Pathwise Differentiation of Worldline Path Integrals

TL;DR

This paper introduces novel methods for accurately computing derivatives of worldline path integrals, enabling precise calculations of forces, energies, and torques in Casimir physics and potentially other fields.

Contribution

It presents new techniques for differentiating worldline path integrals efficiently, including reweighting and partial-averaging methods, applicable to Casimir force and energy calculations.

Findings

Methods achieve high accuracy in derivative computations.

Numerical results demonstrate efficiency in atom-plane and plane-plane geometries.

Techniques are generalizable beyond worldline applications.

Abstract

The worldline method is a powerful numerical path-integral framework for computing Casimir and Casimir-Polder energies. An important challenge arises when one desires derivatives of path-integral quantities--standard finite-difference techniques, for example, yield results of poor accuracy. In this work we present methods for computing derivatives of worldline-type path integrals of scalar fields to calculate forces, energy curvatures, and torques. In Casimir-Polder-type path integrals, which require derivatives with respect to the source point of the paths, the derivatives can be computed by a simple reweighting of the path integral. However, a partial-averaging technique is necessary to render the differentiated path integral computationally efficient. We also discuss the computation of Casimir forces, curvatures, and torques between macroscopic bodies. Here a different method is used, involving summing over the derivatives of all the intersections with a body; again, a different partial-averaging method makes the path integral efficient. To demonstrate the efficiency of the techniques, we give the results of numerical implementations of these worldline methods in atomplane and plane-plane geometries. Being quite general, the methods here should apply to path integrals outside the worldline context (e.g., financial mathematics).