Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

Abstract

The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that represent geodesic motions on 3D manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. A different Hamiltonian defined on the same deformed coalgebra is also shown to generate a maximally superintegrable geodesic motion on 3D Riemannian and (2 + 1)D relativistic spaces whose sectional curvatures are all constant and equal to z. This approach can be generalized to arbitrary dimension. PACS: 02.30.lk, 02.20.Uw

Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

“…This indeed allowed us to obtain the integrals (7.7) for the N D SW system on the three Riemannian spaces in [9,23] by starting from the quantum deformation of the Euclidean SW system introduced in [1,2]. Furthermore, quantum deformations have been shown [6,7] to give rise to Riemannian and relativistic spaces of non-constant curvature on which SW-and KC-type potentials can be considered [8].…”

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

“…Through an appropriate change of coordinates [33] we find that (4.11) is transformed into the 3D Cayley-Klein metric written in terms of geodesic polar coordinates (r, θ, φ):…”

“…Some of these variable curvature systems in 2D and 3D have been already studied (see [31,32,33]), and we present here the most significant elements for their N D generalizations. We will show that this scheme is quite efficient in order to get explicitly a large family of QMS systems.…”

Superintegrability on sl(2)-coalgebra spaces

“…Two infinite families of N-dimensional (ND) quasi-maximally superintegrable Hamiltonians endowed with a set of (2N − 3) integrals of the motion have been recently introduced in [1][2][3][4][5]. In the first family, the superintegrability properties of all these Hamiltonians are shown to be a consequence of a hidden sl(2) Poisson coalgebra symmetry [1].…”

“…In the first family, the superintegrability properties of all these Hamiltonians are shown to be a consequence of a hidden sl(2) Poisson coalgebra symmetry [1]. The second family is just a q-deformation of the former (see [2][3][4][5] and references therein), and the deformed coalgebra symmetry is given by sl z (2) (q = e z ), the Poisson analogue of the non-standard quantum deformation of sl(2) [6]. As a concrete application of these general results, some of these Hamiltonians can be shown to generate superintegrable geodesic motions on certain curved manifolds (see [1,3,5]) .…”