Atoms, Worldlines, and the Scalar Approximation

TL;DR

This paper critically examines the scalar approximation in the worldline path-integral method for Casimir-Polder interactions, revealing its limitations especially in multi-atom systems and emphasizing the need for more accurate models.

Contribution

The authors develop N-atom Casimir-Polder force expressions within the scalar approximation and demonstrate its failure in three-body systems due to polarization mixing.

Findings

Scalar approximation agrees with exact two-atom results by coincidence.

Scalar worldline method fails drastically for three-atom systems, predicting incorrect forces.

Discrepancies in polarization decomposition between worldline and Green-tensor formalisms.

Abstract

The worldline path-integral method, developed thus far for scalar fields, offers promising computational efficiency in general geometries, However, it relies so far on the scalar approximation that decomposes electromagnetic waves into two independent polarizations. In this work, we investigate different theoretical frameworks of fluctuation-induced effects and analyze the limitations of the worldline path-integral method in modeling multiple-atom Casimir-Polder interactions. In particular, we ask the question: how accurate is the scalar approximation? Using the worldline approach, it appears that a simple sum of the contributions from the two polarizations agrees with the exact Casimir-Polder force for two-atom systems. However, it turns out that this agreement is fortuitous. To enable calculations beyond two atoms via worldlines, we develop general N-atom expressions for the Casimir-Polder force within the scalar approximation. For three-body systems, the scalar worldline method fails drastically, predicting significant discrepancies in both magnitude and sign due to strong polarization mixing. Furthermore, we show that the TE/TM decomposition in the worldline method differs from that of the Green-tensor formalism, and we discuss why this is. This study highlights the inadequacy of scalar worldline models that rely on the polarization-decomposition approximation in general geometries.