Casimir and Polder is explained. The one-photon exchange potential $V_{1\gamma }$ is also studied for systems bound by either short-range or Coulomb forces. Some of the modification of the usual assumptions of dispersion theory required to deal with the latter case are described. An erroneous statement in the literature regarding $V_{1\gamma }$ (in the static limit) is pointed out. The character of $V_{1\gamma }$ and $V_{2\gamma }$ arising from photon exchange between elementary particles is described and contrasted with the atomic case. Some of the advantages of a covariant approach to the problem of interatomic forces are discussed.