Local Casimir Effect for a Scalar Field in Presence of a Point Impurity

TL;DR

This paper investigates the local Casimir effect for a scalar field with a point impurity, calculating the renormalized stress-energy tensor at any point away from the impurity using zeta regularization.

Contribution

It provides a detailed analysis of the local Casimir effect near a point impurity, extending previous work that focused only on global energy aspects.

Findings

Computed the renormalized vacuum expectation value of the stress-energy tensor at any point.

Extended the understanding of local Casimir effects in the presence of point impurities.

Used zeta regularization to handle divergences in the stress-energy tensor.

Abstract

The Casimir effect for a scalar field in presence of delta-type potentials has been investigated for a long time in the case of surface delta functions, modelling semi-transparent boundaries. More recently Albeverio, Cacciapuoti, Cognola, Spreafico and Zerbini [9,10,51] have considered some configurations involving delta-type potentials concentrated at points of $\mathbb{R}^3$; in particular, the case with an isolated point singularity at the origin can be formulated as a field theory on $\mathbb{R}^3\setminus \{\mathbf{0}\}$, with self-adjoint boundary conditions at the origin for the Laplacian. However, the above authors have discussed only global aspects of the Casimir effect, focusing their attention on the vacuum expectation value (VEV) of the total energy. In the present paper we analyze the local Casimir effect with a point delta-type potential, computing the renormalized VEV of the stress-energy tensor at any point of $\mathbb{R}^3\setminus \{\mathbf{0}\}$; to this purpose we follow the zeta regularization approach, in the formulation already employed for different configurations in previous works of ours (see [29-31] and references therein).