  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.
