Zero-point energy differences and many-body dispersion forces

The fully retarded dispersion interaction potentials, including many-body interactions, among neutral molecules are found in a systematic way. The method used relates the total zero-point energy of all the electromagnetic modes with the spectral sum of a linear operator. The difference between the zero-point energies with and without the molecules present is given as a contour integral. From the value of this integral it is possible to extract the N-body dispersion energy by locating those terms which depend on the product of the polarizabilities of those N molecules. The Casimir-Polder pairwise energy is the two-body result. General formulas are found, and the special cases for N=3 and 4 are discussed in detail. The nonretarded interaction potentials are found as the asymptotic limits for small intermolecular separations, and the London and the Axilrod-Teller results are the N=2 and N=3 special cases. The N=4 near-zone limit is presented in its explicit form. It is of interest to note that, for the one-body case, the energy shift given by this method is the nonrelativistic Lamb shift.