Casimir effect due to a single boundary as a manifestation of the Weyl problem

TL;DR

This paper investigates the origin of divergences in the Casimir self-energy of a boundary, linking them to the Weyl problem and eigenvalue distribution, and clarifies how regularization methods relate to physical, measurable effects.

Contribution

It demonstrates that the Casimir self-energy can be decomposed into Weyl terms, a geometrical cutoff-independent term, and an intrinsic term, clarifying the nature of divergences and regularization.

Findings

Casimir self-energy includes Weyl and intrinsic terms.

Regularization fails when Weyl and intrinsic parts cannot be separated.

Resolved the divergence puzzle for Casimir force on a ring and corrected earlier sign errors.

Abstract

The Casimir self-energy of a boundary is ultraviolet-divergent. In many cases the divergences can be eliminated by methods such as zeta-function regularization or through physical arguments (ultraviolet transparency of the boundary would provide a cutoff). Using the example of a massless scalar field theory with a single Dirichlet boundary we explore the relationship between such approaches, with the goal of better understanding the origin of the divergences. We are guided by the insight due to Dowker and Kennedy (1978) and Deutsch and Candelas (1979), that the divergences represent measurable effects that can be interpreted with the aid of the theory of the asymptotic distribution of eigenvalues of the Laplacian discussed by Weyl. In many cases the Casimir self-energy is the sum of cutoff-dependent (Weyl) terms having geometrical origin, and an "intrinsic" term that is independent of the cutoff. The Weyl terms make a measurable contribution to the physical situation even when regularization methods succeed in isolating the intrinsic part. Regularization methods fail when the Weyl terms and intrinsic parts of the Casimir effect cannot be clearly separated. Specifically, we demonstrate that the Casimir self-energy of a smooth boundary in two dimensions is a sum of two Weyl terms (exhibiting quadratic and logarithmic cutoff dependence), a geometrical term that is independent of cutoff, and a non-geometrical intrinsic term. As by-products we resolve the puzzle of the divergent Casimir force on a ring and correct the sign of the coefficient of linear tension of the Dirichlet line predicted in earlier treatments.