# Non-Commutative space-time and Hausdorff dimension

## TL;DR

This paper investigates how non-commutative space-time affects the Hausdorff dimension of quantum particle paths, revealing dependence on deformation parameters and contrasting behaviors between non-relativistic and relativistic cases.

## Contribution

It introduces a detailed analysis of the Hausdorff dimension in non-commutative space-time and explores the impact of non-commutativity on quantum wave functions and uncertainty relations.

## Key findings

- Hausdorff dimension is less than two in non-commutative space-time for non-relativistic particles.
- Relativistic quantum particles show increasing Hausdorff dimension with non-commutative parameter.
- Non-commutative corrections induce spinorial features in the Dirac equation.

## 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.