Poisson structure on a space with linear SU(2) fuzziness

TL;DR

This paper develops a Poisson structure for a noncommutative space with SU(2) fuzziness, analyzing classical systems and solutions like free particles and harmonic oscillators within this framework.

Contribution

It introduces a Poisson structure on a noncommutative space with SU(2) fuzziness and explores classical dynamics in this setting, including explicit solutions.

Findings

Path equations for particles are first-order, similar to commutative models.

Explicit solutions for free particles and harmonic oscillators are derived.

SU(2)-invariant systems exhibit familiar classical behavior despite noncommutativity.

Abstract

The Poisson structure is constructed for a model in which spatial coordinates of configuration space are noncommutative and satisfy the commutation relations of a Lie algebra. The case is specialized to that of the group SU(2), for which the counterpart of the angular momentum, as well as the Euler parameterization of the phase space are introduced. SU(2)-invariant classical systems are discussed, and it is observed that the path of particle can be obtained by the solution of a first-order equation, as the case with such models on commutative spaces. The examples of free particle, rotationally-invariant potentials, and specially the isotropic harmonic oscillator are investigated in more detail.