Noncommutative Phase Spaces on Aristotle group

TL;DR

This paper constructs noncommutative phase spaces using coadjoint orbits of extended Aristotle groups, illustrating how magnetic fields induce noncommuting momenta in a two-dimensional setting.

Contribution

It introduces a novel geometric framework for noncommutative phase spaces based on group extensions and coadjoint orbits, linking magnetic fields with noncommutativity.

Findings

Noncommutative momenta arise from group-theoretic constructions.

Magnetic fields naturally induce noncommutativity in the phase space.

The approach provides a geometric interpretation of minimal coupling in quantum mechanics.

Abstract

We realize noncommutative phase spaces as coadjoint orbits of extensions of the Aristotle group in a two-dimensional space. Through these constructions the momenta of the phase spaces do not commute due to the presence of a naturally introduced magnetic field. These cases correspond to the minimal coupling of the momentum with a magnetic potential.