Anomalous Paths in Quantum Mechanical Path-Integrals

TL;DR

This paper explores how certain lattice modifications in quantum path integrals, which vanish in the classical limit, can significantly alter the geometry of quantum paths, leading to fractal, superdiffusive, or Lévy flight behaviors.

Contribution

It introduces novel lattice actions that induce non-trivial quantum path geometries, including paths with arbitrary fractal dimensions and superdiffusive properties.

Findings

Paths can have fractal dimensions between 1 and 2.

A critical fractal dimension of 2 is identified.

Certain modifications lead to superdiffusive Lévy flights.

Abstract

We investigate modifications of the discrete-time lattice action, for a quantum mechanical particle in one spatial dimension, that vanish in the na\"ive 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 L\'{e}vy flights. We also consider the jaggedness of the paths, and show that this gives an independent classification of lattice theories.