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 concret...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2007
Main Authors:	Ragnisco, O., Ballesteros, A., Herranz, F.J., Musso, F.
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Language:	English
Published:	Інститут математики НАН України 2007
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Cite this:	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature / O. Ragnisco, Á. Ballesteros, F.J. Herranz, F. Musso // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 42 назв. — англ.

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spelling	Ragnisco, O. Ballesteros, A. Herranz, F.J. Musso, F. 2019-02-16T08:10:54Z 2019-02-16T08:10:54Z 2007 Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature / O. Ragnisco, Á. Ballesteros, F.J. Herranz, F. Musso // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 42 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J35; 17B37 https://nasplib.isofts.kiev.ua/handle/123456789/147788 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. This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). This work was partially supported by the Ministerio de Educaci´on y Ciencia (Spain, Project FIS2004-07913), by the Junta de Castilla y Le´on (Spain, Project VA013C05), and by the INFN–CICyT (Italy–Spain). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
spellingShingle	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature Ragnisco, O. Ballesteros, A. Herranz, F.J. Musso, F.
title_short	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
title_full	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
title_fullStr	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
title_full_unstemmed	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
title_sort	quantum deformations and superintegrable motions on spaces with variable curvature
author	Ragnisco, O. Ballesteros, A. Herranz, F.J. Musso, F.
author_facet	Ragnisco, O. Ballesteros, A. Herranz, F.J. Musso, F.
publishDate	2007
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	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.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147788
citation_txt	Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature / O. Ragnisco, Á. Ballesteros, F.J. Herranz, F. Musso // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 42 назв. — англ.
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first_indexed	2025-11-25T20:31:25Z
last_indexed	2025-11-25T20:31:25Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 026, 20 pages Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature? Orlando RAGNISCO †, Ángel BALLESTEROS ‡, Francisco J. HERRANZ ‡ and Fabio MUSSO † † Dipartimento di Fisica, Università di Roma Tre and Instituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy E-mail: ragnisco@fis.uniroma3.it, musso@fis.uniroma3.it ‡ Departamento de F́ısica, Universidad de Burgos, E-09001 Burgos, Spain E-mail: angelb@ubu.es, fjherranz@ubu.es Received November 12, 2006, in final form January 22, 2007; Published online February 14, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/026/ Abstract. An infinite family of quasi-maximally superintegrable Hamiltonians with a com- mon 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 Hamil- tonian 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 parame- ter z. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry. Key words: integrable systems; quantum groups; curvature; contraction; harmonic oscillator; Kepler–Coulomb; hyperbolic; de Sitter 2000 Mathematics Subject Classification: 37J35; 17B37 1 Introduction The set of known maximally superintegrable systems on the N -dimensional (ND) 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 ND 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 ND spaces with variable curvature. In order to fix language conventions, we recall that an ND completely integrable Hamilto- nian H(N) is called maximally superintegrable (MS) if there exists a set of (2N − 2) globally ?