Phases in noncommutative quantum mechanics on (pseudo)sphere

TL;DR

This paper compares noncommutative quantum mechanics on (pseudo)spheres with planar systems, revealing a critical point where the system becomes effectively one-dimensional and identifying two distinct phases related to the sign of a key parameter.

Contribution

It introduces the concept of a critical point and phase structure in (pseudo)spherical NCQM, connecting it to known planar NCQM phases and highlighting new geometric effects.

Findings

Identification of a critical point where the system becomes one-dimensional

Existence of two phases distinguished by the sign of ppa

Connection between spherical and planar NCQM phase structures

Abstract

We compare the non-commutative quantum mechanics (NCQM) on sphere and the discrete part of the spectrum of NCQM on pseudosphere (Lobachevsky plane, or $AdS_2$) in the presence of a constant magnetic field $B$ with planar NCQM. We show, that (pseudo)spherical NCQM has a ``critical point'', where the system becomes effectively one-dimensional, and two different ``phases'', which the phases of the planar system originate from, specified by the sign of the parameter $\kappa=1-B\theta$. The ``critical point'' of (pseudo)spherical NCQM corresponds to the $\kappa\to\infty $ point of conventional planar NCQM, and to the ``critical point'' $\kappa=0$ of the so-called ``exotic'' planar NCQM, with a symplectic coupling of the (commutative) magnetic field.