PT-symmetric noncommutative spaces with minimal volume uncertainty relations

TL;DR

This paper develops a systematic method to connect q-deformed oscillator algebras with noncommutative space algebras, imposing PT-symmetry to explore minimal volume uncertainty relations in phase space.

Contribution

It introduces a procedure to relate q-deformed oscillators to noncommutative spaces with PT-symmetry, analyzing minimal uncertainty relations and explicit models.

Findings

Derived noncommutative space algebras with PT-symmetry

Identified minimal length, area, and volume uncertainty relations

Presented explicit models on decomposed noncommutative spaces

Abstract

We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.