Casimir Energy of a Semi-Circular Infinite Cylinder

TL;DR

This paper calculates the Casimir energy for a semi-circular cylindrical shell using zeta function techniques, addressing boundary conditions and edge effects, and highlighting the need for renormalization.

Contribution

It provides exact constructions of zeta functions for scalar fields with boundary conditions on a semi-circular cylinder, extending Casimir energy calculations to this geometry.

Findings

Derived explicit zeta functions for scalar fields with Dirichlet and Neumann conditions.

Identified pole contributions due to edges and corners, indicating the necessity of renormalization.

Presented final expressions for Casimir energies in the semi-circular cylindrical geometry.

Abstract

The Casimir energy of a semi-circular cylindrical shell is calculated by making use of the zeta function technique. This shell is obtained by crossing an infinite circular cylindrical shell by a plane passing through the symmetry axes of the cylinder and by considering only a half of this configuration. All the surfaces, including the cutting plane, are assumed to be perfectly conducting. The zeta functions for scalar massless fields obeying the Dirichlet and Neumann boundary conditions on the semi-circular cylinder are constructed exactly. The sum of these zeta functions gives the zeta function for electromagnetic field in question. The relevant plane problem is considered also. In all the cases the final expressions for the corresponding Casimir energies contain the pole contributions which are the consequence of the edges or corners in the boundaries. This implies that further renormalization is needed in order for the finite physical values for vacuum energy to be obtained for given boundary conditions.