XIV International Summer School on Geometry, Mechanics and Control

The XIV International Summer School on Geometry, Mechanics and Control will be held in Burgos, Spain, from July 4 to July 8, 2022. The School is organised by the Geometry, Mechanics and Control (GMC) Network and the Mathematical Physics Group of the Universidad de Burgos.

This school is oriented to young researchers, Ph.D. and postdoctoral students in Mathematics, Physics and Engineering, whose research topics deal with Geometry, Mechanics and Control Theory. In particular, this year the courses of the School will be focused in spectral geometry, Lie systems and their geometric structures, quantum groups and non-commutative geometry and geometric control. These courses are intended to present an up-to-date introduction to these topics, and also to bring the participant´s attention to some open problems, in particular those ones related to applications. Undegraduate students and master students are also welcome. Participants are encouraged to present oral or poster contributions.

Geometrize and conquer: the Klein-Gordon and Dirac equations in Doubly Special Relativity

In this work (arXiv: 2203.12286) we discuss the deformed relativistic wave equations, namely the Klein–Gordon and Dirac equations in a Doubly Special Relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert space for the Klein–Gordon case.

The noncommutative space of light-like worldlines

In this new paper (arXiv:2202.11767), 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.

Quantum perspective

In this article published in the February 2022 number of the New Scientist, the novel structure of space-time arising from a quantum perspective is presented from different viewpoints. Among them, the idea that by exchanging quantum information, observers can create a shared reality, even if it isn’t there from the start, is presented.

Quantum gravity phenomenology at the dawn of the multi-messenger era – A review

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review (arXiv:2111.05659) is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers.

A video-abstract of this review is available here.

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

Recent work showed that κ-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter Λ, the Planck scale quantum deformation parameter κ and the speed of light parameter c to move to the well-studied κ-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review (arXiv:2110.04867), 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.

Total Collisions in the N-Body Shape Space

In this paper, we discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

Through a Black Hole Singularity

In this paper (arXiv:2109.10753) we show that the Kantowski-Sachs model of a Schwarzschild black hole interior can be slightly generalized in order to accommodate spatial metrics of different orientations, and in this formulation the equations of motion admit a variable redefinition that makes the system regular at the singularity. This system will then traverse the singularity in a deterministic way (information will be conserved through it), and evolve into a time-reversed and orientation-flipped Schwarzschild white hole interior.

Fuzzy worldlines with κ-Poincaré symmetries

This paper (arXiv:2109.09699) presents a novel approach to study the properties of models with quantum-deformed relativistic symmetries which 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 κ-Poincaré model, analyzing the fuzzy properties of κ-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.