Irreducible Many-Body Casimir Energies of Intersecting Objects

The vacuum energy of a bosonic field interacting locally with objects is decomposed into irreducible $N$-body parts. The irreducible $N$-body contribution to the vacuum energy is finite if the common intersection $O_1\cap O_2...\cap O_N$ of all $N$ objects $O_i,i=1,..., N$ is empty. I prove that the perturbative expansion of the corresponding irreducible $N$-body spectral function $\tphi^{(N)}(\beta)$ for $\beta\sim 0$ vanishes to all orders even if some of the objects intersect. These irreducible spectral functions and their associated Casimir energies in principle can be computed numerically or approximated semiclassically without regularization or implicit knowledge of the spectrum. They are analytic in the parameters describing the relative orientation and position of the individual objects and remain finite when some, but not all, of the $N$ objects overlap. The Feynman-Kac theorem is used to compute Casimir energies of a massless scalar field with potential scattering and the finiteness of $N$-body Casimir energies is shown explicitly in this case. The irreducible $N$-body contributions to the vacuum energy of a massless scalar field with potential interactions is shown to be negative for an even- and positive for an odd- number of objects. Some simple examples are used to illustrate the analyticity of the $N$-body Casimir energy and its sign. A multiple scattering representation of the irreducible three-body Casimir energy is given. It remains finite when any two of the three objects overlap.