  Vacuum polarization (or Casimir) energies can be straightforwardly computed
from scattering data for static field configurations whose interactions with
the fluctuating field are frequency independent. In effective theories,
however,such interactions are typically frequency dependent. As a consequence,
the relationship between scattering data and the Green's function is modified,
which may or may not induce additional contributions to the vacuum polarization
energy. We discuss several examples that naturally include frequency dependent
interactions: (i) scalar electrodynamics with a static background potential,
(ii) an effective theory that emerges from integrating out a heavy degree of
freedom, and (iii) quantum electrodynamics coupled to a frequency dependent
dielectric material. In the latter case, we argue that introducing dissipation
as required by the Kramers-Kronig relations requires the consideration of the
Casimir energy within a statistical mechanics formalism, while in the absence
of dissipation we can work entirely within field theory, using an alternative
formulation of the energy density.
