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

In this paper (arXiv:2106.03817), 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.

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

In this paper (arXiv:2101.00616), 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.

The group structure of dynamical transformations between quantum reference frames

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part. While such transformations were shown to be symmetries of the system’s Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work (arXiv:2012.15769), we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.

Exact closed-form solution of a modified SIR model

In this new paper (arXiv:2007.16069) the exact analytical solution in closed form of a modified SIR system where recovered individuals are removed from the population is presented. In this dynamical system the populations S(t) and R(t) of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones I(t) are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The dynamics of this modified SIR system is analysed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. Moreover, it is shown that the modified SIR model with time-dependent transmission rate can be also solved in closed form for certain realistic transmission rate functions.

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 (arXiv:2007.09653) 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.

Hamiltonian structure of compartmental epidemiological models

In this new paper (arXiv:2006.00564), any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological dynamics.

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

In this work (arXiv:2003.03921) we derive the non-relativistic c→∞ and ultra-relativistic c→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.

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

In this new paper (arXiv:1912.12878) we show that the Lorentzian Snyder models, together with their non-relativistic (c→∞) and ultra-relativistic (c→0) 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 these 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 non-relativistic and ultra-relativistic 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. 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.

Coreductive Lie bialgebras and dual homogeneous spaces

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M=G/H equipped with an additional Poisson structure π which is compatible with a Poisson-Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra 𝔤=Lie(G), in this new paper (arXiv:1909.01000) we exploit the idea of Lie bialgebra duality in order to introduce the notion of dual homogeneous space of a given homogeneous space M=G/H with respect to the Lie bialgebra δ. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to the dual space thus showing that an even richer duality framework arises. In order to analyse the physical implications of this new duality, the case of M being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding dual reductive and symmetric spaces are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that the dual space is reductive is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally we show that, despite the dual spaces are not endowed in general with an invariant metric, their geometry can be described by making use of K-structures.

The κ-(A)dS noncommutative spacetime

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed in this new paper (arXiv:1905.12358) by quantizing its semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. Under minimal physical assumptions, it is explicitly proven that this is the only possible generalization to the case of non-vanishing cosmological constant of the well-known κ-Minkowski spacetime. 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.