Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

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.


Introduction
The set of known maximally superintegrable systems on the N -dimensional (N D) Euclidean space is very limited: it comprises the isotropic harmonic oscillator with N centrifugal terms (the so-called Smorodinsky-Winternitz (SW) system [1,2]), the Kepler-Coulomb (KC) problem with (N − 1) centrifugal barriers [3] (and some symmetry-breaking generalizations of it [4]), the Calogero-Moser-Sutherland model [5,6,7,8] and some systems with isochronous potentials [9]. Both the SW and the KC systems have integrals quadratic in the momenta, and also both of them have been generalized to spaces with non-zero constant curvature (see [10,11,12,13,14,15,16,17,18,19,20]). In order to complete this brief N D summary, Benenti systems on constant curvature spaces have also to be considered [21], as well as a maximally superintegrable deformation of the SW system that was introduced in [22] by making use of quantum algebras.
More recently, the study of 2D and 3D superintegrable systems on spaces with variable curvature has been addressed [23,24,25,26,27,28,29]. The aim of this paper is to give a general setting, based on quantum deformations, for the explicit construction of certain classes of superintegrable systems on N D spaces with variable curvature.
In order to fix language conventions, we recall that an N D completely integrable Hamiltonian H (N ) is called maximally superintegrable (MS) if there exists a set of (2N − 2) globally defined functionally independent constants of the motion that Poisson-commute with H (N ) . Among them, at least two different subsets of (N − 1) constants in involution can be found. In the same way, a system will be called quasi-maximally superintegrable (QMS) if there are (2N − 3) integrals with the abovementioned properties. All MS systems are QMS ones, and the latter have only one less integral than the maximum possible number of functionally independent ones.
In this paper we present the construction of QMS systems on variable curvature spaces which is just the quantum algebra generalization of a recent approach to N D QMS systems on constant curvature spaces that include the SW and KC as particular cases [30]. 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. Among them, some specific choices for the Hamiltonian can lead to a MS system, for which only the remaining integral has to be explicitly found.
In the the next Section we will briefly summarize the N D constant curvature construction given in [30], that makes use of an sl(2, R) Poisson coalgebra symmetry. The generic variable curvature approach will be obtained in Section 3 through a non-standard quantum deformation of an sl(2, R) Poisson coalgebra. Some explicit 2D and 3D spaces defined through free motion Hamiltonians will be given in Section 4, and the N D generalization of them will be sketched in Section 5. Section 6 is devoted to the introduction of some potentials that generalize the KC and SW ones. A final Section including some comments and open questions closes the paper.

QMS Hamiltonians with sl(2, R) coalgebra symmetry
Let us briefly recall the main result of [30] that provides an infinite family of QMS Hamiltonians. We stress that, although some of these Hamiltonians can be interpreted as motions on spaces with constant curvature, this approach to QMS systems is quite general, and also nonnatural Hamiltonian systems (for instance, those describing static electromagnetic fields) can be obtained.
with H any smooth function and where b i are arbitrary real parameters, is QMS. The (2N − 3) functionally independent and "universal" integrals of motion are explicitly given by

2)
where m = 2, . . . , N and C (N ) = C (N ) . Moreover, the sets of N functions {H (N ) , C (m) } and The proof of this general result is based on the observation that, for any choice of the function H, the Hamiltonian H (N ) has an sl(2, R) Poisson coalgebra symmetry [34] generated by the following Lie-Poisson brackets and comultiplication map: The Casimir function for sl(2, R) reads In fact, the coalgebra approach [34] provides an N -particle symplectic realization of sl(2, R) through the N -sites coproduct of (2.4) living on sl(2, R) ⊗ · · · N ) ⊗ sl(2, R) [22]: where b i are N arbitrary real parameters. This means that the N -particle generators (2.6) fulfil the commutation rules (2.3) with respect to the canonical Poisson bracket. As a consequence of the coalgebra approach, these generators Poisson commute with the (2N − 3) functions (2.2) given by the sets C (m) and C (m) , which are obtained, in this order, from the "left" and "right" m-th coproducts of the Casimir (2.5) with m = 2, 3, . . . , N (see [35] for details). Therefore, any arbitrary function H defined on the N -particle symplectic realization of sl(2, R) (2.6) is of the form (2.1), that is, and defines a QMS Hamiltonian system that Poisson-commutes with all the "universal integrals" C (m) and C (m) . Notice that for arbitrary N there is a single constant of the motion left to assure maximal superintegrability. In this respect, we stress that some specific choices of H comprise maximally superintegrable systems as well, but the remaining integral does not come from the coalgebra symmetry and has to be deduced by making use of alternative procedures.
Let us now give some explicit examples of this construction.

