FM - Artículos de revistaFM - Artículos de revistahttp://uvadoc.uva.es/handle/10324/221552019-06-25T05:27:48Z2019-06-25T05:27:48ZNon-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groupsBallesteros, A.Bruno, N.R.Herranz, F.J.http://uvadoc.uva.es/handle/10324/336422019-06-14T12:43:54Z2017-01-01T00:00:00ZThe κ-deformation of the (2+1)D anti-de Sitter, Poincar´e 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–doubleand the Poisson–Lie structure underlying the κ-deformation are explicitly given, andthe 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.
2017-01-01T00:00:00ZCurved momentum spaces from quantum groups with cosmological constantBallesteros, A.Gubitosi, G.Gutiérrez-Sagredo, I.Herranz, F.J.http://uvadoc.uva.es/handle/10324/336412019-06-14T12:43:54Z2017-01-01T00:00:00ZWe bring the concept that quantum symmetries describe theories with nontrivial momentum
space properties one step further, looking at quantum symmetries of spacetime in presence of
a nonvanishing cosmological constant Lambda. In particular, the momentum space associated to the κ-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by Lambda. Such momentum space includes both the momenta associated to spacetime translations and the ‘hyperbolic’ momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the κ-Poincar´e algebra are smoothly recovered in the limit Lambda → 0, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.
2017-01-01T00:00:00ZQuantum groups and noncommutative spacetimes with cosmological constantBallesteros, A.Gutiérrez-Sagredo, I.Herranz, F.J.Meusburger, C.Naranjo, P.http://uvadoc.uva.es/handle/10324/336402019-06-14T12:43:53Z2017-01-01T00:00:00ZNoncommutative spacetimes are widely believed to model some properties of the quantum
structure of spacetime at the Planck regime. In this contribution the construction of (anti)de
Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter z is related to a Planck scale, and the cosmological constant Lambda plays the role of a second deformation parameter of geometric nature, whose limit Lambda → 0 provides the corresponding noncommutative Minkowski spacetimes.
2017-01-01T00:00:00ZSuperintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separabilityCariñena, J.F.Herranz, F.J.Rañada, M.F.http://uvadoc.uva.es/handle/10324/336392019-06-14T12:43:54Z2017-01-01T00:00:00ZThe Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T +V into a geodesic Hamiltonian 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 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 Ur leading to H_r = T_r +U_r. Secondly, we study the superintegrability of the four Hamiltonians tilde{H}_r = H_r/μ_r, where μ_r is a certain position-dependent mass, that enjoys the same separability as the original system H_r. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.
2017-01-01T00:00:00ZAdS Poisson homogeneous spaces and Drinfel’d doublesBallesteros, A.Meusburger, C.Naranjo, P.http://uvadoc.uva.es/handle/10324/336382019-06-14T12:43:53Z2017-01-01T00:00:00ZThe correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and
Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2,R) ∼= SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2,R) and as a coisotropic one for the others.
We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter
space AdS3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS3 that arise from two Drinfel’d double structures on SO(2, 2). The first one realises AdS3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2,R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS3 as a coisotropic Poisson homogeneous space.
2017-01-01T00:00:00ZLie–Hamilton systems on curved spaces: A geometrical approachHerranz, F.J.de Lucas, J.Tobolski, M.http://uvadoc.uva.es/handle/10324/336372019-06-14T12:43:52Z2017-01-01T00:00:00ZA Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot–Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie–Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.
2017-01-01T00:00:00ZThe kappa-(A)dS quantum algebra in (3+1) dimensionsBallesteros, A.Herranz, F.J.Musso, F.Naranjo, PO.http://uvadoc.uva.es/handle/10324/336362019-06-14T12:43:51Z2017-01-01T00:00:00ZThe quantum duality principle is used to obtain explicitly the Poisson analogue of the κ-(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 Lambda is included as a Poisson–Lie group contraction parameter, and the limit Lambda→ 0 leads to the well-known κ-Poincar´e algebra in the bicrossproduct basis. A twisted version with Drinfel’d double structure of this κ-(A)dS deformation is sketched.
2017-01-01T00:00:00ZPoisson-Lie groups, bi-Hamiltonian systems and integrable deformationsBallesteros, A.Marrero, J.C.Ravanpak, Z.http://uvadoc.uva.es/handle/10324/336352019-06-14T12:43:52Z2017-01-01T00:00:00ZGiven a Lie-Poisson completely integrable bi-Hamiltonian system on R^n, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group G_eta of dimension n, where eta \in R is the deformation parameter. Moreover, we show
that from the two multiplicative (Poisson-Lie) Hamiltonian structures on G_eta that underly the
dynamics of the deformed system and by making use of the group law on G_eta, one may obtain two completely integrable Hamiltonian systems on G_eta x G_eta. By construction, both systems admit reduction, via the multiplication in G_eta, to the deformed bi-Hamiltonian system in G_eta. The previous approach is applied to two relevant Lie-Poisson completely integrable
bi-Hamiltonian systems: the Lorenz and Euler top systems.
