Weyl problem and Casimir effects in spherical shell geometry

TL;DR

This paper analyzes the spectral effects of inserting spherical shells into vacuum, resolving controversies about the universality of Casimir self-energy and connecting it to the Weyl spectral problem, with implications for quantum fields and magnons.

Contribution

It demonstrates the cutoff dependence of Casimir self-energy for scalar fields and its universality for electromagnetic fields, linking spectral theory to Casimir effects in spherical geometries.

Findings

Casimir self-energy for scalar fields depends on cutoff.

Electromagnetic Casimir self-energy is universal.

Non-relativistic Casimir effect due to magnons observed.

Abstract

We compute the generic mode sum that quantifies the effect on the spectrum of a harmonic field when a spherical shell is inserted into vacuum. This encompasses a variety of problems including the Weyl spectral problem and the Casimir effect of quantum electrodynamics. This allows us to resolve several long-standing controversies regarding the question of universality of the Casimir self-energy; the resolution comes naturally through the connection to the Weyl problem. Specifically we demonstrate that in the case of a scalar field obeying Dirichlet or Neumann boundary conditions on the shell surface the Casimir self-energy is cutoff-dependent while in the case of the electromagnetic field perturbed by a conductive shell the Casimir self-energy is universal. We additionally show that an analog non-relativistic Casimir effect due to zero-point magnons takes place when a non-magnetic spherical shell is inserted inside a bulk ferromagnet.