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at http://www.emis.de/journals/SIGMA/LOR2006.html mailto:ragnisco@fis.uniroma3.it mailto:musso@fis.uniroma3.it mailto:angelb@ubu.es mailto:fjherranz@ubu.es http://www.emis.de/journals/SIGMA/2007/026/ http://www.emis.de/journals/SIGMA/LOR2006.html 2 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso 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 ND 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 ND 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 ND 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 ND 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. 2 QMS Hamiltonians with sl(2, R) coalgebra symmetry Let us briefly recall the main result of [30] that provides an infinite family of QMS Hamilto- nians. 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 non- natural Hamiltonian systems (for instance, those describing static electromagnetic fields) can be obtained. Theorem 1 ([30]). Let {q,p} = {(q1, . . . , qN ), (p1, . . . , pN )} be N pairs of canonical variables. The ND Hamiltonian H(N) = H ( q2, p̃2,q · p ) , (2.1) with H any smooth function and q2 = N∑ i=1 q2 i , p̃2 = N∑ i=1 ( p2 i + bi q2 i ) ≡ p2 + N∑ i=1 bi q2 i , q · p = N∑ i=1 qi pi, where bi are arbitrary real parameters, is QMS. The (2N − 3) functionally independent and “universal” integrals of motion are explicitly given by C(m) = m∑ 1≤i<j { (qipj − qjpi)2 + ( bi q2 j q2 i + bj q2 i q2 j )} + m∑ i=1 bi, C(m) = N∑ N−m+1≤i<j { (qipj − qjpi)2 + ( bi q2 j q2 i + bj q2 i q2 j )} + N∑ i=N−m+1 bi, (2.2) where m = 2, . . . , N and C(N) = C(N). Moreover, the sets of N functions {H(N), C(m)} and {H(N), C(m)} (m = 2, . . . , N) are in involution. Quantum Deformations and Variable Curvature 3 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: {J3, J+} = 2J+, {J3, J−} = −2J−, {J−, J+} = 4J3, (2.3) ∆(Jl) = Jl ⊗ 1 + 1⊗ Jl, l = +,−, 3. (2.4) The Casimir function for sl(2, R) reads C = J−J+ − J2 3 . (2.5) 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]: J− = N∑ i=1 q2 i ≡ q2, J+ = N∑ i=1 ( p2 i + bi q2 i ) ≡ p2 + N∑ i=1 bi q2 i , J3 = N∑ i=1 qipi ≡ q · p, (2.6) where bi 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, H(N) = H (J−, J+, J3) = H ( q2,p2 + N∑ i=1 bi q2 i ,q · p ) , and defines a QMS Hamiltonian system that Poisson-commutes with all the “universal integ- rals” 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. 2.1 Free motion on Riemannian spaces of constant curvature It is immediate to realize that the kinetic energy T of a particle on the ND Euclidean space EN directly arises through the generator J+ in the symplectic realization (2.6) with all bi = 0: H = T = 1 2 J+ = 1 2 p2. Now the interesting point is that the kinetic energy on ND Riemannian spaces with constant curvature κ can be expressed in Hamiltonian form as a function of the ND symplectic realization of the sl(2, R) generators (2.6). In fact, this can be done in two different ways [30]: HP = T P = 1 2 (1 + κJ−)2 J+ = 1 2 ( 1 + κq2 )2 p2, HB = T B = 1 2 (1 + κJ−) ( J+ + κJ2 3 ) = 1 2 (1 + κq2) ( p2 + κ(q · p)2 ) . (2.7) The function HP is just the kinetic energy for a free particle on the spherical SN (κ > 0) and hyperbolic HN (κ < 0) spaces when this is expressed in terms of Poincaré coordinates q and canonical momenta p (coming from a stereographic projection in RN+1); on the other hand HB corresponds to Beltrami coordinates and momenta (central projection). By construction, both Hamiltonians are QMS ones since they Poisson-commute with the integrals (2.2). 4 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso 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 bi’s: H = T (J+, J−, J3) + V(J−). The Hamiltonians that we will obtain in this way are the curved counterpart of the Euclidean sys- tems, and through different values of the curvature κ we will simultaneously cover the cases SN (κ > 0), HN (κ < 0), and EN (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 bi will be expressed in ambient coordinates xi [30]: Poincaré: xi = 2qi 1 + κq2 ; Beltrami: xi = qi√ 1 + κq2 . 