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.
|Original language||English (US)|
|Journal||Physical Review D - Particles, Fields, Gravitation and Cosmology|
|State||Published - Jan 14 2011|
ASJC Scopus subject areas
- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)