Global vs local Casimir effect

TL;DR

This paper investigates the Casimir effect using algebraic quantum field theory, analyzing how boundary conditions influence the global force and local energy density, revealing divergences and universal limits in different boundary scenarios.

Contribution

It introduces a controlled approximation of boundary conditions via interaction models and analyzes the resulting effects on Casimir forces and energy densities in the sharp boundary limit.

Findings

Neumann case: total force diverges in the sharp boundary limit.

Dirichlet case: local energy density has a universal limit.

Discrepancy in force behavior is due to order of limits and integration.

Abstract

This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH) we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.