Non-Commutative space-time and Hausdorff dimension

Abstract

We study the Hausdorff dimension of the path of a quantum particle in non-commutative space-time. We show that the Hausdorff dimension depends on the deformation parameter $a$ and the resolution $\Delta x$ for both non-relativistic and relativistic quantum particle. For the non-relativistic case, it is seen that Hausdorff dimension is always less than two in the non-commutative space-time. For relativistic quantum particle, we find the Hausdorff dimension increases with the non-commutative parameter, in contrast to the commutative space-time. We show that non-commutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space-time. By imposing self-similarity condition on the path of non-relativistic and relativistic quantum particle in non-commutative space-time, we derive the corresponding generalised uncertainty relation.

16 pages, 3 figures, minor changes, to appear in IJMPA

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• Gravitational tests of the Generalized Uncertainty Principle
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• Fractal properties of quantum spacetime
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• Diffusion on $\kappa$-Minkowski space
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• Hausdorff dimension of a particle path in a quantum manifold
• MICZ Kepler Systems in Noncommutative Space and Duality of Force Laws
• Dimensional flow in the kappa-deformed space-time

Cited by in corpus (4)

• Compact stars in quantum spacetime
• Emergence of maximal acceleration from non-commutativity of space-time
• Maximal acceleration in non-commutative space-time and its implications
• $\kappa$-deformed power spectrum and modified Unruh temperature