arXiv subjects

Francisco J. Herranz

Publications and source records attributed to Francisco J. Herranz.

At least 55 records · Page 3Linked to original sources

arXiv2017

Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

The $κ$-deformation of the (2+1)D anti-de Sitter, Poincaré and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel'd-double and the Poisson-Lie structure underlying the $κ$-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the $κ$-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of "duality" between the cosmological constant and the Planck scale is also envisaged.

hep-th↗

arXiv2017

Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians ${\cal T}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential ${\cal U}_r$ leading to ${\cal H}_r={\cal T}_r +{\cal U}_r$. Secondly, we study the superintegrability of the four Hamiltonians $\widetilde{\cal H}_r= {\cal H}_r/ μ_r$, where $μ_r$ is a certain position-dependent mass, that enjoys the same separability as the original system ${\cal H}_r$. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.

math-ph↗

arXiv2016

The kappa-(A)dS quantum algebra in (3+1) dimensions

The quantum duality principle is used to obtain explicitly the Poisson analogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant $Λ$ is included as a Poisson-Lie group contraction parameter, and the limit $Λ\to 0$ leads to the well-known kappa-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this kappa-(A)dS deformation is sketched.

hep-th↗

arXiv2016

The anisotropic oscillator on curved spaces: A new exactly solvable model

We present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies $ω_x$ and $ω_y$. The new curved Hamiltonian ${H}_κ$ depends on the curvature $κ$ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of ${H}_κ$ relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies $ω_x: ω_y$, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the $κ\to 0$ limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known $1:1$ and $2:1$ anisotropic curved oscillators are recovered as particular cases of ${H}_κ$, meanwhile all the remaining $ω_x: ω_y$ curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian $\hat {H}_κ$ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the Hamiltonian $\hat {H}_κ$ turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as an analytical deformation of the Euclidean eigenvalues in terms of both the curvature $κ$ and the Planck constant $\hbar$. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

quant-ph↗

arXiv2016

Factorization approach to superintegrable systems: Formalism and applications

The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super)integrable anisotropic oscillators on the sphere are constructed. The Tremblay-Turbiner-Winternitz system on the Euclidean plane is also studied from this viewpoint.

math-ph↗

arXiv2015

The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

math-ph↗

arXiv2015

An integrable Henon-Heiles system on the sphere and the hyperbolic plane

We construct a constant curvature analogue on the two-dimensional sphere ${\mathbf S}^2$ and the hyperbolic space ${\mathbf H}^2$ of the integrable Hénon-Heiles Hamiltonian $\mathcal{H}$ given by $$ \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ Ω\left( q_{1}^{2}+ 4 q_{2}^{2}\right) +α\left( q_{1}^{2}q_{2}+2 q_{2}^{3}\right) , $$ where $Ω$ and $α$ are real constants. The curved integrable Hamiltonian $\mathcal{H}_κ$ so obtained depends on a parameter $κ$ which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system $\mathcal{H}$ is smoothly obtained in the zero-curvature limit $κ\to 0$. On the other hand, the Hamiltonian $\mathcal{H}_κ$ that we propose can be regarded as an integrable perturbation of a known curved integrable $1:2$ anisotropic oscillator. We stress that in order to obtain the curved Hénon-Heiles Hamiltonian $\mathcal{H}_κ$, the preservation of the full integrability structure of the flat Hamiltonian $\mathcal{H}$ under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani-Dorizzi-Grammaticos (RDG) series $\cal{V}_{n}$ of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved RDG potentials $\cal{V}_{κ, n} $ on ${\mathbf S}^2$ and ${\mathbf H}^2$ will be also explicitly presented.

nlin.SI↗

arXiv2015

Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

We present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding principle towards the description of (3+1) gravity. To this aim, a canonical classical $r$-matrix arising from a Drinfel'd double structure for the three (3+1) Lorentzian algebras is obtained. This $r$-matrix turns out to be a twisted version of the one corresponding to the (3+1) $κ$-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the $κ$-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

gr-qc↗

arXiv2015

A curved Henon-Heiles system and its integrable perturbations

The constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon-Heiles Hamiltonian $\mathcal{H}$ given by $$ \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ Ω\left(q_{1}^{2}+ 4 q_{2}^{2}\right) +α\left(q_{1}^{2}q_{2}+2 q_{2}^{3}\right), $$ where $Ω$ and $α$ are real constants, is revisited. The resulting integrable curved Hamiltonian, $\mathcal{H}_κ$, depends on a parameter $κ$ which is just the curvature of the underlying space and allows one to recover $\mathcal{H}$ under the smooth flat/Euclidean limit $κ\to 0$. This system can be regarded as an integrable cubic perturbation of a specific curved $1:2$ anisotropic oscillator, which was already known in the literature. The Ramani-Dorizzi-Grammaticos (RDG) series of potentials associated to $\mathcal{H}_κ$ is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of $\mathcal{H}$ that are separable in parabolic coordinates. Integrable perturbations of $\mathcal{H}_κ$ are also fully presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hamiltonian $\mathcal{H}$. It will be explicitly shown that the latter perturbations can be understood as the "negative index" counterpart of the curved RDG series of potentials. Furthermore, it is shown that the integrability of the curved Hénon-Heiles Hamiltonian $\mathcal{H}_κ$ is preserved under the simultaneous addition of curved analogues of "positive" and "negative" families of RDG potentials.

nlin.SI↗

arXiv2014

Exactly solvable deformations of the oscillator and Coulomb systems and their generalization

We present two maximally superintegrable Hamiltonian systems ${\cal H}_λ$ and ${\cal H}_η$ that are defined, respectively, on an $N$-dimensional spherically symmetric generalization of the Darboux surface of type III and on an $N$-dimensional Taub-NUT space. Afterwards, we show that the quantization of ${\cal H}_λ$ and ${\cal H}_η$ leads, respectively, to exactly solvable deformations (with parameters $λ$ and $η$) of the two basic quantum mechanical systems: the harmonic oscillator and the Coulomb problem. In both cases the quantization is performed in such a way that the maximal superintegrability of the classical Hamiltonian is fully preserved. In particular, we prove that this strong condition is fulfilled by applying the so-called conformal Laplace-Beltrami quantization prescription, where the conformal Laplacian operator contains the usual Laplace-Beltrami operator on the underlying manifold plus a term proportional to its scalar curvature (which in both cases has non-constant value). In this way, the eigenvalue problems for the quantum counterparts of ${\cal H}_λ$ and ${\cal H}_η$ can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation (in terms of the parameters $λ$ and $η$) of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out that the maximal degeneracy of both systems is preserved under deformation. Finally, new further multiparametric generalizations of both systems that preserve their superintegrability are envisaged.

quant-ph↗

arXiv2014

From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes

It is shown that the canonical classical $r$-matrix arising from the Drinfel'd double structure underlying the two-fold centrally extended (2+1) Galilean and Newton-Hooke Lie algebras (with either zero or non-zero cosmological constant $Λ$, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical $r$-matrices associated with the Drinfel'd double structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and Newton-Hooke Drinfel'd doubles is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting 'absolute time' coordinate ${\hat x}_0$ together with two noncommutative space coordinates $({\hat x}_1,{\hat x}_2)$, whose commutator is a function of the cosmological constant $Λ$ and of the (central) 'quantum time' coordinate ${\hat x}_0$. Thus, the Chern-Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

gr-qc↗

arXiv2014

A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian $H_κ$, we will make use of a group theoretical approach in which the curvature $κ$ of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. It turns out that when the oscillator parameters $Ω_1$ and $Ω_2$ are such that $Ω_2=4Ω_1$, the system turns out to be the well-known superintegrable 1:2 oscillator on $S^2$ and $H^2$. Nevertheless, numerical integration of the trajectories of $H_κ$ suggests that for other values of the parameters $Ω_1$ and $Ω_2$ the system is not superintegrable. In this way, we support the conjecture that for each commensurate (and thus superintegrable) $m:n$ Euclidean oscillator there exists a two-parametric family of curved integrable (but not superintegrable) oscillators that turns out to be superintegrable only when the parameters are tuned to the $m:n$ commensurability condition.

nlin.SI↗

arXiv2014

An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric

In this paper we quantize the $N$-dimensional classical Hamiltonian system $H= \frac{|q|}{2(η+ |q|)} p^2-\frac{k}{η+|q|}$, that can be regarded as a deformation of the Coulomb problem with coupling constant $k$, that it is smoothly recovered in the limit $η\to 0$. Moreover, the kinetic energy term in $H$ is just the one corresponding to an $N$-dimensional Taub-NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the Hamiltonian $H$ is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines the usual Laplace-Beltrami operator on the Taub-NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form for positive values of $η$ and $k$, and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analysed, and the quantization problem for negative values of $η$ and/or $k$ is also sketched.

math-ph↗

arXiv2014

Twisted (2+1) $κ$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant $Λ$ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like $κ$-AdS and dS quantum algebras; their flat limit $Λ\to 0$ leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear $Λ$-deformation of the $κ$-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

math-ph↗

arXiv2014

A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant

We show that the Drinfel'd double associated to the standard quantum deformation $sl_η(2,R)$ is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter $η$ related to the cosmological constant $Λ=-η^2$. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case $Λ\neq 0$ are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear $Λ$-deformation of the Minkowskian one.

math-ph↗

arXiv2013

A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system

The $N$-dimensional quantum Hamiltonian $ \hat{H} = -\frac{\hbar^2 {|\mathbf{q} } | }{2(η+| {\mathbf{q}} |)} {\mathbf{\nabla}}^2 - \frac{k}{η+ |{\mathbf{q}} |} $ is shown to be exactly solvable for any real positive value of the parameter $η$. Algebraically, this Hamiltonian system can be regarded as a new maximally superintegrable $η$-deformation of the $N$-dimensional Kepler-Coulomb Hamiltonian while, from a geometric viewpoint, this superintegrable Hamiltonian can be interpreted as a system on an $N$-dimensional Riemannian space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation parameter $η$ and of the coupling constant $k$.

math-ph↗

arXiv2013

On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

Quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincaré algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the κ-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.

hep-th↗

arXiv2013

Superintegrable quantum oscillator and Kepler-Coulomb systems on curved spaces

An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.

quant-ph↗