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 HP = T P + V ( 4J− (1− κJ−)2 ) = 1 2 ( 1 + κq2 )2 p2 + V ( 4q2 (1− κq2)2 ) + N∑ i=1 2bi x2 i , HB = T B + V (J−) = 1 2 (1 + κq2) ( p2 + κ(q · p)2 ) + V ( q2 ) + N∑ i=1 bi 2x2 i , (2.8) 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: HP = T P + 4ω2J− (1− κJ−)2 = 1 2 ( 1 + κq2 )2 p2 + 4ω2q2 (1− κq2)2 + N∑ i=1 2bi x2 i , HB = T B + ω2J− = 1 2 (1 + κq2) ( p2 + κ(q · p)2 ) + ω2q2 + N∑ i=1 bi 2x2 i . This is a MS Hamiltonian and the remaining constant of the motion can be chosen from any of the following N functions: IP i = ( pi(1− κq2) + 2κ(q · p)qi )2 + 8ω2q2 i (1− κq2)2 + bi (1− κq2)2 q2 i , IB i = (pi + κ(q · p)qi) 2 + 2ω2q2 i + bi/q2 i , i = 1, . . . , N. Quantum Deformations and Variable Curvature 5 • 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 HP = T P − k ( 4J− (1− κJ−)2 )−1/2 = 1 2 ( 1 + κq2 )2 p2 − k (1− κq2) 2 √ q2 + N∑ i=1 2bi x2 i , HB = T B − kJ −1/2 − = 1 2 (1 + κq2) ( p2 + κ(q · p)2 ) − k√ q2 + N∑ i=1 bi 2x2 i . This is again a MS system provided that, at least, one bi = 0. In this case the remaining constant of the motion turns out to be LP i = N∑ l=1 ( pl(1− κq2) + 2κ(q · p)ql ) (qlpi − qipl) + kqi 2 √ q2 − N∑ l=1;l 6=i bl qi(1− κq2) q2 l , LB i = N∑ l=1 (pl + κ(q · p)ql) (qlpi − qipl) + kqi√ q2 − N∑ l=1;l 6=i bl qi q2 l . (2.9) If another bj = 0, then LP,B j is also a new constant of the motion. In this way the proper curved KC system [37] (with all the bi’s equal to zero) is obtained, and in that case (2.9) are just the N components of the Laplace–Runge–Lenz vector on SN (κ > 0) and HN (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. 3 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. Theorem 2. Let {q,p} = {(q1, . . . , qN ), (p1, . . . , pN )} be N pairs of canonical variables. The ND Hamiltonian H(N) z = Hz ( q2, p̃2 z, (q · p)z ) , (3.1) where Hz is any smooth function and q2 = N∑ i=1 q2 i , p̃2 z = N∑ i=1 ( sinh zq2 i zq2 i p2 i + zbi sinh zq2 i ) ezK (N) i (q2), (q · p)z = N∑ i=1 sinh zq2 i zq2 i qipi ezK (N) i (q2), with K (h) i (q2) = − i−1∑ k=1 q2 k + h∑ l=i+1 q2 l , (3.2) 6 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso is QMS for any choice of the function H and for arbitrary real parameters bi. The (2N − 3) functionally independent and “universal” integrals of the motion are given by C(m) z = m∑ 1≤i<j Qz ij ezK (m) ij (q2) + m∑ i=1 bi e2zK (m) i (q2), Cz,(m) = N∑ N−m+1≤i<j Qz ij ezK̃ (N−m+1) ij (q2) + N∑ i=N−m+1 bi e2zK̃ (N−m+1) i (q2), (3.3) where m = 2, . . . , N , C (N) z = Cz,(N), and K (h) ij (q2) = K (h) i (q2) + K (h) j (q2) = −2 i−1∑ k=1 q2 k − q2 i + q2 j + 2 h∑ l=j+1 q2 l , K̃ (h) i (q2) = − i−1∑ k=h q2 k + N∑ l=i+1 q2 l , K̃ (h) ij (q2) = K̃ (h) i (q2) + K̃ (h) j (q2) = −2 i−1∑ k=h q2 k − q2 i + q2 j + 2 N∑ l=j+1 q2 l , Qz ij = { sinh zq2 i zq2 i sinh zq2 j zq2 j (qipj − qjpi) 2 + ( bi sinh zq2 j sinh zq2 i + bj sinh zq2 i sinh zq2 j )} , with i < j. Moreover, the sets of N functions {H(N) z , C (m) z } and {H(N) z , Cz,(m)} (m = 2, . . . , N) are in involution. 3.1 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, slz(2, R), coming (under a certain symplectic realiza- tion) from the non-standard quantum deformation of sl(2, R) [38, 39] where z is the deformation parameter (q = ez). 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 slz(2, R) Poisson coalgebra is given by the following deformed Poisson brackets and coproduct [22]: {J3, J+} = 2J+ cosh zJ−, {J3, J−} = −2 sinh zJ− z , {J−, J+} = 4J3, (3.4) ∆z(J−) = J− ⊗ 1 + 1, ∆z(Jl) = Jl ⊗ ezJ− + e−zJ− ⊗ Jl, l = +, 3. (3.5) The Casimir function for slz(2, R) reads Cz = sinh zJ− z J+ − J2 3 . (3.6) A one-particle symplectic realization of (3.4) is given by J (1) − = q2 1, J (1) + = sinh zq2 1 zq2 1 p2 1 + zb1 sinh zq2 1 , J (1) 3 = sinh zq2 1 zq2 1 q1p1, where b1 is a real parameter that labels the representation through Cz = b1. Quantum Deformations and Variable Curvature 7 Now the essential point is the fact that the coalgebra approach [34] provides the corresponding N -particle symplectic realization of slz(2, R) through the N -sites coproduct of (3.5) living on slz(2, R)⊗ · · ·N) ⊗ slz(2, R) [22]: J (N) − = N∑ i=1 q2 i ≡ q2, J (N) 3 = N∑ i=1 sinh zq2 i zq2 i qipi ezK (N) i (q2) ≡ (q · p)z, J (N) + = N∑ i=1 ( sinh zq2 i zq2 i p2 i + zbi sinh zq2 i ) ezK (N) i (q2) ≡ p̃2 z, (3.7) where K (N) i (q2) is defined in (3.2) and bi are N arbitrary real parameters that label the rep- resentation 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 {f, g} = N∑ i=1 ( ∂f ∂qi ∂g ∂pi − ∂g ∂qi ∂f ∂pi ) . Therefore the Hamiltonian (3.1) is obtained through an arbitrary smooth function Hz defined on the N -particle symplectic realization of the generators of slz(2, R): H(N) z = Hz ( J (N) − , J (N) + , J (N) 3 ) = Hz ( q2, p̃2 z, (q · p)z ) . (3.8) By construction [34], the functions (3.7) Poisson