Noncommutative 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 ∆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.

Compact stars in quantum spacetime

“…In addition to the derivation of the constraints on the deformation parameter a, we are able to fix the choice of realization of κ-spacetime by utilizing the generalised uncertainty relation obtained in [44]. If we compare the expression of the pressure given in equation (38) for degenerate Fermi gas using statistical mechanics and the pressure obtained by utilizing GUP as given in equation (51), we find that the realization should be e ap k 0 j = -, for both the expressions to be equivalent.…”

“…In addition to the derivation of the constraints on the deformation parameter a, we are able to fix the choice of realization of κ-spacetime by utilizing the generalised uncertainty relation obtained in [34]. If we compare the expression of the pressure given in eqn.…”

“…We now look for the effect of generalised uncertainty principle on the pressure. We use a generalised uncertainty relation consistent with κ-deformed theories, as shown in [34], i.e., (∆x)(∆p)(1 +…”

Maximal acceleration in non-commutative space-time and its implications

“…In this section we derive the κ-deformed corrections to maximal acceleration, valid up to first order in a, using the κ-deformed uncertainty relation between position and momenta. We begin with the uncertainty relation between energy and a function of time [33,34] We know that the uncertainty in the velocity of the particle cannot exceed its maximum attainable velocity and from special theory of relativity this maximum attainable velocity should be less than the velocity of light, i.e, (∆v In [52] the κ-deformed uncertainty relation between ∆x and ∆p has been derived up to first order in a, by imposing self-similarity condition on the path of relativistic quantum particle in κ-space-time and it is given by ∆x∆p…”