Vacuum Energy and Casimir Force in a Presence of Skin-depth Dependent Boundary Condition

TL;DR

This paper investigates how a skin-depth dependent boundary condition affects vacuum energy and Casimir force for a massive scalar field, revealing the importance of surface contributions and their relation to bulk energy.

Contribution

It introduces a novel boundary condition involving a skin-depth parameter and analyzes its impact on vacuum energy, including surface and bulk contributions, with a focus on their interplay and analytical properties.

Findings

Casimir energy depends on mass, cavity size, and skin-depth.

Surface potential energy dominates when cavity size is comparable to skin-depth.

Vacuum energy remains finite and analytically dependent on space dimension.

Abstract

The vacuum energy-momentum tensor (EMT) and the vacuum energy corresponding to massive scalar field on $\Re_{t}\times [0,l] \times \Re^{D-2}$ space-time with boundary condition involving a dimensional parameter ($\delta$) are found. The dependent on the cavity size $l$ Casimir energy $\wt E_{C}$ is a uniquely determinable function of mass $m$, size $l$ and "skin-depth" $\delta$. This energy includes the "bulk" and the surface (potential energy) contributions. The latter dominates when $l \sim \delta$. Taking the surface potential energy into account is crucial for the coincidence between the derivative $-\d \wt E_{C}/\d l$ and the $ll$-component of the vacuum EMT. Casimir energy $\wt E_C$ and the bulk contribution to it are interconnected through Legendre transformation, in which the quantity $\delta^{-1}$ is conjugate to the vacuum surface energy multiplied by $\delta$. The surface singularities of the vacuum EMT do not depend on $l$ and, for even $D$, $\delta =0$ or $\infty$, possess finite interpretation. The corresponding vacuum energy is finite and retains known analytical dependence on the dimension $D$.