Quantum mechanics on noncommutative plane and sphere from constrained systems

TL;DR

This paper demonstrates how quantum mechanics on noncommutative spaces can be derived from constrained systems, providing formulations for particles on NC plane and sphere, and exploring different variable choices affecting commutative or NC brackets.

Contribution

It introduces Lagrangian actions for quantum systems on NC plane and sphere, linking constrained systems to NC geometry and quantization with the Moyal product.

Findings

Quantum mechanics on NC spaces derived from constrained systems.

Lagrangian formulations for particles on NC and commutative spheres.

Multiple variable choices lead to either commutative or NC brackets.

Abstract

It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere.