Minimal and maximal lengths from position-dependent non-commutativity

Abstract

Fring et al. in Ref.[1] have introduced a new set of noncommutative space-time commutation relations in two space dimensions. It had been shown that any fundamental objects introduced in this space-space noncommutativity are string-like. Taking this result into account, we generalize the seminal work of Fring et al to the case that there is also a maximal length from position-dependent noncommutativity and a minimal momentum arising from generalized versions of Heisenberg's uncertainty relations. The existence of maximal length is related to the presence of an extra, first order term in particle's length that provides the basic difference of our analysis with theirs. This maximal length breaks up the well known singularity problem of space time. We establish different representations of this noncommutative space and finally we study some basic and interesting quantum mechanical systems in these new variables.

Thermodynamics of Ideal Gas at Planck Scale with Strong Quantum Gravity Measurement

“…The interesting physical consequence we found is that the existence of maximal length in this theory brings a lot of new features to the Hilbert space representation and agrees with the similar perturbative approaches predicted by DSR theories [12,14,15]. In this paper [1], we predicted that this concept of maximal length could induce strong graviton localizations and could be the approach candidate for the measurement of quantum gravity with low energies. In the present paper to make evidence of this prediction, we investigate in 1D, the outgoing of this GUP model with maximal length in the statistics of the canonical and microcanonical ensembles of an ideal gas at the Planck scale.…”

“…Since the Planck scale marks the frontier of our universe, then the existence of magnetic fields at this scale can only be from a multiverse that bounces at these minimal positions l min . As we have predicted in our previous work [1], this scenario lets break up the big bang singularity. This present part of the paper is currently under investigation and will be the object of further exhaustive study.…”

“…In this prophetic paper [16], Perivolaropoulos also predicted the simultaneous existence of maximal and minimal position uncertainties. More recently, without any formal knowledge of this result, we introduced a version of position-dependent noncommutative space in two-dimensional (2D) configuration spaces which, for simultaneous measurement lead to minimal and maximal lengths of quantum gravity [1]. The interesting physical consequence we found is that the existence of maximal length in this theory brings a lot of new features to the Hilbert space representation and agrees with the similar perturbative approaches predicted by DSR theories [12,14,15].…”

“…The quantum gravity according to Kempf et Al formalism is manifested by the quadratic deformation of the Heisenberg algebra [2]. Recently, we proposed a new version of position-dependent deformed Heisenberg algebra in two-dimensional configuration spaces that introduces a simultaneous presence of maximal and minimal position uncertainties [1]. We define this algebra as follows:…”

“…In the next section, we review in 1D, the GUP with the maximal length and its deformed translation symmetry [1]. Section 3, is devoted to the study of a non-relativistic quantum particle in an infinite square well potential at the Planck scale.…”