commute with the (2N − 3) functions (3.3) given by the sets C (m) z and Cz,(m), which are obtained from the “left” and “right” m-th copro- ducts of the Casimir (3.6) with m = 2, 3, . . . , N [35]. For instance, the C (m) z integrals are nothing but C(m) z = sinh zJ (m) − z J (m) + − ( J (m) 3 )2 , and the right ones Cz,(m) can be obtained through an appropriate permutation of the labelling of the lattice sites (note that these integrals depend on the canonical coordinates running from (N −m + 1) up to N). Thus H (N) z Poisson commutes with the (2N − 3) integrals and, further- more, the coalgebra symmetry also ensures that each of the subsets {C(2) z , . . . , C (N) z ,H (N) z } and {Cz,(2), . . . , Cz,(N),H (N) z } consists of N functions in involution. In order to prove the functional independence of the 2N−2 functions {C(2) z , C (3) z , . . . , C (N) z ≡ Cz,(N), Cz,(N−1), . . . , Cz,(2),H (N) z } it suffices to realize that such functions are just deformations in the deformation parameter z of the sl(2, R) integrals given by (2.2), and the latter (which are recovered when z → 0) are indeed functionally independent. Thus, we conclude that any arbitrary function Hz (3.8) defines a QMS Hamiltonian system. 3.2 The N = 2 case In order to illustrate the previous construction, let us explicitly write the 2-particle symplectic realization of slz(2, R) (3.7): J (2) − = q2 1 + q2 2, J (2) 3 = sinh zq2 1 zq2 1 ezq2 2q1p1 + sinh zq2 2 zq2 2 e−zq2 1q2p2, J (2) + = sinh zq2 1 zq2 1 ezq2 2p2 1 + sinh zq2 2 zq2 2 e−zq2 1p2 2 + zb1 sinh zq2 1 ezq2 2 + zb2 sinh zq2 2 e−zq2 1 . 8 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso In this case there is a single (left and right) constant of the motion: C(2) z = sinh zJ (2) − z J (2) + − ( J (2) 3 )2 . After some straightforward computations this integral can be expressed as C(2) z = sinh zq2 1 zq2 1 sinh zq2 2 zq2 2 (q1p2 − q2p1) 2 ez(q2 2−q2 1) + b1e2zq2 2 + b2e−2zq2 1 + ( b1 sinh zq2 2 sinh zq2 1 + b2 sinh zq2 1 sinh zq2 2 ) ez(q2 2−q2 1). (3.9) By construction, this constant of the motion will Poisson-commute with all the Hamiltonians H(2) z = Hz ( J (2) − , J (2) + , J (2) 3 ) . 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 ≡ Cz,(2); this fact does not exclude that there could be some specific choices for Hz for which an additional integral does exist. When N ≥ 3, Theorem 2 will always provide QMS Hamiltonians. 4 Free motion on 2D and 3D curved manifolds 4.1 2D curved manifolds Throughout this Section we will consider only free motion. Therefore we shall take the symplectic realization with b1 = b2 = 0 in order to avoid centrifugal potential terms. In general, we can consider an infinite family of integrable (and quadratic in the momenta) free N = 2 motions with slz(2, R) coalgebra symmetry through Hamiltonians of the type H(2) z = 1 2 J (2) + f ( zJ (2) − ) , (4.1) where f is an arbitrary smooth function such that lim z→0 f ( zJ (2) − ) = 1, that is, lim z→0 H (2) z = 1 2(p2 1 + p2 2). We shall explore in the sequel some specific choices for f , and we shall analyse the spaces generated by them. 4.1.1 An integrable case Of course, the simplest choice will be just to set f ≡ 1 [31]: HI z = 1 2 J (2) + = 1 2 ( sinh zq2 1 zq2 1 ezq2 2p2 1 + sinh zq2 2 zq2 2 e−zq2 1p2 2 ) . (4.2) Hence the kinetic energy T I z (qi, pi) coming from HI z is T I z (qi, q̇i) = 1 2 ( zq2 1 sinh zq2 1 e−zq2 2 q̇2 1 + zq2 2 sinh zq2 2 ezq2 1 q̇2 2 ) , (4.3) and defines a geodesic flow on a 2D Riemannian space with signature diag(+,+) and metric given by: ds2 I = 2zq2 1 sinh zq2 1 e−zq2 2 dq2 1 + 2zq2 2 sinh zq2 2 ezq2 1 dq2 2. (4.4) Quantum Deformations and Variable Curvature 9 The Gaussian curvature K for this space can be computed through K = −1 √ g11g22 { ∂ ∂q1 ( 1 √ g11 ∂ √ g22 ∂q1 ) + ∂ ∂q2 ( 1 √ g22 ∂ √ g11 ∂q2 )} , and turns out to be non-constant and negative: K(q1, q2; z) = −z sinh ( z(q2 1 + q2 2) ) . 