Lorentzian Poisson homogeneous spaces, quantum groups and noncommutative spacetimes

El objetivo de esta Tesis Doctoral es el estudio de ciertas deformaciones cuánticas de los grupos cinemáticos Lorentzianos (Poincaré y (anti-)de Sitter), sus espacios homogéneos cuánticos asociados, y algunas de sus consecuencias físicas. En particular, se construye el espacio no conmutativo de kappa-(Anti-)de Sitter. Además, se propone un método para introducir no-conmutatividad en el espacio de geodésicas de un espacio-tiempo no conmutativo, estudiando en detalle el ejemplo de kappa-Poincaré. Así mismo, se estudia el espacio de momentos curvo asociado a la deformación de kappa-(Anti-)de Sitter previamente introducida. Además, se estudian sistemáticamente las estructuras de doble de Drinfel’d para los grupos de Poincaré en 2+1 dimensiones y Euclídeo en 3 dimensiones, junto con los grupos de Poisson-Lie a ellas asociadas. Finalmente se profundiza en la noción de dualidad para espacios homogéneos de Poisson, mostrando su relación con ciertas propiedades de las relaciones de incertidumbre que presentan los espacios no conmutativos asociados

On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications

In 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.Comment: 11 pages, 6 figures. New figures. Updated to match the published version in Physics Letters

Shannon information entropy for a quantum nonlinear oscillator on a space of non-constant curvature

The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of a non-negative parameter λ which is directly related to the curvature of the underlying space. In this paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular, analytical results for the Shannon entropy in the position space can be found in the N-dimensional case, and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered in the limit of vanishing curvature λ → 0. However, the Fourier transform of the Darboux III wave functions cannot be computed in exact form, thus preventing the analytical study of the information entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one and three-dimensional cases and we have found that by increasing the absolute value of the negative curvature (through a larger λ parameter) the information entropy in position space increases, while in momentum space it becomes smaller. This result is indeed consistent with the spreading properties of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of the entropies in position and momentum spaces has been also analysed in terms of the curvature: for all excited states such total entropy decreases with λ, but for the ground state the total entropy is minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.This work has been partially supported by Agencia Estatal de Investigación (Spain) under grant PID2019-106802GB-I00/AEI/ 10.13039/501100011033, by the Regional Government of Castilla y León (Junta de Castilla y León, Spain) and by the Spanish Ministry of Science and Innovation MICIN and the European Union NextGenerationEU/PRTR, as well as the contribution of the European Cooperation in Science and Technology through the COST Action CA18108. The authors acknowledge A. Najafizade for useful discussions at the early stages of this work, and also the Referee for several relevant comments and suggestions

Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions

Curved momentum spaces associated to the $\kappa$-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 $\kappa$-deformation with non-vanishing cosmological constant. The $\kappa$-de Sitter and $\kappa$-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 $\kappa$-Poincar\'e curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.Comment: v2: Version accepted for publication on Phys. Rev.

Cayley-Klein Poisson Homogeneous Spaces

Producción CientíficaThe nine two-dimensional Cayley–Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson–Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley–Klein geometries. The kinematical interpretation for the semiRiemannian and pseudo-Riemannian Cayley–Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.Ministerio de Ciencia, Innovación y Universidades (grant MTM2016-79639-P)European Cooperation in Science and Technology (COST Action MP1405 QSPACE

Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions

Producción CientíficaCurved 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.Ministerio de Economía, Industria y Competitividad (projects MTM2013-43820-P / MTM2016-79639-P)Junta de Castilla y León (projects BU278U14 / VA057U16)European Cooperation in Science and Technology (Action MP1405 QSPACE