Noncommutative quantum mechanics -- a perspective on structure and spatial extent

TL;DR

This paper investigates the structure and spatial extent in noncommutative quantum mechanics, revealing how extended objects and corrections to classical quantities naturally emerge, with a focus on local and nonlocal descriptions.

Contribution

It introduces a dual local and nonlocal framework for noncommutative quantum mechanics, demonstrating the emergence of structured objects and corrections to classical conserved quantities.

Findings

Extended objects naturally emerge in noncommutative quantum mechanics.

Local description reveals detailed structure of quantum states.

Corrections to energy and angular momentum are identified.

Abstract

We explore the notion of spatial extent and structure, already alluded to in earlier literature, within the formulation of quantum mechanics on the noncommutative plane. Introducing the notion of average position and its measurement, we find two equivalent pictures: a constrained local description in position containing additional degrees of freedom, and an unconstrained nonlocal description in terms of the position without any other degrees of freedom. Both these descriptions have a corresponding classical theory which shows that the concept of extended, structured objects emerges quite naturally and unavoidably there. It is explicitly demonstrated that the conserved energy and angular momentum contain corrections to those of a point particle. We argue that these notions also extend naturally to the quantum level. The local description is found to be the most convenient as it manifestly displays additional information about structure of quantum states that is more subtly encoded in the nonlocal, unconstrained description. Subsequently we use this picture to discuss the free particle and harmonic oscillator as examples.