The Dirichlet Casimir Problem

TL;DR

This paper investigates the limitations of traditional Casimir force calculations by analyzing the vacuum polarization energy in quantum field theory with background fields, revealing divergences that depend on material properties and challenge idealized boundary condition models.

Contribution

It demonstrates that imposing Dirichlet boundary conditions via background fields leads to divergences in Casimir energy in two and three dimensions, highlighting the importance of material properties.

Findings

Casimir energy diverges as boundary conditions become idealized.

Divergences depend on material properties, not captured by ideal boundary conditions.

Force calculations remain unaffected by these divergences.

Abstract

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.