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 λ2 6= 0: cosh(λ1ρ) = exp { z(q2 1 + q2 2) } , sin2(λ2θ) = exp { 2zq2 1 } − 1 exp { 2z(q2 1 + q2 2) } − 1 , where z = λ2 1 and λ2 can take either a real or a pure imaginary value. Note that the new variable cosh(λ1ρ) 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 ρ → 2(q2 1 + q2 2), sin2(λ2θ) → q2 1 q2 1 + q2 2 . Thus ρ can be interpreted as a radial coordinate and θ is either a circular (λ2 real) or a hy- perbolic angle (λ2 imaginary). Notice that in the latter case, say λ2 = i, the coordinate q1 is imaginary and can be written as q1 = iq̃1 where q̃1 is a real coordinate; then ρ → 2(q2 2 − q̃2 1) which corresponds to a relativistic radial distance. Therefore the introduction of the additional parameter λ2 will allow us to obtain Lorentzian metrics. In this new coordinates, the metric (4.4) reads ds2 I = 1 cosh(λ1ρ) ( dρ2 + λ2 2 sinh2(λ1ρ) λ2 1 dθ2 ) = 1 cosh(λ1ρ) ds2 0, where ds2 0 is just the metric of the 2D Cayley–Klein spaces in terms of geodesic polar coordi- nates [40, 41] provided that we identify z = λ2 1 ≡ −κ1 and λ2 2 ≡ κ2; hence λ2 determines the signature of the metric. The Gaussian curvature turns out to be K(ρ) = −1 2 λ2 1 sinh2(λ1ρ) cosh(λ1ρ) . In this way we find the following spaces, whose main properties are summarized in Table 1: • When λ2 is real, we get a 2D deformed sphere S2 z (z < 0), and a deformed hyperbolic or Lobachewski space H2 z (z > 0). • When λ2 is imaginary, we obtain a deformation of the (1+1)D anti-de Sitter spacetime AdS1+1 z (z < 0) and of the de Sitter one dS1+1 z (z > 0). • In the non-deformed case z → 0, the Euclidean space E2 (λ2 real) and Minkowskian spacetime M1+1 (λ2 imaginary) are recovered. Accordingly, the kinetic energy (4.3) is transformed into T I z (ρ, θ; ρ̇, θ̇) = 1 2 cosh(λ1ρ) ( ρ̇2 + λ2 2 sinh2(λ1ρ) λ2 1 θ̇2 ) , 10 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso Table 1. Metric and Gaussian curvature of the 2D spaces with slz(2, R) coalgebra symmetry for different values of the deformation parameter z = λ2 1 and signature parameter λ2. 2D deformed Riemannian spaces (1 + 1)D deformed relativistic spacetimes • Deformed sphere S2 z • Deformed anti-de Sitter spacetime AdS1+1 z z = −1; (λ1, λ2) = (i, 1) z = −1; (λ1, λ2) = (i, i) ds2 = 1 cos ρ ( dρ2 + sin2 ρ dθ2) ds2 = 1 cos ρ ( dρ2 − sin2 ρ dθ2) K = − sin2 ρ 2 cos ρ K = − sin2 ρ 2 cos ρ • Euclidean space E2 • Minkowskian spacetime M1+1 z = 0; (λ1, λ2) = (0, 1) z = 0; (λ1, λ2) = (0, i) ds2 = dρ2 + ρ2dθ2 ds2 = dρ2 − ρ2dθ2 K = 0 K = 0 • Deformed hyperbolic space H2 z • Deformed de Sitter spacetime dS1+1 z z = 1; (λ1, λ2) = (1, 1) z = 1; (λ1, λ2) = (1, i) ds2 = 1 cosh ρ ( dρ2 + sinh2 ρ dθ2) ds2 = 1 cosh ρ ( dρ2 − sinh2 ρ dθ2) K = − sinh2 ρ 2 cosh ρ K = − sinh2 ρ 2 cosh ρ and the free motion Hamiltonian (4.2) is written as H̃I z = 1 2 cosh(λ1ρ) ( p2 ρ + λ2 1 λ2 2 sinh2(λ1ρ) p2 θ ) , where H̃I z = 2HI z. There is a unique constant of the motion C (2) z ≡ Cz,(2) (3.9) which in terms of the new phase space is simply given by C̃z = p2 θ, provided that C̃z = 4λ2 2C (2) z . This allows us to apply a radial-symmetry reduction: H̃I z = 1 2 cosh(λ1ρ) p2 ρ + λ2 1 cosh(λ1ρ) 2λ2 2 sinh2(λ1ρ) C̃z. We remark that the explicit integration of the geodesic motion on all these spaces can be explicitly performed in terms of elliptic integrals. 4.1.2 The superintegrable case A MS Hamiltonian is given by HMS z = 1 2 J (2) + ezJ (2) − = 1 2 ( sinh zq2 1 zq2 1 ezq2 1e2zq2 2p2 1 + sinh zq2 2 zq2 2 ezq2 2p2 2 ) , since there exists an additional (and functionally independent) constant of the motion [22]: Iz = sinh zq2 1 2zq2 1 ezq2 1p2 1. (4.5) Quantum Deformations and Variable Curvature 11 This choice corresponds to the kinetic energy T MS z (qi, q̇i) = 1 2 ( zq2 1 sinh zq2 1 e−zq2 1e−2zq2 2 q̇2 1 + zq2 2 sinh zq2 2 e−zq2 2 q̇2 2 ) , whose associated metric is ds2 MS = 2zq2 1 sinh zq2 1 e−zq2 1e−2zq2 2 dq2 1 + 2zq2 2 sinh zq2 2 e−zq2 2 dq2 2. Surprisingly enough, the computation of the Gaussian curvature K for ds2 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 λ2) transforms the metric into ds2 MS = dr2 + λ2 2 sin2(λ1r) λ2 1 dθ2, which exactly coincides with the metric of the Cayley–Klein spaces written in geodesic polar coordinates (r, θ) provided that now z = λ2 1 ≡ κ1 and λ2 2 ≡ κ2. Obviously, after this change of variables the geodesic motion can be reduced to a “radial” 1D system: H̃MS z = 1 2 p2 r + λ2 1 2λ2 2 sin2(λ1r) C̃z, where H̃MS z = 2HMS z and C̃z = p2 θ is, as in the previous case, the usual generalized momentum for the θ coordinate. 4.1.3 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 K(x) = z ( f ′(x) cosh x + ( f ′′(x)− f(x)− f ′2(x) f(x) ) sinhx ) , where x ≡ zJ− = z(q2 1 + q2 2), f ′ = df(x) dx and f ′′ = d2f(x) dx2 . 