Measuring the Hausdorff Dimension of Quantum Mechanical Paths

TL;DR

This paper uses Monte Carlo simulations to measure the Hausdorff dimension of quantum paths, revealing universality classes and implications for quantum field theory and solid state physics.

Contribution

It introduces a method to determine the Hausdorff dimension of quantum paths and uncovers a universal value of 2 for local potentials, extending to velocity-dependent actions.

Findings

Hausdorff dimension is 2 for local potentials

Velocity-dependent actions yield a dimension of α/(α-1) for α > 2

Implications for fractal paths in quantum field theory and solid state physics

Abstract

We measure the propagator length in imaginary time quantum mechanics by Monte Carlo simulation on a lattice and extract the Hausdorff dimension $d_{H}$. We find that all local potentials fall into the same universality class giving $d_{H}=2$ like the free motion. A velocity dependent action ($S \propto \int dt \mid \vec{v} \mid^{\alpha}$) in the path integral (e.g. electrons moving in solids, or Brueckner's theory of nuclear matter) yields $d_{H}=\frac{\alpha }{\alpha - 1}$ if $\alpha > 2$ and $d_{H}=2$ if $\alpha \leq 2$. We discuss the relevance of fractal pathes in solid state physics and in $QFT$, in particular for the Wilson loop in $QCD$.