Van der Waals chain: a simple model for Casimir forces in dielectrics

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.