Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

TL;DR

This paper introduces a family of superintegrable Hamiltonians with quantum algebra symmetries, generating geodesic motions on spaces with variable or constant curvature, and demonstrates their properties and generalizations.

Contribution

It presents a new class of superintegrable Hamiltonians linked to quantum algebra symmetries, with applications to geodesic motions on curved spaces and potential for higher-dimensional generalizations.

Findings

Hamiltonians exhibit superintegrability due to quantum sl(2,R) symmetry.

Generated geodesic motions on manifolds with variable and constant curvature.

Framework can be extended to arbitrary dimensions using coalgebra symmetry.

Abstract

An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. Moreover, another Hamiltonian in this family is shown to generate geodesic motions on Riemannian and relativistic spaces all of whose sectional curvatures are constant and equal to the deformation parameter z. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry.