Measuring the Hausdorff Dimension of Quantum Mechanical Paths

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 ($S \propto \int dt \mid \vec{v} \mid^α$) in the path integral (e.g. electrons moving in solids, or Brueckner's theory of nuclear matter) yields $d_{H}=\frac{α}{α- 1}$ if $α> 2$ and $d_{H}=2$ if $α\leq 2$. We discuss the relevance of fractal pathes in solid state physics and in $QFT$, in particular for the Wilson loop in $QCD$.

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