A Theory of the Casimir Effect for Compact Regions

TL;DR

This paper develops a rigorous mathematical framework for the Casimir effect, focusing on regularization-independent contributions and exploring specific geometries like plates and spheres, with insights into the cube case.

Contribution

It introduces a precise theoretical approach to the Casimir effect, emphasizing the role of Ramanujan sums and proposing an Ansatz for the cube's force.

Findings

Verified the hypothesis for parallel plates

Identified a cutoff-free version of the Casimir effect

Proposed an Ansatz for the cube's Casimir force

Abstract

We develop a mathematically precise framework for the Casimir effect. Our working hypothesis, verified in the case of parallel plates, is that only the regularization-independent Ramanujan sum of a given asymptotic series contributes to the Casimir pressure. As an illustration, we treat two cases: parallel plates, identifying a previous cutoff free version (by G. Scharf and W. W.) as a special case, and the sphere.We finally discuss the open problem of the Casimir force for the cube. We propose an Ansatz for the exterior force and argue why it may provide the exact solution, as well as an explanation of the repulsive sign of the force.