Hausdorff dimension of a particle path in a quantum manifold

TL;DR

This paper investigates the fractal properties of a quantum particle's path in a manifold influenced by quantum gravity, revealing how minimal length scales affect its Hausdorff dimension and self-similarity.

Contribution

It introduces the impact of quantum-gravity-induced minimal length on the Hausdorff dimension of quantum paths, linking quantum mechanics and manifold fluctuations.

Findings

Hausdorff dimension reflects quantum uncertainty and manifold fluctuations

Minimal length breaks the self-similarity of quantum paths

Hausdorff and spectral dimensions depend on resolution loss

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.