Solvable Models on Noncommutative Spaces with Minimal Length Uncertainty Relations

TL;DR

This paper explores models in higher-dimensional noncommutative spaces with minimal length uncertainty, relating q-deformed oscillators to noncommutative algebra, and constructs coherent states with classical-like properties.

Contribution

It introduces a procedure to connect q-deformed oscillator algebra to noncommutative space algebra and analyzes physical implications using PT-symmetry and coherent states.

Findings

Operators obey algebras with minimal length uncertainty

Constructed coherent states saturate uncertainty relations

Classical-like behavior demonstrated by coherent states

Abstract

Our main focus is to explore different models in noncommutative spaces in higher dimensions. We provide a procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of noncommutative spaces, first by utilising the perturbation theory, later in an exact manner. In many cases the operators are not Hermitian, therefore we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent. Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. The qualitative behaviour is found to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures.