  The effects of quantum fluctuations in fields confined by background
configurations may be simply and transparently computed using the Green's
function approach pioneered by Schwinger. Not only can total energies and
surface forces be computed in this way, but local energy densities, and in
general, all components of the vacuum expectation value of the energy-momentum
tensor may be calculated. For simple geometries this approach may be carried
out exactly, which yields insight into what happens in less tractable
situations. In this talk I will concentrate on the example of a scalar field in
a circular cylindrical delta-function background. This situation is quite
similar to that of a spherical delta-function background. The local energy
density in these cases diverges as the surface of the background is approached,
but these divergences are integrable. The total energy is finite in strong
coupling, but in weak coupling a divergence occurs in third order. This
universal feature is shown to reflect a divergence in the energy associated
with the surface, the integrated local energy density within the shell itself,
which surface energy should be removable by a process of renormalization.
