Hausdorff dimension of a particle path in a quantum manifold
Abstract
After recalling the concept of the Hausdorff dimension, we study the fractal properties of a quantum particle path. As a novelty we consider the possibility for the space where the particle propagates to be endowed with a quantum-gravity-induced minimal length. We show that the Hausdorff dimension accounts for both the quantum mechanics uncertainty and manifold fluctuations. In addition the presence of a minimal length breaks the self-similarity property of the erratic path of the quantum particle. Finally we establish a universal property of the Hausdorff dimension as well as the spectral dimension: They both depend on the amount of resolution loss which affects both the path and the manifold when quantum gravity fluctuations occur.
Keywords
Cite
@article{arxiv.1009.3267,
title = {Hausdorff dimension of a particle path in a quantum manifold},
author = {Piero Nicolini and Benjamin Niedner},
journal= {arXiv preprint arXiv:1009.3267},
year = {2011}
}
Comments
7 pages, 4 figure, updated version which matches that published on Physical Review D