Comments on the Sign and Other Aspects of Semiclassical Casimir Energies

TL;DR

This paper analyzes the semiclassical approach to Casimir energies, clarifying the role of periodic rays, subtractions, and boundary effects, and discusses how to estimate the sign of the Casimir energy without full calculations.

Contribution

It provides explicit methods for calculating semiclassical Casimir energies, relates zeta function regularization poles to spectral subtractions, and extends the analysis to manifolds with boundaries and deformations.

Findings

Semiclassical Casimir energy matches zeta function regularization results.

The sign of Casimir energy can be estimated from shortest periodic rays.

Boundary contributions depend on boundary conditions and geometry.

Abstract

The Casimir energy of a massless scalar field is semiclassically given by contributions due to classical periodic rays. The required subtractions in the spectral density are determined explicitly. The so defined semiclassical Casimir energy coincides with that obtained using zeta function regularization in the cases studied. Poles in the analytic continuation of zeta function regularization are related to non-universal subtractions in the spectral density. The sign of the Casimir energy of a scalar field on a smooth manifold is estimated by the sign of the contribution due to the shortest periodic rays only. Demanding continuity of the Casimir energy under small deformations of the manifold, the method is extended to integrable systems. The Casimir energy of a massless scalar field on a manifold with boundaries includes contributions due to periodic rays that lie entirely within the boundaries. These contributions in general depend on the boundary conditions. Although the Casimir energy due to a massless scalar field may be sensitive to the physical dimensions of manifolds with boundary, its sign can in favorable cases be inferred without explicit calculation of the Casimir energy.