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Abstract: After recalling the concept of the Hausdorff dimension, we study the fractalproperties of a quantum particle path. As a novelty we consider the possibilityfor the space where the particle propagates to be endowed with aquantum-gravity-induced minimal length. We show that the Hausdorff dimensionaccounts for both the quantum mechanics uncertainty and manifold fluctuations.In addition the presence of a minimal length breaks the self-similarityproperty of the erratic path of the quantum particle. Finally we establish auniversal property of the Hausdorff dimension as well as the spectraldimension: They both depend on the amount of resolution loss which affects boththe path and the manifold when quantum gravity fluctuations occur.



Author: Piero Nicolini, Benjamin Niedner

Source: https://arxiv.org/







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