Scalar Casimir Energies of Tetrahedra

TL;DR

This paper calculates the scalar Casimir energies for three specific tetrahedral cavities, providing explicit mode summation results and analyzing their dependence on geometric features under different boundary conditions.

Contribution

It presents the first direct evaluation of Casimir energies for tetrahedral cavities using mode summation with known eigenmodes, including analysis of divergences and geometric dependence.

Findings

Correct Weyl divergences obtained, confirming mode counting accuracy.

Finite quantum energies extracted unambiguously due to lack of curvature.

Systematic behavior of energies in relation to geometric invariants explored.

Abstract

New results for scalar Casimir self-energies arising from interior modes are presented for the three integrable tetrahedral cavities. Since the eigenmodes are all known, the energies can be directly evaluated by mode summation, with a point-splitting regulator, which amounts to evaluation of the cylinder kernel. The correct Weyl divergences, depending on the volume, surface area, and the corners, are obtained, which is strong evidence that the counting of modes is correct. Because there is no curvature, the finite part of the quantum energy may be unambiguously extracted. Dirichlet and Neumann boundary conditions are considered and systematic behavior of the energy in terms of geometric invariants is explored.