Hausdorff dimension of a particle path in a quantum manifold

Details Hausdorff dimension of a particle path in a quantum manifold Hausdorff dimension of a particle path in a quantum manifold

2010-09-16

arxiv.org/abs/1009.3267v2

Phys.Rev.D83:024017,2011

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

7 pages, 4 figure, updated version which matches that published on Physical Review D

Scientific paper

10.1103/PhysRevD.83.024017

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.

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