Jaggedness of Path Integral Trajectories

TL;DR

This paper introduces the concept of jaggedness for path integral trajectories, showing it is scale invariant and useful for classifying relevant paths, especially in numerical simulations.

Contribution

It defines jaggedness as a new property of path integral trajectories, demonstrating its scale invariance, self-averaging, and utility in classifying relevant paths in the continuum limit.

Findings

Jaggedness is scale invariant and self-averaging.

Relevant paths in the continuum limit have jaggedness 1/2.

Jaggedness is useful for assessing trajectory algorithms in simulations.

Abstract

We define and investigate the properties of the jaggedness of path integral trajectories. The new quantity is shown to be scale invariant and to satisfy a self-averaging property. Jaggedness allows for a classification of path integral trajectories according to their relevance. We show that in the continuum limit the only paths that are not of measure zero are those with jaggedness 1/2, i.e. belonging to the same equivalence class as random walks. The set of relevant trajectories is thus narrowed down to a specific subset of non-differentiable paths. For numerical calculations, we show that jaggedness represents an important practical criterion for assessing the quality of trajectory generating algorithms. We illustrate the obtained results with Monte Carlo simulations of several different models.