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arXiv · 1704.07105

Non-Commutative space-time and Hausdorff dimension

V. Anjana·E. Harikumar·A. K. Kapoor
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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.

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V. Anjana, E. Harikumar, A. K. Kapoor. 2017-04-24. Non-Commutative space-time and Hausdorff dimension. https://doi.org/10.1142/s0217751x17501834

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