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 K/z = f ′ coshx + ( f ′′ − f − (f ′)2/f ) sinhx = f ( g coshx + (g′ − 1) sinh x ) . If we now require K to be a constant we get the equation K ′ = 0 ≡ 2y coshx + y′ sinhx = 0, where y := 2g′ + g2 − 1. The solution for this equation yields y = A sinh2 x , where A is a constant, and solving for g, we get for F := f 1 2 the equation F ′′ = 1 4 ( 1 + A sinh2 x ) F, 12 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso whose general solution is (A := λ(λ− 1)): F = (sinhx)λ { C1 ( sinh(x/2) )(1−2λ) + C2 ( cosh(x/2) )(1−2λ) } , where C1 and C2 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 Hz = 1 2 J+e±zJ− , and the curvature of their associated spaces is K = ±z. 4.2 3D curved manifolds The study of the 3D case follows exactly the same pattern. The three-particle symplectic realization of slz(2, R) (with all bi = 0) is obtained from (3.7): J (3) − = q2 1 + q2 2 + q2 3 ≡ q2, J (3) + = sinh zq2 1 zq2 1 p2 1 ezq2 2ezq2 3 + sinh zq2 2 zq2 2 p2 2 e−zq2 1ezq2 3 + sinh zq2 3 zq2 3 p2 3 e−zq2 1e−zq2 2 , J (3) 3 = sinh zq2 1 zq2 1 q1p1ezq2 2ezq2 3 + sinh zq2 2 zq2 2 q2p2e−zq2 1ezq2 3 + sinh zq2 3 zq2 3 q3p3e−zq2 1e−zq2 2 . By construction, these generators Poisson-commute with the three integrals {C(2) z , C (3) z ≡ Cz,(3), Cz,(2)} given in (3.3): C(2) z = sinh zq2 1 zq2 1 sinh zq2 2 zq2 2 (q1p2 − q2p1) 2 e−zq2 1ezq2 2 , Cz,(2) = sinh zq2 2 zq2 2 sinh zq2 3 zq2 3 (q2p3 − q3p2) 2 e−zq2 2ezq2 3 , C(3) z = sinh zq2 1 zq2 1 sinh zq2 2 zq2 2 (q1p2 − q2p1) 2 e−zq2 1ezq2 2e2zq2 3 (4.6) + sinh zq2 1 zq2 1 sinh zq2 3 zq2 3 (q1p3 − q3p1) 2 e−zq2 1ezq2 3 + sinh zq2 2 zq2 2 sinh zq2 3 zq2 3 (q2p3 − q3p2) 2 e−2zq2 1e−zq2 2ezq2 3 . 4.2.1 QMS free motion: non-constant curvature If we now consider the kinetic energy Tz(qi, q̇i) coming from the Hamiltonian Hz(qi, pi) = 1 2 J (3) + , (4.7) it corresponds to the free Lagrangian [33] Tz = 1 2 ( zq2 1 sinh zq2 1 e−zq2 2e−zq2 3 q̇2 1 + zq2 2 sinh zq2 2 ezq2 1e−zq2 3 q̇2 2 + zq2 3 sinh zq2 3 ezq2 1ezq2 2 q̇2 3 ) , Quantum Deformations and Variable Curvature 13 that defines a geodesic flow on a 3D Riemannian space with metric ds2 = 2zq2 1 sinh zq2 1 e−zq2 2e−zq2 3 dq2 1 + 2zq2 2 sinh zq2 2 ezq2 1e−zq2 3 dq2 2 + 2zq2 3 sinh zq2 3 ezq2 1ezq2 2 dq2 3. The corresponding sectional curvatures Kij are K12 = z 4 e−zq2 ( 1 + e2zq2 3 − 2e2zq2 ) , K13 = z 4 e−zq2 ( 2− e2zq2 3 + e2zq2 2e2zq2 3 − 2e2zq2 ) , K23 = z 4 e−zq2 ( 2− e2zq2 2e2zq2 3 − 2e2zq2 ) . The following nice expression for the scalar curvature K is found: K12 + K13 + K23 = −5 2 z sinh(zq2) = K/2. Once again, the radial symmetry can be explicitly emphasized through new canonical coor- dinates (ρ, θ, φ) defined by: cosh2(λ1ρ) = e2zq2 , sinh2(λ1ρ) cos2(λ2θ) = e2zq2 1e2zq2 2 ( e2zq2 3 − 1 ) , sinh2(λ1ρ) sin2(λ2θ) cos2 φ = e2zq2 1 ( e2zq2 2 − 1 ) , sinh2(λ1ρ) sin2(λ2θ) sin2 φ = e2zq2 1 − 1, (4.8) where z = λ2 1 and λ2 6= 0 is the additional signature parameter, that will allow for the presence of relativistic spaces. Under this change of variables, the metric is transformed into ds2 = 1 cosh(λ1ρ) ( dρ2 + λ2 2 sinh2(λ1ρ) λ2 1 ( dθ2 + sinh2(λ2θ) λ2 2 dφ2 )) . This is just the metric of the 3D Riemannian and relativistic spacetimes written in geodesic polar coordinates and multiplied by a global factor 1/cosh(λ1ρ) that encodes the information concerning the variable curvature of the space. Sectional and scalar curvatures are now written in the form K12 = K13 = −1 2 λ2 1 sinh2(λ1ρ) cosh(λ1ρ) , K23 = K12/2, K = −5 2 λ2 1 sinh2(λ1ρ) cosh(λ1ρ) . Therefore, according to the values of (λ1, λ2) we have obtained a deformation of the 3D sphere (i, 1), hyperbolic (1, 1), de Sitter (1, i) and anti-de Sitter (i, i) spaces. The “classical” limit z → 0 corresponds to a zero-curvature limit leading to the proper Euclidean (0, 1) and Minkowskian (0, i) spaces. The QMS Hamiltonian (4.7), that determines the free motion on the above spaces, and its three integrals of the motion (4.6) are written in terms of the new canonical coordinates (ρ, θ, φ) and conjugated momenta (pρ, pθ, pφ) as H̃z = 1 2 cosh(λ1ρ) ( p2 ρ + λ2 1 λ2 2 sinh2(λ1ρ) ( p2 θ + λ2 2 sin2(λ2θ) p2 φ )) , C̃(2) z = p2 φ, C̃(3) z = p2 θ + λ2 2 sin2(λ2θ) p2 φ, C̃z,(2) = ( cos φ pθ − λ2 sinφ pφ tan(λ2θ) )2 , (4.9) provided that H̃z = 2Hz, C̃ (2) z = 4C (2) z , C̃z,(2) = 4λ2 2Cz,(2) and C̃ (3) z = 4λ2 2C (3) z . 