Boundary conditions and the entropy bound

TL;DR

This paper investigates how boundary conditions affect the entropy-to-energy bound for a quantum scalar field, revealing potential violations near specific boundary parameter values.

Contribution

It provides a detailed analysis of Robin boundary conditions, explicitly computes the relevant zeta function, and identifies parameter regions where the entropy bound is violated.

Findings

Bound is violated near certain boundary parameters.

Explicit calculation of the zeta function at these points.

Constraints on boundary interactions are proposed.

Abstract

The entropy-to-energy bound is examined for a quantum scalar field confined to a cavity and satisfying Robin condition on the boundary of the cavity. It is found that near certain points in the space of the parameter defining the boundary condition the lowest eigenfrequency (while non-zero) becomes arbitrarily small. Estimating, according to Bekenstein and Schiffer, the ratio $S/E$ by the $\zeta$-function, $(24\zeta (4))^{1/4}$, we compute $\zeta (4)$ explicitly and find that it is not bounded near those points that signals violation of the bound. We interpret our results as imposing certain constraints on the value of the boundary interaction and estimate the forbidden region in the parameter space of the boundary conditions.