Vacuum and thermal fluctuations of a scalar field with point interactions

TL;DR

This paper analyzes vacuum and thermal fluctuations of a scalar field interacting with point-like obstacles, deriving explicit thermodynamic and Casimir energy expressions, and demonstrating that forces are always attractive.

Contribution

It introduces a rigorous self-adjoint Laplacian framework and a convergent Born series for Casimir energy, highlighting multiple-scattering effects and non-local interactions.

Findings

Casimir forces are always attractive for identical obstacles.

Explicit low- and high-temperature behaviors of thermodynamic observables are derived.

A convergent Born series expansion identifies multiple-scattering as the vacuum force mechanism.

Abstract

We investigate the vacuum and thermal fluctuations of a neutral massless scalar field living in Minkowski spacetime and interacting with a finite number of point-like obstacles, modelled by zero-range potentials. The system is described rigorously in terms of self-adjoint realizations of the Laplacian, under assumptions ensuring the absence of instabilities. Using the relative zeta-function technique, we determine the renormalized connected partition function and derive explicit expressions for the thermodynamic observables, characterizing both their low- and high-temperature behaviours. Furthermore, we derive of a convergent Born series expansion for the Casimir energy, which identifies multiple-scattering processes as the mechanism underlying vacuum forces. The latter decompose into pairwise contributions directed along the lines joining the obstacles, with intensities depending non-locally on the full configuration. We also present some numerical results for identical obstacles, indicating that the Casimir forces are always attractive in this context.