Pathwise differentiation of worldline path integrals

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 atom-plane and plane-plane geometries. Being quite general, the methods here should apply to path integrals outside the worldline context (e.g., financial mathematics).