The Casimir effect for pistons with transmittal boundary conditions

Abstract

This work focuses on the analysis of the Casimir effect for pistons subject to transmittal boundary conditions. In particular we consider, as piston configuration, a direct product manifold of the type $I\times N$ where $I$ is a closed interval of the real line and $N$ is a smooth compact Riemannian manifold. By utilizing the spectral zeta function regularization technique, we compute the Casimir energy of the system and the Casimir force acting on the piston. Explicit results for the force are provided when the manifold $N$ is a $d$-dimensional ball.Comment: 20 pages, LaTe

Casimir pistons with general boundary conditions

“…It is clear from the above definition that M has dimension D = d + 1. The piston configuration can be obtained from the manifold M following the construction detailed in [20,22]. The two chambers of the piston are realized by dividing the manifold M with a cross-sectional manifold N a at the point a ∈ (0, L).…”

“…In fact, suitable boundary conditions can be utilized to describe physical properties of real materials. Some results regarding the Casimir effect with general boundary conditions have been obtained, for instance, in [3] in the ambit of parallel plates and in [20,21,22] in regards to piston configurations. It is important to mention, for completeness, that real materials could be modeled by smooth potentials with compact support rather than boundaries (see e.g.…”

“…This work is mainly aimed at generalizing the results, obtained in [20,22], for the Casimir effect in piston configurations. We consider a piston configuration of the type I × N where I ⊂ R is a closed interval of the real line and N is a smooth compact Riemannian manifold with or without boundary ∂N .…”

Gluing Formula for Casimir Energies

“…In order to control these divergencies, different regularization techniques have been applied [4,5,8,14,16,17,29,35,40]. However, they typically lead to different divergencies which raises the question of their interpretation.…”

“…Many configurations, such as flat pistons at zero temperature [11][12][13][14][15][16][17][18][19][20] or finite temperatures [21][22][23][24], as well as curved pistons [25][26][27][28][29], have been analyzed on the basis of the spectrum of a Laplace-type operator associated with M 1 and M 2 . It is the aim of this article to introduce a completely new perspective on this type of analysis.…”