  The Casimir problem is usually posed as the response of a fluctuating quantum
field to externally imposed boundary conditions. In reality, however, no
interaction is strong enough to enforce a boundary condition on all frequencies
of a fluctuating field. We construct a more physical model of the situation by
coupling the fluctuating field to a smooth background potential that implements
the boundary condition in a certain limit. To study this problem, we develop
general new methods to compute renormalized one--loop quantum energies and
energy densities. We use analytic properties of scattering data to compute
Green's functions in time--independent background fields at imaginary momenta.
Our calculational method is particularly useful for numerical studies of
singular limits because it avoids terms that oscillate or require cancellation
of exponentially growing and decaying factors. To renormalize, we identify
potentially divergent contributions to the Casimir energy with low orders in
the Born series to the Green's function. We subtract these contributions and
add back the corresponding Feynman diagrams, which we combine with counterterms
fixed by imposing standard renormalization conditions on low--order Green's
functions. The resulting Casimir energy and energy density are finite
functionals for smooth background potentials. In general, however, the Casimir
energy diverges in the boundary condition limit. This divergence is real and
reflects the infinite energy needed to constrain a fluctuating field on all
energy scales; renormalizable quantum field theories have no place for ad hoc
surface counterterms. We apply our methods to simple examples to illustrate
cases where these subtleties invalidate the conclusions of the boundary
condition approach.
