Self-Stress on a Dielectric Ball and Casimir-Polder Forces

TL;DR

This paper refutes recent claims that the Casimir self-energy of a dielectric sphere begins with a linear term in permittivity deviation, demonstrating that such a term originates from bulk energy contributions and highlighting the importance of proper stress tensor integration.

Contribution

The paper clarifies the correct origin of the linear term in dielectric Casimir self-energy, correcting previous misconceptions and emphasizing the role of the transverse stress tensor.

Findings

Linear term in permittivity deviation is due to bulk energy, not self-stress.

Proper subtraction of bulk terms is essential for accurate Casimir energy.

Omission of transverse stress tensor integral can lead to incorrect conclusions.

Abstract

It has always been conventionally understood that, in the dilute limit, the Casimir energy of interaction between bodies or the Casimir self-energy of a dielectric body could be identified with the sum of the van der Waals or Casimir-Polder energies of the constituents of the bodies. Recently, this proposition for self-energies has been challenged by Avni and Leonhardt [Ann.\ Phys.\ {\bf 395}, 326 (2018)], who find that the energy or self-stress of a homogeneous dielectric ball with permittivity $\varepsilon$ begins with a term of order $\varepsilon-1$. Here we demonstrate that this cannot be correct. The only possible origin of a term linear in $\varepsilon-1$ lies in the bulk energy, that energy which would be present if either the material of the body, or of its surroundings, filled all space. Since Avni and Leonhardt correctly subtract the bulk terms, the linear term they find likely arises from their omission of an integral over the transverse stress tensor.