Measuring the Hausdorff Dimension of Quantum Mechanical Paths

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 $\alpha \le 2$. We discuss the relevance of fractal pathes in solid state physics and in $QFT$, in particular for the Wilson loop in $QCD$.

Comments: uuencoded and compressed shell archive file. 8 pages with 7 figures