Minimal and maximal lengths from position-dependent noncommutativity

TL;DR

This paper extends previous work on position-dependent noncommutative spaces by introducing maximal length and minimal momentum, which help resolve singularities and alter quantum system representations.

Contribution

It generalizes noncommutative space models to include maximal length and minimal momentum, providing new representations and insights into quantum systems.

Findings

Maximal length breaks space-time singularities

New representations of noncommutative space are established

Quantum systems are analyzed in the new framework

Abstract

Fring and al in their paper entitled "Strings from position-dependent noncommutativity" have introduced a new set of noncommutative space commutation relations in two space dimensions. It had been shown that any fundamental objects introduced in this space-space non-commutativity are string-like. Taking this result into account, we generalize the seminal work of Fring and al to the case that there is also a maximal length from position-dependent noncommutativity and 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.