MEASURING THE HAUSDORFF DIMENSION OF QUANTUM-MECHANICAL PATHS

We measure the propagator length in imaginary time quantum mechanics b y Monte Carlo simulation on a lattice and extract the Hausdorff dimens ion d(H). We find that all local potentials fall into the same univers ality class, giving d(H) = 2 as for the free motion. A velocity-depend ent term in the action of the path integral, which occurs for electron s moving in solid states or in Brueckner's theory of nuclear matter, h as been investigated. S proportional to integral dt\upsilon\(alpha) yi elds d(H) = alpha/(alpha - 1) if alpha > 2 and d(H) = 2 if alpha less than or equal to 2. We discuss the relevance of fractal paths in solid state physics and in QFT, in particular for the Wilson loop in QCD.