arXiv subjects

Angel Ballesteros

Publications and source records attributed to Angel Ballesteros.

At least 19 recordsLinked to original sources

arXiv2026

The Fock-Darwin-Darboux system: eigenstates, information entropies and dispersion-like measures

The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, R\'enyi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.

quant-ph↗

arXiv2026

Universal $T$-matrices for quantum Poincar\'e groups: contractions and quantum reference frames

Universal $T$-matrices, or Hopf algebra dual forms, for quantum groups are revisited, and their contraction theory is developed. As a first illustrative example, the (1+1) timelike $\kappa$-Poincar\'e $T$-matrix is explicitly worked out. Afterwards, motivated by recent results on the role of the Hopf algebra dual form of a quantum (1+1) centrally extended Galilei group as the algebraic object underlying non-relativistic quantum reference frame transformations, a new quantum deformation of the (1+1) centrally extended Poincar\'e Lie algebra is obtained, and its universal $T$-matrix is presented. Finally, the Hopf algebra dual form contraction is applied to this Poincar\'e $T$-matrix, showing that its corresponding non-relativistic counterpart is precisely the Galilei $T$-matrix associated with quantum reference frames. In this way, the Poincar\'e Hopf algebra dual form introduced here stands as a natural candidate for describing the symmetry structure of relativistic quantum reference frame transformations. In the appropriate basis, the associated quantum Poincar\'e group is recognized, remarkably, as a non-trivial central extension of the (1+1) spacelike $\kappa$-Poincar\'e dual Hopf algebra.

math.QA↗

arXiv2025

R\'enyi and Tsallis information entropies for the Darboux III quantum nonlinear oscillator

The Darboux III oscillator is an exactly solvable $N$-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. Its one-dimensional version can be seen as a position dependent mass system whose mass function $\mu = (1 + \lambda x^2)$ depends on the nonlinearity parameter $\lambda$, such that in the limit $\lambda \to 0$ the harmonic oscillator is recovered. In this paper, a detailed study of the entropic moments and of the R\'enyi and Tsallis information entropies for the quantum version of the one-dimensional Darboux III oscillator is presented. In particular, analytical expressions for the aforementioned quantities in position space are obtained. Since the Fourier transform of the Darboux III wave functions does not admit a closed form expression, a numerical analysis of these quantities has been performed. Throughout the paper the interplay between the entropy parameter $\alpha$ and the nonlinearity parameter $\lambda$ is analysed, and known results for the Shannon entropy of the Darboux III and for the R\'enyi and Tsallis entropies of the harmonic oscillator are recovered in the limits $\alpha \to 1$ and $\lambda \to 0$, respectively. Finally, motivated by the strong non-linear effects arising when large values of $\lambda$ and/or highly excited states are considered, an approximation to the probability density function valid in those regimes is presented. From it, an analytical approximation to the probability density in momentum space can be obtained, and some of the previously observed effects arising from the interplay between $\alpha$ and $\lambda$ can be explained.

quant-ph↗

arXiv2025

Quantum Galilei group as quantum reference frame transformations

Quantum groups have been widely explored as a tool to encode possible nontrivial generalisations of reference frame transformations, relevant in quantum gravity. In quantum information, it was found that the reference frames can be associated to quantum particles, leading to quantum reference frames transformations. The connection between these two frameworks is still unexplored, but if clarified it will lead to a more profound understanding of symmetries in quantum mechanics and quantum gravity. Here, we establish a correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at first order in the quantum deformation parameter. This is found once the quantum group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame. These results allow us to show that quantum reference frame transformations are physically relevant when the state of the quantum reference frame is in a quantum superposition of semiclassical states. We conjecture that the all-order quantum Galilei group describes quantum reference frame transformations between more general quantum states of the quantum reference frame.

quant-ph↗

arXiv2025

Deformations of the symmetric subspace of qubit chains

The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group $SU(2)$, whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group $\mathcal{U}_q(\mathfrak{su}(2))$. We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.

quant-ph↗

arXiv2024

Noncommutative Lightcones from Quantum SO(2,1) Conformal Groups

Five new families of noncommutative lightcones in 2+1 dimensions are presented as the quantizations of the inequivalent Poisson homogeneous structures that emerge when the lightcone is constructed as a homogeneous space of the SO(2,1) conformal group. Each of these noncommutative lightcones maintains covariance under the action of the respective quantum deformation of the SO(2,1) conformal group. We discuss the role played by SO(2,1) automorphisms in the classification of inequivalent Poisson homogeneous lightcones, as well as the geometric aspects of this construction. The localization properties of the novel quantum lightcones are analyzed and shown to be deeply connected with the geometric features of the Poisson homogeneous spaces.

hep-th↗

arXiv2024

Integrable deformations of Rikitake systems, Lie bialgebras and bi-Hamiltonian structures

Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie-Poisson Hamiltonian structures, which are considered linearizations of Poisson--Lie structures on certain (dual) Lie groups. By taking into account that there exists a one-to one correspondence between Poisson--Lie groups and Lie bialgebra structures, a number of deformed Poisson coalgebras can be obtained, which allow the construction of integrable deformations of coupled Rikitake systems. Moreover, the integrals of the motion for these coupled systems can be explicitly obtained by means of the deformed coproduct map. The same procedure can be also applied when the initial system is bi-Hamiltonian with respect to two different Lie-Poisson algebras. In this case, to preserve a bi-Hamiltonian structure under deformation, a common Lie bialgebra structure for the two Lie-Poisson structures has to be found. Coupled dynamical systems arising from this bi-Hamiltonian deformation scheme are also presented, and the use of collective `cluster variables', turns out to be enlightening in order to analyse their dynamical behaviour. As a general feature, the approach here presented provides a novel connection between Lie bialgebras and integrable dynamical systems.

math.DS↗

arXiv2023

Noncommutative spacetimes versus noncommutative spaces of geodesics

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an 'ambiguity' can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from 'extended' noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the $\kappa$-Poincar\'e and $\kappa$-Galilei algebras.

hep-th↗

arXiv2022

The Dunkl oscillator on a space of nonconstant curvature: an exactly solvable quantum model with reflections

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions, defined as a $\lambda-$deformation of the N-dimensional Dunkl oscillator. This deformation can be interpreted either as the introduction of a non-constant curvature related to $\lambda$ on the underlying space or, equivalently, as a Dunkl oscillator with a position-dependent mass function. This new quantum model is shown to be exactly solvable in arbitrary dimension N, and its eigenvalues and eigenfunctions are explicitly presented. Moreover, it is shown that in the two-dimensional case both the Darboux III and the Dunkl oscillators can be separately coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of a position-dependent mass and the Dunkl derivatives on the structure of the Landau levels can be explicitly studied. Finally, the whole 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the $\lambda$-deformation and the magnetic field is explicitly illustrated.

quant-ph↗

arXiv2022

A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincar\'e

In this contribution we present a general procedure that allows the construction of noncommutative spaces with quantum group invariance as the quantization of their associated coisotropic Poisson homogeneous spaces coming from a coboundary Lie bialgebra structure. The approach is illustrated by obtaining in an explicit form several noncommutative spaces from (3+1)D (A)dS and Poincar\'e coisotropic Lie bialgebras. In particular, we review the construction of the $\kappa$-Minkowski and $\kappa$-(A)dS spacetimes in terms of the cosmological constant $\Lambda$. Furthermore, we present all noncommutative Minkowski and (A)dS spacetimes that preserved a quantum Lorentz subgroup. Finally, it is also shown that the same setting can be used to construct the three possible 6D $\kappa$-Poincar\'e spaces of time-like. Some open problems are also addressed.

math-ph↗

arXiv2022

$\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines

The noncommutative spacetimes associated to the $\kappa$-Poincar\'e relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $\kappa$-Galilei and $\kappa$-Carroll symmetries as contractions of the corresponding $\kappa$-Poincar\'e space and we show that these three spaces are defined by different algebras. In particular, the $\kappa$-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the $\kappa$-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.

hep-th↗

arXiv2022

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 $\lambda$ 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 $\lambda \to 0$. 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 $\lambda$ 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 $\lambda$, but for the ground state the total entropy is minimised when $\lambda$ vanishes, and the corresponding uncertainty relation is always fulfilled.

math-ph↗

arXiv2022

All noncommutative spaces of $\kappa$-Poincar\'e geodesics

Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light- versions of the $\kappa$-Poincar\'e quantum symmetry in (3+1) dimensions. Remarkably enough, only for the light-like (or null-plane) $\kappa$-Poincar\'e deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations both the quantum time-like and space-like geodesics can be defined, but not the light-like one. This obstruction comes from the constraint imposed by the coisotropy condition for the corresponding deformation with respect to the isotropy subalgebra associated to the given space of geodesics, since all these quantum spaces are constructed as quantizations of the corresponding classical coisotropic Poisson homogeneous spaces. The known quantum space of geodesics on the light cone is given by a five-dimensional homogeneous quadratic algebra, and the six nocommutative spaces of time-like and space-like geodesics are explicitly obtained as six-dimensional nonlinear algebras. Five out of these six spaces are here presented for the first time, and Darboux generators for all of them are found, thus showing that the quantum deformation parameter $\kappa^{-1}$ plays exactly the same algebraic role on quantum geodesics as the Planck constant $\hbar$ plays in the usual phase space description of quantum mechanics.

hep-th↗

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) $\kappa$-deformation of the (3+1) Poincar\'e 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 $\kappa^{-1}$. Moreover, it is shown that the usual time-like $\kappa$-deformation of the Poincar\'e 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 $\kappa$-deformation.

hep-th↗

arXiv2021

Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries

Recent work showed that $\kappa$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincar\'e algebra of special-relativistic symmetries, one can toggle the curvature parameter $\Lambda$, the Planck scale quantum deformation parameter $\kappa$ and the speed of light parameter $c$ to move to the well-studied $\kappa$-Poincar\'e algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters.

gr-qc↗

arXiv2021

Fuzzy worldlines with $\kappa$-Poincar\'e symmetries

A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the $\kappa$-Poincar\'e model, analyzing the fuzzy properties of $\kappa$-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.

hep-th↗

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 Poincar\'e 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.

math-ph↗

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.

math-ph↗