14 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso Furthermore the set of three functions {C̃(2) z , C̃ (3) z , H̃z}, which characterizes the complete integrability of the Hamiltonian, allows us to write three equations, each of them depending on a canonical pair: C̃(2) z (φ, pφ) = p2 φ, C̃(3) z (θ, pθ) = p2 θ + λ2 2 sin2(λ2θ) C̃(2) z , H̃z(ρ, pρ) = 1 2 cosh(λ1ρ) ( p2 ρ + λ2 1 λ2 2 sinh2(λ1ρ) C̃(3) z ) . (4.10) Therefore the Hamiltonian is separable and reduced to a 1D radial system. 4.2.2 MS free motion: constant curvature The following choice for the Hamiltonian HMS z = 1 2 J (3) + ezJ (3) − , yields a MS system since it has four (functionally independent) constants of motion, the three universal integrals (4.6) together with Iz (4.5). In fact, this the 3D version of the Hamiltonian described in Section 4.1.2. The associated kinetic energy is T MS z = 1 2 ( zq2 1 sinh zq2 1 ezq2 1 q̇2 1 + zq2 2 sinh zq2 2 e2zq2 1ezq2 2 q̇2 2 + zq2 3 sinh zq2 3 e2zq2 1e2zq2 2ezq2 3 q̇2 3 ) , and the underlying metric reads ds2 MS = 2zq2 1 sinh zq2 1 ezq2 1 dq2 1 + 2zq2 2 sinh zq2 2 e2zq2 1ezq2 2 dq2 2 + 2zq2 3 sinh zq2 3 e2zq2 1e2zq2 2ezq2 3 dq2 3. (4.11) This space is again a Riemannian one with constant sectional and scalar curvatures given by Kij = z, K = 6z. 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): ds2 MS = dr2 + λ2 2 sin2(λ1r) λ2 1 ( dθ2 + sin2(λ2θ) λ2 2 dφ2 ) . Therefore, according to the values of (λ1, λ2), 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, H̃MS z = 2HMS z , is written as H̃MS z = 1 2 ( p2 r + λ2 1 λ2 2 sin2(λ1r) ( p2 θ + λ2 2 sin2(λ2θ) p2 φ )) , and the four functionally independent integrals are given by (4.9) and Ĩz = ( λ2 sin(λ2θ) sinφ pr + λ1 cos(λ2θ) sinφ tan(λ1r) pθ + λ1λ2 cos φ tan(λ1r) sin(λ2θ) pφ )2 , where Ĩz = 4λ2 2Iz. The two sets {H̃MS z , C̃ (2) z , C̃ (3) z } and {H̃MS z , C̃z,(2), Ĩz} consist of three functions in involution. Similarly to (4.10), this Hamiltonian is also separable and can be reduced to a 1D system: H̃MS z (r, pr) = 1 2 ( p2 r + λ2 1 λ2 2 sin2(λ1r) C̃(3) z ) . Quantum Deformations and Variable Curvature 15 5 ND 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 ND geodesic motion that, in the simplest case, reads: Hz = 1 2 J (N) + = 1 2 N∑ i=1 sinh zq2 i zq2 i p2 i exp ( −z i−1∑ k=1 q2 k + z N∑ l=i+1 q2 l ) . The geometric characterization of the underlying ND 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 Hz = 1 2 N∑ i=1 sz(q2 i ) p2 i exp z N∑ k=1;k 6=i sgn(k − i)q2 k , where sz(q2 i ) = sinh zq2 i /(zq2 i ) and sgn(k − i) is the sign of the difference k − i, we get again a free Lagrangian: Tz = 1 2 N∑ i=1 (q̇i)2 exp ( −z N∑ k=1;k 6=i sgn(k − i)q2 k ) sz(q2 i ) , with the corresponding (diagonal) metric given by ds2 = N∑ i=1 gii(q) dq2 i , gii(q) = exp ( −z N∑ k=1;k 6=i sgn(k − i)q2 k ) sz(q2 i ) . (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] (ξ0, ξ1, . . . , ξN ). They play a similar role to the ambient coordinates arising when ND Riemannian spaces of constant curvature are embedded within RN+1. • N “intrinsic” variables (ρ, θ2, . . . , θN ) which describe the ND space itself. They are the analogous to the geodesic polar coordinates on ND Riemannian spaces of constant curva- ture [10, 11]. The above coordinates are defined in terms of the initial qi by: ξ2 0 = cosh2(λ1ρ) := N∏ i=1 exp(2zq2 i ), ξ2 k = sinh2(λ1ρ) k∏ j=2 sin2 θj cos2 θk+1 := N−k∏ i=1 exp(2zq2 i ) ( exp(2zq2 N−k+1)− 1 ) , (5.2) ξ2 N = sinh2(λ1ρ) N∏ j=2 sin2 θj := exp(2zq2 1)− 1, 16 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso where z = λ2 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 λ2 (which would have been associated with θ2). This definition is the ND generalization of the change of coordinates (4.8) given in the 3D case with θ = θ2, φ = θ3 and λ2 = 1. Clearly, the N + 1 collective variables are not independent and they fulfil a pseudosphere relation (of hyperbolic type): ξ2 0 − N∑ k=1 ξ2 k = 1. The geodesic flow in the canonical coordinates (ρ, θ) and momenta (ρ, pθ) is then given by the Hamiltonian H̃z = 2Hz: H̃z = 1 2 cosh(λ1ρ) p2 ρ + λ2 1 sinh2(λ1ρ) N∑ i=2 i−1∏ j=2 1 sin2 θj  p2 θi , and the (left) integrals of the motion C̃ (m) z = 4C (m) z are found to be C̃(m) z = N∑ i=N−m+2  i−1∏ j=N−m+2 1 sin2 θj  p2 θi , m = 2, . . . , N. By taking into account the N functions {H̃z, C̃ (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: C̃(2) z (θN , pθN ) = p2 θN , C̃(m) z (θN−m+2, pθN−m+2 ) = p2 θN−m+2 + C (m−1) z sin2 θN−m+2 , m = 3, . . . , N, H̃z(ρ, pρ) = 1 2 cosh(λ1ρ) ( p2 ρ + λ2 1 sinh2(λ1ρ) C̃(N) z ) . 