Minimal and maximal lengths from position-dependent non-commutativity

Abstract

Fring et al (2010 J. Phys. A: Math. Theor. 43 345401) 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.