Geometrical interpretation of the Casimir effect

TL;DR

This paper presents a geometric approach to calculating Casimir forces, clarifies the universal and non-universal contributions in various geometries, and provides explicit calculations and interpretations for different boundary configurations.

Contribution

It offers a new geometric interpretation and efficient computation method for Casimir effects across diverse boundary geometries, including curved cases.

Findings

Universal force in planar geometry is finite and well-defined.

Divergent non-universal terms are linked to boundary self-energy.

Explicit calculations distinguish universal and non-universal Casimir contributions.

Abstract

Casimir forces are a manifestation of the change in the zero-point energy of the vacuum caused by the insertion of boundaries. We show how the Casimir force can be efficiently computed by consideration of the vacuum fluctuations that are suppressed by the boundaries, and rederive the scalar Casimir effects for a series of the Dirichlet geometries. For the planar case a finite universal force is automatically found. Consistent with other calculations of the effect, for curved geometries divergent (non-universal) expressions are encountered. They are interpreted geometrically following Candelas and Deutsch (1979) as largely due to the divergent self-energy of the boundary contributing to the force. This viewpoint is supported by explicit calculations for a wedge-circular arc geometry in two dimensions where non-universal and universal contributions into the effect can be unambiguously separated. We also give a heuristic derivation of the purely geometrical expression (Sen, 1981) for the non-universal piece of the Casimir energy due to an arbitrary smooth two-dimensional Dirichlet boundary of a compact region.