The vacuum energy with non-ideal boundary conditions via an approximate functional equation

TL;DR

This paper develops an approximate analytical method to compute vacuum and Casimir energies for scalar fields with non-ideal boundary conditions, using functional equations related to the Riemann zeta and Epstein zeta functions.

Contribution

It introduces an asymptotic expansion approach based on approximate functional equations to evaluate vacuum energies with non-ideal boundary conditions.

Findings

Derived vacuum energy expressions for scalar fields with non-ideal boundaries.

Extended the method to rectangular geometries using Epstein zeta-function approximations.

Provided a framework for analyzing conductivity corrections in Casimir energy calculations.

Abstract

We discuss the vacuum energy of a quantized scalar field in the presence of classical surfaces, defining bounded domains $\Omega \subset {\mathbb{R}}^{d}$, where the field satisfies ideal or non-ideal boundary conditions. For the electromagnetic case, this situation describes the conductivity correction to the zero-point energy. Using an analytic regularization procedure, we obtain the vacuum energy for a massless scalar field at zero temperature in the presence of a slab geometry $\Omega=\mathbb R^{d-1}\times[0, L]$ with Dirichlet boundary conditions. To discuss the case of non-ideal boundary conditions, we employ an asymptotic expansion, based on an approximate functional equation for the Riemann zeta-function, where finite sums outside their original domain of convergence are defined. Finally, to obtain the Casimir energy for a massless scalar field in the presence of a rectangular box, with lengths $L_{1}$ and $L_{2}$, i.e., $\Omega=[0,L_{1}]\times[0,L_{2}]$ with non-ideal boundary conditions, we employ an approximate functional equation of the Epstein zeta-function.