Hausdorff dimension of a particle path in a quantum manifold

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.

Figure 1

Construction of the Koch curve. At each step, the middle third of each interval is replaced by the other two sides of an equilateral triangle.

Figure 2

Schematic view of the geometrical structure of the particle path. In (a) we have the classical regime and in (b) the quantum-mechanical regime. Upon further magnification in (c), the path exhibits the same structure.

Figure 3

The Hausdorff dimension of a particle path depending on $\Delta x/\ell$ for different numbers of spacetime dimensions.

Figure 4

Schematic view of the geometrical structure of the quantum particle path in (a) the classical regime, (b) the quantum-mechanical regime, (c),(d) the Planckian regime, and (e) the trans-Planckian regime.