Strings from position-dependent noncommutativity

TL;DR

This paper introduces a novel two-dimensional noncommutative space with position-dependent relations, explores their symmetries, and analyzes minimal length scales, applying these concepts to simple quantum models.

Contribution

It presents a new class of position-dependent noncommutative relations, constructs their Hermitian counterparts, and investigates their physical implications and simple model solutions.

Findings

Objects have a fundamental string-like length scale.

PT-like symmetries are preserved in the deformed space.

Minimal lengths and momenta are derived from generalized uncertainty relations.

Abstract

We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PTlike symmetries, i.e.antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenberg's uncertainty relations and find that any object in this two dimensional space is string like, i.e.having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT-symmetric deformations and the harmonic oscillator.