In this paper (arXiv:2304.08843), by using the theory of Lie-Hamilton systems, formal generalized stochastic Hamiltonian systems that enlarge a recently proposed stochastic SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamitonian models based on the book and oscillator algebras, denoted respectively by 𝔟2 and 𝔥4. The last generalization corresponds to a SIS system possessing the so-called two-photon algebra symmetry 𝔥6, according to the embedding chain 𝔟2⊂𝔥4⊂𝔥6, for which an exact solution cannot generally be found, but a nonlinear superposition rule is explicitly given.
The beauty of curved momentum space
In this paper (arXiv:2303.08220), we will discuss the notion of curved momentum space, as it arises in the discussion of noncommutative or doubly special relativity theories. We will illustrate it with two simple examples, the Casimir effect in anti-Snyder space and the introduction of fermions in doubly special relativity. We will point out the existence of intriguing results, which suggest nontrivial connections with spectral geometry and Hopf algebras.
A New Perspective on Doubly Special Relativity
Doubly special relativity considers a deformation of the special relativistic kinematics parametrized by a high-energy scale, in such a way that it preserves a relativity principle. When this deformation is assumed to be applied to any interaction between particles, one faces some inconsistencies. In order to avoid them, in this paper we propose a new perspective where the deformation affects only the interactions between elementary particles. A consequence of this proposal is that the deformation cannot modify the special relativistic energy–momentum relation of a particle.
Gaussian curvature effects on graphene quantum dots
In the last few years, much attention has been paid to the exotic properties that graphene nanostructures exhibit, especially those emerging upon deforming the material. In this new paper we present a study of the mechanical and electronic properties of bent hexagonal graphene quantum dots employing density functional theory. We explore three different kinds of surfaces with Gaussian curvature exhibiting different shapes—spherical, cylindrical, and one-sheet hyperboloid—used to bend the material, and several boundary conditions regarding what atoms are forced to lay on the chosen surface. In each case, we study the curvature energy and two quantum regeneration times (classic and revival) for different values of the curvature radius. A strong correlation between Gaussian curvature and these regeneration times is found, and a special divergence is observed for the revival time for the hyperboloid case, probably related to the pseudo-magnetic field generated by this curvature being capable of causing a phase transition.
The Dunkl oscillator on a space of nonconstant curvature: an exactly solvable quantum model with reflections
In this paper (arXiv:2212.13575) we introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions, defined as a λ−deformation of the N-dimensional Dunkl oscillator. This deformation can be interpreted either as the introduction of a non-constant curvature related to λ 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 λ-deformation and the magnetic field is explicitly illustrated.
A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincaré
In this contribution (arXiv:2212.11750) 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 Poincaré coisotropic Lie bialgebras. In particular, we review the construction of the κ-Minkowski and κ-(A)dS spacetimes in terms of the cosmological constant Λ. 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 κ-Poincaré spaces of time-like. Some open problems are also addressed.
κ-Galilean and κ-Carrollian noncommutative spaces of worldlines
The noncommutative spacetimes associated to the κ-Poincaré relativistic symmetries and their “non-relativistic” (Galilei) and “ultra-relativistic” (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work (arXiv:2212.01125), 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 κ-Galilei and κ-Carroll symmetries as contractions of the corresponding κ-Poincaré space and we show that these three spaces are defined by different algebras. In particular, the κ-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the κ-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.
Deformed relativistic symmetry principles
We review (arXiv:2211.11684) the main features of models where relativistic symmetries are deformed at the Planck scale. We cover the motivations and links to other quantum gravity approaches. We describe in some detail the most studied theoretical frameworks, including Hopf algebras, relative locality, and other scenarios with deformed momentum space geometry. We discuss possible phenomenological consequences, and point out current open questions.
New class of plane waves for κ-noncommutative Quantum Field Theory
In this work (arXiv:2211.11627) we discuss the construction of a free scalar quantum field theory on κ-Minkowski noncommutative spacetime. We do so in terms of κ-Poincaré-invariant N-point functions, i.e. multilocal functions which respect the deformed symmetries of the spacetime. As shown in a previous paper by some of us, this is only possible for a lightlike version of the commutation relations, which allow the construction of a covariant algebra of N points that generalizes the κ-Minkowski commutation relations. We solve the main shortcoming of our previous approach, which prevented the development of a fully covariant quantum field theory: the emergence of a non-Lorentz-invariant boundary of momentum space. To solve this issue, we propose to “extend” momentum space by introducing a class of new Fourier modes and we prove that this approach leads to a consistent definition of the Pauli-Jordan function, which turns out to be undeformed with respect to the commutative case. We finally address the quantization of our scalar field and obtain a deformed, κ-Poincaré-invariant, version of the bosonic oscillator algebra.
Quantum Euler angles and agency-dependent spacetime
Quantum gravity is expected to introduce quantum aspects into the description of reference frames. In this paper (arXiv:2211.11347) we set the stage for exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to the new paradigm of agency-dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.