Pathwise Differentiation of Worldline Path Integrals

• URL: http://arxiv.org/abs/2407.06470v2
• Date: Sun, 14 Jul 2024 22:12:32 GMT
• Title: Pathwise Differentiation of Worldline Path Integrals
• Authors: Jonathan B. Mackrory, He Zheng, Daniel A. Steck,
• Abstract summary: 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. We also discuss the computation of Casimir forces, curvatures, and torques between macroscopic bodies.
• Score: 1.1014050587473097
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

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