Casimir energy for a Regular Polygon with Dirichlet Boundaries

TL;DR

This paper calculates the Casimir energy for scalar fields in regular polygons with Dirichlet boundaries, deriving an expansion in 1/N, comparing with known geometries, and extending to higher-dimensional spaces and polygonal cylinders.

Contribution

It introduces a novel expansion for eigenvalues of polygons in terms of circle eigenvalues and applies it to compute Casimir energies for polygons and related geometries.

Findings

Eigenvalues expanded in 1/N series involving zeta functions

Casimir energy for polygons compared with square case

Extension of results to higher-dimensional spaces and polygonal cylinders

Abstract

We study the Casimir energy of a scalar field for a regular polygon with N sides. The scalar field obeys Dirichlet boundary conditions at the perimeter of the polygon. The polygon eigenvalues $\lambda_N$ are expressed in terms of the Dirichlet circle eigenvalues $\lambda_C$ as an expansion in $\frac{1}{N}$ of the form, $\lambda_N = \lambda_C (1 + \frac{4\zeta(2)}{N^2} + \frac{4\zeta(3)}{N^3} + \frac{28\zeta(4)}{N^4}+...)$. A comparison follows between the Casimir energy on the polygon with N=4 found with our method and the Casimir energy of the scalar field on a square. We generalize the result to spaces of the form $R^d\times P_N$, with $P_N$ a N-polygon. By the same token, we find the electric field energy for a "cylinder" of infinite length with polygonal section. With the method we use and in view of the results, it stands to reason to assume that the Casimir energy of $D$-balls has the same sign with the Casimir energy of regular shapes homeomorphic to the $D$-ball. We sum up and discuss our results at the end of the article.