Noncommutative space-time and Hausdorff dimension

Abstract

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space-time. We show that the Hausdorff dimension depends on the deformation parameter a and the resolution x for both nonrelativistic and relativistic quantum particle. For the nonrelativistic case, it is seen that Hausdorff dimension is always less than 2 in the noncommutative space-time. For relativistic quantum particle, we find the Hausdorff dimension increases with the noncommutative parameter, in contrast to the commutative space-time. We show that noncommutative 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 nonrelativistic and relativistic quantum particle in noncommutative space-time, we derive the corresponding generalized uncertainty relation.