2017-01-01T00:00:00ZOn Hamiltonians with position-dependent mass from Kaluza-Klein compactificationsBallesteros, A.Gutiérrez-Sagredo, I.Naranjo, P.http://uvadoc.uva.es/handle/10324/336342019-06-14T12:43:51Z2017-01-01T00:00:00ZIn a recent paper (J.R. Morris, Quant. Stud. Math. Found. 2 (2015) 359), an inhomogeneous
compactification of the extra dimension of a five-dimensional Kaluza-Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. Here we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Moreover, it is also shown how the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.
2017-01-01T00:00:00ZExtended noncommutative Minkowski spacetimes and hybrid gauge symmetriesBallesteros, A.Mercati, F.http://uvadoc.uva.es/handle/10324/336332019-06-14T12:43:51Z2018-01-01T00:00:00ZWe study the Lie bialgebra structures that can be built on the one-dimensional central extension
of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman–Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra.
2018-01-01T00:00:00ZA unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)Ballesteros, A.Campoamor-Stursberg, R.Fernandez-Saiz, E.Herranz, F.J.de Lucas, J.http://uvadoc.uva.es/handle/10324/336322019-06-14T12:43:53Z2018-01-01T00:00:00ZBased on a recently developed procedure to construct Poisson-Hopf deformations of Lie–Hamilton systems, a novel unified approach to nonequivalent deformations of Lie–Hamilton systems on the real plane with a Vessiot–Guldberg Lie algebra isomorphic to sl(2) is proposed. This, in particular, allows us to define a notion of Poisson–Hopf systems in dependence of a
parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of sl(2) Lie–Hamilton systems. Our results cover deformations of the Ermakov system, Milne–Pinney, Kummer–Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie–Hamilton systems and their deformations for other Vessiot–Guldberg Lie algebras and their deformations.
2018-01-01T00:00:00ZCurved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensionsBallesteros, A.Gubitosi, G.Gutiérrez-Sagredo, IHerranz, F.J.http://uvadoc.uva.es/handle/10324/336312019-06-14T12:43:52Z2018-01-01T00:00:00ZCurved momentum spaces associated to the k-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the k-deformation with non-vanishing cosmological constant. The k-de Sitter and k-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with SO(4, 4) invariance. Such spaces are made of the momenta associated to spacetime translations and the ‘hyperbolic’ momenta associated to boost transformations. The known k-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.
2018-01-01T00:00:00ZPoisson-Hopf algebra deformations of Lie-Hamilton systemsBallesteros, A.Campoamor-Stursberg, R.Fernández-Saiz, E.Herranz, F.J.de Lucas, J.http://uvadoc.uva.es/handle/10324/336302019-06-14T12:43:51Z2018-01-01T00:00:00ZHopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation
of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.
2018-01-01T00:00:00ZGlobal versus local superintegrability of nonlinear oscillatorsAnco, S.C.Ballesteros, A.Gandarias, M.L.http://uvadoc.uva.es/handle/10324/336292019-06-14T12:43:53Z2019-01-01T00:00:00ZLiouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are
used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that super-integrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.
2019-01-01T00:00:00ZDomain walls in a non-linear S2-sigma model with homogeneous quartic polynomial potentialAlonso-Izquierdo, A.Balseyro Sebastián, A.J.González León, M.A.http://uvadoc.uva.es/handle/10324/336272019-06-14T12:43:54Z2018-01-01T00:00:00ZIn this paper the domain wall solutions of a Ginzburg-Landau non-linear S2-sigma hybrid model are exactly calculated. There exist two types of basic domain walls and two families of composite domain walls. The domain wall solutions have been identified by using a Bogomolny arrangement in a system of sphero-conical coordinates on the sphere S2. The stability of all the domain walls is also investigated.
2018-01-01T00:00:00ZCompleteness and Nonclassicality of Coherent States for Generalized Oscillator AlgebrasZelaya, K.Rosas-Ortiz, O.Blanco-Garcia, Z.Cruz y Cruz, S.http://uvadoc.uva.es/handle/10324/336262019-06-14T12:43:53Z2017-01-01T00:00:00ZThe purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are 𝑃-represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer 𝐺-function. This property automatically defines the delta distribution as the 𝑃-representation of such states.Then, in principle, theremust be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1, 1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2), we use a beamsplitter to showthat the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them.We also show that these states lack second-order coherence. That is, although the 𝑃-representation of the nonlinear coherent states is a delta function, they are not full coherent.Therefore, the systems associated with the generalized oscillator algebras cannot
be considered “classical” in the context of the quantum theory of optical coherence.
2017-01-01T00:00:00Z