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 ND spaces with constant curva- ture can be obtained by considering the MS Hamiltonian HMS z = 1 2 J (N) + ezJ (N) − = 1 2 ezq2 N∑ i=1 sinh zq2 i zq2 i p2 i exp ( −z i−1∑ k=1 q2 k + z N∑ l=i+1 q2 l ) . (5.3) Under a suitable change of coordinates, similar to (5.2) but involving a different radial coordi- nate r instead of ρ, this Hamiltonian leads to the MS geodesic motion on SN , HN and EN in the proper geodesic polar coordinates which can be found in [10, 11]. 6 QMS potentials As we have just noticed, we can also consider more general ND QMS Hamiltonians based on slz(2, R) (3.7) by considering arbitrary bi’s (contained in J+) and adding some functions depend- ing on J−; hereafter we drop the index “(N)” in the generators. The family of Hamiltonians Quantum Deformations and Variable Curvature 17 that we consider has the form (see [32] for the 2D construction): Hz = 1 2 J+ f(zJ−) + U(zJ−), where the arbitrary smooth functions f and U are such that lim z→0 U(zJ−) = V(J−), lim z→0 f(zJ−) = 1. This, in turn, means that lim z→0 Hz = 1 2 p2 + V(q2) + N∑ i=1 bi 2q2 i , recovering the superposition of a central potential V(J−) ≡ V(q2) with N centrifugal terms on EN [36]. Such a “flat” system has a (non-deformed) sl(2, R) coalgebra symmetry as given in Theorem 1. We recall that the function f(zJ−) gives us the type of curved background, which is character- ized by the metric ds2/f(zq2) where ds2 is the variable curvature metric associated to Hz = 1 2J+ and given in (5.1). The two special cases with f(zJ−) = e±zJ− give rise to Riemannian spaces of constant sectional curvatures, all equal to ±z (as (5.3)). In particular, QMS deformations of the ND SW system would be given by any U such that lim z→0 U(zJ−) = ωJ−, and for the ND generalized KC potential we can consider U functions such that lim z→0 U(zJ−) = −k/ √ J−. In both cases centrifugal type potentials come from the bi’s terms contained in J+f(zJ−). With account of the geometrical arguments, the following QMS SW system on spaces with non-constant curvature (5.1) has been proposed in [32]: HSW z = 1 2 J+ + ω sinh zJ− z , while a candidate for a generalized KC system on such spaces is given by the formula [32]: HKC z = 1 2 J+ − k √ 2z e2zJ− − 1 e2zJ− . In the constant curvature case, the approach here presented allows us to recover the known results for the SW potential on Riemannian spaces with constant curvature, as well as their generalization to relativistic spaces (whenever the signature parameter λ2 is considered). In particular, the MS SW system on ND spaces of constant curvature is given by the Hamil- tonian HMS,SW z = 1 2 J+ezJ− + ω sinh zJ− z ezJ− ≡ HSW z ezJ− , and the additional constant of the motion that provides the MS property reads Iz = sinh zq2 1 2zq2 1 ezq2 1p2 1 + zb1 2 sinh zq2 1 ezq2 1 + ω 2z e2zq2 1 . The corresponding results for the generalized KC system are currently under investigation. 18 O. Ragnisco, Á. Ballesteros, F.J. Herranz and F. Musso 7 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). 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Gen. 36 (2003), 10505–10518, math-ph/0307013. http://arxiv.org/abs/math-ph/9910041 http://arxiv.org/abs/math-ph/0110019 http://arxiv.org/abs/math-ph/0307013 1 Introduction 2 QMS Hamiltonians with sl(2,R) coalgebra symmetry 2.1 Free motion on Riemannian spaces of constant curvature 2.2 Superintegrable potentials on Riemannian spaces of constant curvature 3 QMS Hamiltonians with quantum deformed sl(2,R) coalgebra symmetry 3.1 The proof 3.2 The N=2 case 4 Free motion on 2D and 3D curved manifolds 4.1 2D curved manifolds 4.1.1 An integrable case 4.1.2 The superintegrable case 4.1.3 A more general case 4.2 3D curved manifolds 4.2.1 QMS free motion: non-constant curvature 4.2.2 MS free motion: constant curvature 5 ND spaces with variable curvature 6 QMS potentials 7 Concluding remarks References

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

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