Free motion on Riemannian spaces of constant curvature
It is immediate to realize that the kinetic energy T of a particle on the N D Euclidean space E N directly arises through the generator J + in the symplectic realization (2.6) with all b i = 0: Now the interesting point is that the kinetic energy on N D Riemannian spaces with constant curvature κ can be expressed in Hamiltonian form as a function of the N D symplectic realization of the sl(2, R) generators (2.6). In fact, this can be done in two different ways [30]: The function H P is just the kinetic energy for a free particle on the spherical S N (κ > 0) and hyperbolic H N (κ < 0) spaces when this is expressed in terms of Poincaré coordinates q and canonical momenta p (coming from a stereographic projection in R N +1 ); on the other hand H B corresponds to Beltrami coordinates and momenta (central projection). By construction, both Hamiltonians are QMS ones since they Poisson-commute with the integrals (2.2).

Superintegrable potentials on Riemannian spaces of constant curvature
QMS potentials V on constant curvature spaces can now be constructed by adding some suitable functions depending on J − to (2.7) and by considering arbitrary centrifugal terms that come from symplectic realizations of the J + generator with generic b i 's: The Hamiltonians that we will obtain in this way are the curved counterpart of the Euclidean systems, and through different values of the curvature κ we will simultaneously cover the cases S N (κ > 0), H N (κ < 0), and E N (κ = 0).
In order to motivate the choice of the potential functions V(J − ), it is important to recall that in the constant curvature analogues of the oscillator and KC problems the Euclidean radial distance r is just replaced by the function 1 √ κ tan( √ κ r) (see [30] for the expression of this quantity in terms of Poincaré and Beltrami coordinates). Also, for the sake of simplicity, the centrifugal terms coming from the symplectic realization with arbitrary b i will be expressed in ambient coordinates x i [30]: Beltrami: Special choices for V(J − ) lead to the following systems, that are always expressed in both Poincaré and Beltrami phase spaces: • A curved Evans system. The constant curvature generalization of a 3D Euclidean system with radial symmetry [36] would be given by where V is an arbitrary smooth function that determines the central potential; the specific dependence on J − of V corresponds to the square of the radial distance in each coordinate system.
• The curved Smorodinsky-Winternitz system [10,11,12,13,14,15]. Such a system is just the Higgs oscillator [16,17] with angular frequency ω (that arises as the argument of V in (2.8)) plus the corresponding centrifugal terms: This is a MS Hamiltonian and the remaining constant of the motion can be chosen from any of the following N functions: • A curved generalized Kepler-Coulomb system [12,13,14,18,19,20]. The curved KC potential with real constant k together with N centrifugal terms would be given by This is again a MS system provided that, at least, one b i = 0. In this case the remaining constant of the motion turns out to be is also a new constant of the motion. In this way the proper curved KC system [37] (with all the b i 's equal to zero) is obtained, and in that case (2.9) are just the N components of the Laplace-Runge-Lenz vector on S N (κ > 0) and H N (κ < 0).
We also stress that all these examples share the same set of constants of the motion (2.2), although the geometric meaning of the canonical coordinates and momenta can be different.

QMS Hamiltonians with quantum deformed sl(2, R) coalgebra symmetry
Here we will show that a generalization of the construction presented in the previous Section can be obtained through a quantum deformation of sl(2, R), yielding QMS systems for certain spaces with variable curvature. Let us now state the general statement that provides a superintegrable deformation of Theorem 1.
where H z is any smooth function and

2)
is QMS for any choice of the function H and for arbitrary real parameters b i . The (2N − 3) functionally independent and "universal" integrals of the motion are given by

The proof
The proof is based on the fact that, for any choice of the function H, the Hamiltonian H (N ) z has a deformed Poisson coalgebra symmetry, sl z (2, R), coming (under a certain symplectic realization) from the non-standard quantum deformation of sl(2, R) [38,39] where z is the deformation parameter (q = e z ). If we perform the limit z → 0 in all the results given in Theorem 2, we shall exactly recover Theorem 1. Here we sketch the main steps of this construction, referring to [22,35] for further details.
We recall that the non-standard sl z (2, R) Poisson coalgebra is given by the following deformed Poisson brackets and coproduct [22]: The Casimir function for sl z (2, R) reads (3.6) A one-particle symplectic realization of (3.4) is given by Now the essential point is the fact that the coalgebra approach [34] provides the corresponding N -particle symplectic realization of sl z (2, R) through the N -sites coproduct of (3.5) living on sl z (2, R) ⊗ · · · N ) ⊗ sl z (2, R) [22]: where K (N ) i (q 2 ) is defined in (3.2) and b i are N arbitrary real parameters that label the representation on each "lattice" site. This means that the N -particle generators (3.7) fulfil the commutation rules (3.4) with respect to the canonical Poisson bracket Therefore the Hamiltonian (3.1) is obtained through an arbitrary smooth function H z defined on the N -particle symplectic realization of the generators of sl z (2, R): Thus, we conclude that any arbitrary function H z (3.8) defines a QMS Hamiltonian system.

