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∝∫dt∣
v
⃗
∣
α
) in the path integral (e.g. electrons moving in solids, or Brueckner's theory of nuclear matter) yields
d
H
=
α
α−1
if
α>2
and
d
H
=2
if
α≤2
. We discuss the relevance of fractal pathes in solid state physics and in
QFT
, in particular for the Wilson loop in
QCD
.