Casimir Forces between Arbitrary Compact Objects

We develop an exact method for computing the Casimir energy between arbitrary compact objects, either dielectrics or perfect conductors. The energy is obtained as an interaction between multipoles, generated by quantum current fluctuations. The objects’ shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly. A low frequency expansion yields the energy as a series in the ratio of the objects’ size to their separation. As an example, we obtain this series for two dielectric spheres and the full interaction at all separations for perfectly conducting spheres.

Figure 1

Casimir energy of two metal spheres, divided by the PFA estimate ${\mathcal{E}}_{\mathrm{PFA}}=-({\pi }^3/1440)\hslash cR/(L-2R{)}^2$, which holds only in the limit $R/L\to 1/2$. The label $l$ denotes the multipole order of truncation. The curves $l=\infty$ are obtained by extrapolation. The Casimir-Polder curve is the leading term of Eq. (13). Inset: Convergence at short separations.