The N = 2 case
In order to illustrate the previous construction, let us explicitly write the 2-particle symplectic realization of sl z (2, R) (3.7): In this case there is a single (left and right) constant of the motion: After some straightforward computations this integral can be expressed as By construction, this constant of the motion will Poisson-commute with all the Hamiltonians .
Note that in the N = 2 case quasi-maximal superintegrability means only integrability, i.e., the only constant given by Theorem 2 is just C (2) z ≡ C z, (2) ; this fact does not exclude that there could be some specific choices for H z for which an additional integral does exist. When N ≥ 3, Theorem 2 will always provide QMS Hamiltonians.
where f is an arbitrary smooth function such that lim z→0 f zJ . We shall explore in the sequel some specific choices for f , and we shall analyse the spaces generated by them.

An integrable case
Of course, the simplest choice will be just to set f ≡ 1 [31]: The Gaussian curvature K for this space can be computed through and turns out to be non-constant and negative: Therefore, the underlying 2D space is of hyperbolic type and endowed with a "radial" symmetry.
Let us now consider the following change of coordinates that includes a new parameter λ 2 = 0: where z = λ 2 1 and λ 2 can take either a real or a pure imaginary value. Note that the new variable cosh(λ 1 ρ) is a collective variable, a function of ∆(J − ); its role will be specified later. On the other hand, the zero-deformation limit z → 0 is in fact the flat limit K → 0, since in this limit Thus ρ can be interpreted as a radial coordinate and θ is either a circular (λ 2 real) or a hyperbolic angle (λ 2 imaginary). Notice that in the latter case, say λ 2 = i, the coordinate q 1 is imaginary and can be written as q 1 = iq 1 whereq 1 is a real coordinate; then ρ → 2(q 2 2 −q 2 1 ) which corresponds to a relativistic radial distance. Therefore the introduction of the additional parameter λ 2 will allow us to obtain Lorentzian metrics.
In this new coordinates, the metric (4.4) reads where ds 2 0 is just the metric of the 2D Cayley-Klein spaces in terms of geodesic polar coordinates [40,41] provided that we identify z = λ 2 1 ≡ −κ 1 and λ 2 2 ≡ κ 2 ; hence λ 2 determines the signature of the metric. The Gaussian curvature turns out to be In this way we find the following spaces, whose main properties are summarized in Table 1: • When λ 2 is real, we get a 2D deformed sphere S 2 z (z < 0), and a deformed hyperbolic or Lobachewski space H 2 z (z > 0). • When λ 2 is imaginary, we obtain a deformation of the (1+1)D anti-de Sitter spacetime AdS 1+1 z (z < 0) and of the de Sitter one dS 1+1 z (z > 0).
Accordingly, the kinetic energy (4.3) is transformed into and the free motion Hamiltonian (4.2) is written as There is a unique constant of the motion C (2) z ≡ C z,(2) (3.9) which in terms of the new phase space is simply given by z . This allows us to apply a radial-symmetry reduction: We remark that the explicit integration of the geodesic motion on all these spaces can be explicitly performed in terms of elliptic integrals.

The superintegrable case
A MS Hamiltonian is given by Surprisingly enough, the computation of the Gaussian curvature K for ds 2 MS gives that K = z. Therefore, we are dealing with a space of constant curvature which is just the deformation parameter z. In [31] it was shown that a certain change of coordinates (that includes the signature parameter λ 2 ) transforms the metric into which exactly coincides with the metric of the Cayley-Klein spaces written in geodesic polar coordinates (r, θ) provided that now z = λ 2 1 ≡ κ 1 and λ 2 2 ≡ κ 2 . Obviously, after this change of variables the geodesic motion can be reduced to a "radial" 1D system: where H MS z = 2H MS z and C z = p 2 θ is, as in the previous case, the usual generalized momentum for the θ coordinate.

A more general case
At this point, one could wonder whether there exist other choices for the Hamiltonian yielding constant curvature. In fact, let us consider the generic Hamiltonian (4.1) depending on f . If we compute the general expression for the 2D Gaussian curvature in terms of the function f (x) we find that Thus, in general, we obtain spaces with variable curvature. In order to characterize the constant curvature cases, we can define g := f ′ /f and write If we now require K to be a constant we get the equation where y := 2g ′ + g 2 − 1.
The solution for this equation yields where A is a constant, and solving for g, we get for F := f 1 2 the equation whose general solution is (A := λ(λ − 1)): where C 1 and C 2 are two integration constants. Therefore, many different solutions lead to 2D constant curvature spaces. However, we must impose as additional condition that lim x→0 f = 1. In this way we obtain that only the cases with A = 0 are possible, that is, either λ = 1 or λ = 0. Hence the two elementary solutions are just the Hamiltonians and the curvature of their associated spaces is K = ±z.

