Francisco J. Herranz

Maximal superintegrability on N-dimensional curved spaces

A unified algebraic construction of the classical Smorodinsky–Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO(N, 1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space N+1, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N + 1 oscillators. Furthermore, each Lie algebra generator provides an integral of motion and a set of 2N − 1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky–Winternitz system is shown for any value of the curvature.

Conformal symmetries of spacetimes

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley–Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton–Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Möbius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley–Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional ‘conformal ambient space’, which in turn leads to an explicit description of the ‘conformal completion’ or compactification of the nine spaces.In this paper, we give a unified and global new approach to the study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, and both Newton–Hooke and Galilean). We obtain general expressions within a Cayley–Klein framework, holding simultaneously for all these nine spaces, whose cycles (including geodesics and circles) are explicitly characterized in a new way. The corresponding cycle-preserving symmetries, which give rise to (Mobius-like) conformal Lie algebras, together with their differential realizations are then deduced without having to resort to solving the conformal Killing equations. We show that each set of three spaces with the same signature type and any curvature have isomorphic conformal algebras; these are related through an apparently new conformal duality. Laplace and wave-type differential equations with conformal algebra symmetry are finally constructed.

Quantum mechanics on spaces of nonconstant curvature: The oscillator problem and superintegrability

The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N-dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N-dimensional space with nonconstant curvature.

Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries

A simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two-dimensional Cayley-Klein geometries. Their central extensions are also considered under this unified pattern. In both cases some fundamental properties characterizing the classical CK geometries (as the existence of a set of commuting involutions, contractions and dualities relationships), remain in the quantum version.

Maximal superintegrability of the generalized Kepler–Coulomb system on N-dimensional curved spaces

The superposition of the Kepler–Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable (Verrier and Evans 2008 J. Math. Phys. 49 022902) by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper, we present the generalization of this result to the N-dimensional spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler–Coulomb potential together with N centrifugal terms, is shown to be endowed with 2N − 1 functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler–Coulomb potential, with its associated quadratic Laplace–Runge–Lenz N-vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic) and coalgebra symmetries in all these systems is highlighted.

A new “null-plane” quantum Poincaré algebra

Abstract A new quantum deformation, which we call null-plate, of the (3+1) Poincare algebra is obtained. The algebraic properties of the classical null-plane description are generalized to this quantum deformation. In particular, the classical isotopy subalgebra of the null-plane is deformed into a Hopf subalgebra, and deformed spin operators having classical commutation rules can be defined. Quantum Hamiltonian, mass and position operators are studied, and the null-plane evolution is expressed in terms of a deformed Schrodinger equation.

A maximally superintegrable system on an n-dimensional space of nonconstant curvature

Abstract We present a novel Hamiltonian system in n dimensions which admits the maximal number 2 n − 1 of functionally independent, quadratic first integrals. This system turns out to be the first example of a maximally superintegrable Hamiltonian on an n -dimensional Riemannian space of nonconstant curvature, and it can be interpreted as the intrinsic Smorodinsky–Winternitz system on such a space. Moreover, we provide three different complete sets of integrals in involution and solve the equations of motion in closed form.

Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky–Winternitz and generalized Kepler–Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results presented here are a consequence of the Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated with Poincare and Beltrami coordinates.

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.

Cayley-Klein algebras as graded contractions of so(N+1)

We study Z2(X)N graded contractions of the real compact simple Lie algebra so(N+1), and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.