  Casimir forces are conventionally computed by analyzing the effects of
boundary conditions on a fluctuating quantum field. Although this analysis
provides a clean and calculationally tractable idealization, it does not always
accurately capture the characteristics of real materials, which cannot
constrain the modes of the fluctuating field at all energies. We study the
vacuum polarization energy of renormalizable, continuum quantum field theory in
the presence of a background field, designed to impose a Dirichlet boundary
condition in a particular limit. We show that in two and three space
dimensions, as a background field becomes concentrated on the surface on which
the Dirichlet boundary condition would eventually hold, the Casimir energy
diverges. This result implies that the energy depends in detail on the
properties of the material, which are not captured by the idealized boundary
conditions. This divergence does not affect the force between rigid bodies, but
it does invalidate calculations of Casimir stresses based on idealized boundary
conditions.
