Abbreviated Journal Title
Phys. Lett. B
BROWNIAN-MOTION; DIMENSION; TIME; Physics, Multidisciplinary
We investigate modifications of the discrete-time lattice action, for a quantum mechanical particle in one spatial dimension, that vanish in the naive continuum limit but which, nevertheless, induce non-trivial effects due to quantum fluctuations. These effects are seen to modify the geometry of the paths contributing to the path-integral describing the time evolution of the particle, which we investigate through numerical simulations. In particular, we demonstrate the existence of a modified lattice action resulting in paths with any fractal dimension, d(f), between one and two. We argue that d(f) = 2 is a critical value, and we exhibit a type of lattice modification where the fluctuations in the position of the particle becomes independent of the time step, in which case the paths are interpreted as superdiffusive Levy flights. We also consider the jaggedness of the paths, and show that this gives an independent classification of lattice theories.
Physics Letters B
Grimsmo, Arne L.; Klauder, John R.; and Skagerstam, Bo-Sture K., "Anomalous paths in quantum mechanical path-integrals" (2013). Faculty Bibliography 2010s. 4057.