Hausdorff dimension of a particle path in a quantum manifold
arXiv:1009.3267 · doi: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.
7 pages, 4 figure, updated version which matches that published on Physical Review D
References in corpus (10)
- Noncommutative geometry inspired charged black holes
- Noncommutative Black Hole Thermodynamics
- Non-commutative geometry inspired higher-dimensional charged, black holes
- Charged rotating noncommutative black holes
- Fractal properties of quantum spacetime
- Noncommutative Inspired Black Holes in Extra Dimensions
- Trace Anomaly in Quantum Spacetime Manifold
- The decay-time of non-commutative micro-black holes
- Un-spectral dimension and quantum spacetime phases
- Minimal Scales from an Extended Hilbert Space