  The Casimir force between dielectric bodies is well-understood, but not the
Casimir force inside a dielectric, in particular its renormalization. We
develop and analyse a simple model for the Casimir forces inside a medium that
is completely free of renormalization and show then how renormalization
emerges. We consider a one-dimensional chain of point particles interacting
with each other by scattering the zero-point fluctuations of the
electromagnetic field confined to one dimension. We develop a fast, efficient
algorithm for calculating the forces on each particle and apply it to study the
macroscopic limit of infinitely many, infinitely weak scatterers. The force
density converges for piece-wise homogeneous media, but diverges in
inhomogeneous media, which would cause instant collapse in theory. We argue
that short-range counter forces in the medium prevent this collapse in reality.
Their effect appears as the renormalization of the Casimir stress in
dielectrics. Our simple model also allows us to derive an elementary analogue
of the trace anomaly of quantum fields in curved space.
