arXiv subjects

Francisco J. Herranz

Publications and source records attributed to Francisco J. Herranz.

At least 37 records · Page 2Linked to original sources

arXiv2022

The noncommutative space of light-like worldlines

The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) $κ$-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter $κ^{-1}$. Moreover, it is shown that the usual time-like $κ$-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like $κ$-deformation.

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arXiv2021

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

The complete classification of classical $r$-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincaré groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists three classes of such $r$-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

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arXiv2021

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

The Cayley-Klein (CK) formalism is applied to the real algebra ${so}(5)$ by making use of four graded contraction parameters describing in a unified setting 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel'd-Jimbo real Lie bialgebra for ${so}(5)$ together with its Drinfel'd double structure, we obtain the corresponding CK bialgebra and the CK $r$-matrix coming from a Drinfel'd double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring to deal with real structures, it comes out that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we find 14 classical $r$-matrices coming from Drinfel'd doubles, obtaining new results for the de Sitter ${so}(4,1)$ and anti-de Sitter ${so}(3,2)$ and for some of their contractions. These geometric results are exhaustively applied onto the (3+1)D kinematical algebras, not only considering the usual (3+1)D spacetime but also the 6D space of lines. We establish different assignations between the geometrical CK generators and the kinematical ones which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature $Λ$ and speed of light $c$. We finally obtain four classes of kinematical $r$-matrices together with their noncommutative spacetimes and spaces of lines, comprising all $κ$-deformations as particular cases.

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arXiv2021

Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra

The formalism for Poisson-Hopf (PH) deformations of Lie-Hamilton systems is refined in one of its crucial points concerning applications, namely the obtention of effective and computationally feasible PH deformed superposition rules for prolonged PH deformations of Lie-Hamilton systems. The two new notions here proposed are a generalization of the standard superposition rules and the concept of diagonal prolongations for Lie systems, which are consistently recovered under the non-deformed limit. Using a technique from superintegrability theory, we obtain a maximal number of functionally independent constants of the motion for a generic prolonged PH deformation of a Lie-Hamilton system, from which a simplified deformed superposition rule can be derived. As an application, explicit deformed superposition rules for prolonged PH deformations of Lie-Hamilton systems based on the oscillator Lie algebra ${h}_4$ are computed. Moreover, by making use that the main structural properties of the book subalgebra ${b}_2$ of ${h}_4$ are preserved under the PH deformation, we consider prolonged PH deformations based on ${b}_2$ as restrictions of those for ${h}_4$-Lie-Hamilton systems, thus allowing the study of prolonged PH deformations of the complex Bernoulli equations, for which both the constants of the motion and the deformed superposition rules are explicitly presented.

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arXiv2020

Generalized noncommutative Snyder spaces and projective geometry

Given a group of kinematical symmetry generators, one can construct a compatible noncommutative spacetime and deformed phase space by means of projective geometry. This was the main idea behind the very first model of noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework, spacetime coordinates are the translation generators over a manifold that is symmetric under the required generators, while momenta are projective coordinates on such a manifold. In these proceedings we review the construction of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the freedom left by this construction in the choice of the physical momenta, because of different available choices of projective coordinates. In particular, we derive a quasi-canonical structure for both the Euclidean and Lorentzian Snyder noncommutative models such that their phase space algebra is diagonal although no longer quadratic.

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arXiv2020

Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry

We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-)de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton--Hooke) spaces. This exchange between the Galilei and Carroll limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit $c\to \infty$. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.

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arXiv2020

The $κ$-Newtonian and $κ$-Carrollian algebras and their noncommutative spacetimes

We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $κ$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $κ$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $κ$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $κ$-Newtonian and $κ$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $κ$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $κ$, the curvature parameter $η$ and the speed of light parameter $c$.

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arXiv2019

Quantum groups, non-commutative Lorentzian spacetimes and curved momentum spaces

The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $Λ$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $κ$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $κ$-deformation, while curved momentum spaces are defined as orbits generated by the $κ$-dual of the Hopf algebra of quantum symmetries. Throughout this construction we use kinematical coordinates, in terms of which the physical interpretation becomes more transparent, and the cosmological constant $Λ$ is included as an explicit parameter whose $Λ\rightarrow 0$ limit provides the Minkowskian case. The generalization of these results to the physically relevant (3+1)-dimensional deformation is also commented.

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arXiv2019

The $κ$-(A)dS noncommutative spacetime

The (3+1)-dimensional $κ$-(A)dS noncommutative spacetime is explicitly constructed by quantizing its semiclassical counterpart, which is the $κ$-(A)dS Poisson homogeneous space. This turns out to be the only possible generalization of the well-known $κ$-Minkowski spacetime to the case of non-vanishing cosmological constant, under the condition that the time translation generator of the corresponding quantum (A)dS algebra is primitive. Moreover, the $κ$-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the $κ$-Minkowski spacetime. When expressed in ambient coordinates, the quantum $κ$-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.

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arXiv2019

Curvature as an integrable deformation

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon-Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched

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arXiv2019

Noncommutative spaces of worldlines

The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the $κ$-Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of $κ$-Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter $κ^{-1}$ plays the very same role as the Planck constant $\hbar$ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

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arXiv2018

Cayley-Klein Poisson homogeneous spaces

The 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 semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.

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arXiv2018

The Poincaré group as a Drinfel'd double

The eight nonisomorphic Drinfel'd double (DD) structures for the Poincaré Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincaré group are also identified and constructed, while in (3+1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré $r$-matrix whose properties are analysed. Some of these $r$-matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2+1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of $\mathfrak{sl}(2)$ with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1+1) and (2+1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré $r$-matrices through the vanishing cosmological constant limit is also analysed.

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arXiv2018

Drinfel'd double structures for Poincaré and Euclidean groups

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel'd double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel'd double structures of the $\mathfrak{so}(3,1)$ Lie algebra is presented.

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arXiv2018

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

Curved momentum spaces associated to the $κ$-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 $κ$-deformation with non-vanishing cosmological constant. The $κ$-de Sitter and $κ$-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 $κ$-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.

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arXiv2018

A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)

Based 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 $\mathfrak{sl}(2)$ 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 $\mathfrak{sl}(2)$ 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.

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arXiv2017

Poisson-Hopf algebra deformations of Lie-Hamilton systems

Hopf 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 $\mathfrak{sl}(2)$ 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.

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arXiv2017

Lie-Hamilton systems on curved spaces: A geometrical approach

A 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.

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