Path Integral over Reparametrizations: Levy Flights versus Random Walks

TL;DR

This paper explores the path integral over reparametrizations in string theory, revealing that typical trajectories resemble Levy flights with fractal properties, and confirms their relevance to QCD scattering amplitudes through numerical simulations.

Contribution

It demonstrates that the trajectories in the path integral are Levy flights with zero Hausdorff dimension, providing new insights into their structure and implications for string and QCD theories.

Findings

Trajectories are Levy flights, not Brownian motion.

Hausdorff dimension of trajectories is zero.

Numerical results support analytical predictions for QCD amplitudes.

Abstract

We investigate the properties of the path integral over reparametrizations (= the boundary value of the Liouville field in open string theory). Discretizing the path integral, we apply the Metropolis-Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Levy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby the discretization and heuristic consideration of QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective-string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.