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Hausdorff dimension of a particle path in a quantum manifold

Piero Nicolini^* and Benjamin Niedner^†,‡

Frankfurt Institute for Advanced Studies (FIAS), Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany

^*nicolini@th.physik.uni-frankfurt.de
^†Present address: Theoretical Physics Group, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom.
^‡bjn10@imperial.ac.uk

Phys. Rev. D 83, 024017 – Published 14 January, 2011

DOI: https://doi.org/10.1103/PhysRevD.83.024017

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.

Article Text

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