3D curved manifolds
The study of the 3D case follows exactly the same pattern. The three-particle symplectic realization of sl z (2, R) (with all b i = 0) is obtained from (3.7): (4.10) Therefore the Hamiltonian is separable and reduced to a 1D radial system.

MS free motion: constant curvature
The following choice for the Hamiltonian This space is again a Riemannian one with constant sectional and scalar curvatures given by 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): Therefore, according to the values of (λ 1 , λ 2 ), this metric provides the 3D sphere (1, 1), Euclidean (0, 1), hyperbolic (i, 1), anti-de Sitter (1, i), Minkowskian (0, i), and de Sitter (i, i) spaces. Now the MS Hamiltonian,H MS

N D spaces with variable curvature
The generalization to arbitrary dimension is obtained through the same procedure, and the starting point is the QMS Hamiltonian for the N D geodesic motion that, in the simplest case, reads: The geometric characterization of the underlying N D curved spaces follows the same path as in the 2D and 3D cases described in the previous sections. If we write the above Hamiltonian as where s z (q 2 i ) = sinh zq 2 i /(zq 2 i ) and sgn(k − i) is the sign of the difference k − i, we get again a free Lagrangian: with the corresponding (diagonal) metric given by .

(5.1)
It turns out that the most suitable way to understand the nature of the problem as well as to enforce separability is to consider two sets of new coordinates: • N + 1 "collective" variables [42] (ξ 0 , ξ 1 , . . . , ξ N ). They play a similar role to the ambient coordinates arising when N D Riemannian spaces of constant curvature are embedded within R N +1 .
• N "intrinsic" variables (ρ, θ 2 , . . . , θ N ) which describe the N D space itself. They are the analogous to the geodesic polar coordinates on N D Riemannian spaces of constant curvature [10,11].
The above coordinates are defined in terms of the initial q i by: where z = λ 2 1 , k = 1, . . . , N − 1, and hereafter a product k j such that j > k is assumed to be equal to 1. Notice also that for the sake of simplicity we have not introduced the additional signature parameter λ 2 (which would have been associated with θ 2 ). This definition is the N D generalization of the change of coordinates (4.8) given in the 3D case with θ = θ 2 , φ = θ 3 and λ 2 = 1.
Clearly, the N + 1 collective variables are not independent and they fulfil a pseudosphere relation (of hyperbolic type): The geodesic flow in the canonical coordinates (ρ, θ) and momenta (ρ, p θ ) is then given by the HamiltonianH z = 2H z : By taking into account the N functions {H z ,C (m) z }, we obtain the following set of N equations, each of them depending on a single canonical pair, which shows the reduction of the system to a 1D problem: We stress that these models can be extended to incorporate appropriate interactions with an external central field, preserving superintegrability. This will be achieved by modifying the Hamiltonian by adding an arbitrary function of J − , as we shall see in the next Section. Finally we remark that the corresponding generalization to N D spaces with constant curvature can be obtained by considering the MS Hamiltonian Under a suitable change of coordinates, similar to (5.2) but involving a different radial coordinate r instead of ρ, this Hamiltonian leads to the MS geodesic motion on S N , H N and E N in the proper geodesic polar coordinates which can be found in [10,11].

QMS potentials
As we have just noticed, we can also consider more general N D QMS Hamiltonians based on sl z (2, R) (3.7) by considering arbitrary b i 's (contained in J + ) and adding some functions depending on J − ; hereafter we drop the index "(N )" in the generators. The family of Hamiltonians that we consider has the form (see [32] for the 2D construction):

Concluding remarks
The main message that we would like to convey to the scientific community through the present paper is that "Superintegrable Systems are not rare!". Indeed, in our approach they turn out to be a natural manifestation of coalgebra symmetry: as such, they can be equally well constructed on a flat or on a curved background, the latter being possibly equipped with a variable curvature. Moreover, and, we would say, quite remarkably the construction holds for an arbitrary number of dimensions.
In that perspective, the most interesting problems that are still open are in our opinion the following ones: 1. The explicit integration of the equations of motion for (at least some) of the prototype examples we have introduced in the previous sections; 2. The construction of the quantum-mechanical counterpart of our approach.
As for the former point, partial results have already been obtained, and a detailed description of the most relevant examples will be published soon. The latter point, in particular as far as the non-standard deformation of sl(2, R) is concerned, is however more subtle and deserves careful investigation (which is actually in progress). In fact, first of all one has to find a proper ∞-dimensional representation of such a non-standard deformation in terms of linear operators acting on a suitably defined Hilbert space, ensuring self-adjointness of the Hamiltonians; second, and certainly equally important, at least in some physically interesting special cases one would like to exhibit the explicit solution of the corresponding spectral problem.