TL;DR
This paper explores how restricting a charged particle's motion to the lowest energy level on a sphere with a magnetic monopole leads to noncommutative angular variables and introduces a noncommuting spherical product, revealing algebraic structures similar to angular momentum.
Contribution
It extends noncommutative geometry concepts from Landau levels to spherical geometries with magnetic monopoles, defining a noncommuting spherical product and related algebra.
Findings
Noncommutativity arises in angular variables due to magnetic monopoles.
A spherical star product is defined based on the algebraic structure.
Solutions for dynamics with angular potentials are obtained.
Abstract
Restricting the states of a charged particle to the lowest Landau level introduces a noncommutativity between Cartesian coordinate operators. This idea is extended to the motion of a charged particle on a sphere in the presence of a magnetic monopole. Restricting the dynamics to the lowest energy level results in noncommutativity for angular variables and to a definition of a noncommuting spherical product. The values of the commutators of various angular variables are not arbitrary but are restricted by the discrete magnitude of the magnetic monopole charge. An algebra, isomorphic to angular momentum, appears. This algebra is used to define a spherical star product. Solutions are obtained for dynamics in the presence of additional angular dependent potentials.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.