We derive the non-relativistic ?→∞ and ultra-relativistic c→0 limits of the ?-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 ?-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the ?-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 ?-Newtonian and ?-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the ?-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter ?, the curvature parameter ? and the speed of light parameter ?.
We show that the Lorentzian Snyder models, together with their non-relativistic (?→∞) and ultra-relativistic (?→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.
Next week in Barcelona there will be the first Network Activity of the COST Action CA18108 QG-MM “Quantum gravity phenomenology in the multi-messenger approach”. The objective of the meeting is to initiate a discussion among the different communities involved in the project, with the immediate aim of reviewing the present experimental and theoretical status in the search of quantum gravity signatures in the phenomenology of the different cosmic messengers. I will give a review talk for the WG1 – Theoretical frameworks for quantum gravity effects below the Planck energy.
I am editing a Special Issue for the journal Symmetry, together with Angel Ballesteros and Francisco Herranz. We are accepting submissions until 15th of March 2020. Below is the Special Issue description. Please feel free to contact me if you have any question.
Quantum groups appeared during the eighties as the underlying algebraic symmetries of several two-dimensional integrable models. They are noncommutative generalizations of Lie groups endowed with a Hopf algebra structure, and the possibility of defining noncommutative spaces that are covariant under quantum group (co)actions soon provided a fruitful link with noncommutative geometry. At the same time, when quantum group analogues of the Lie groups of spacetime symmetries (Galilei, Poincare’ and (anti-) de Sitter) were constructed, they attracted the attention of quantum gravity researchers. In fact, they provided a possible mathematical framework to model the “quantum” geometry of space–time and the quantum deformations of its kinematical symmetries at the Planck scale, where nontrivial features are expected to arise because of the interplay between gravity and quantum theory.
This Special Issue is open to contributions dealing with any of the many facets of quantum group symmetry and their generalizations. On the more formal side, possible topics include the theory of Poisson–Lie groups and Poisson homogeneous spaces as the associated semiclassical objects; Hopf algebras; the classification of quantum groups and spaces, their representation theory and its connections with q-special functions; the construction of noncommutative differential calculi; and the theory of quantum bundles. On application side, possible topics are: classical and quantum integrable models with quantum group invariance; the applications of quantum groups in different (2+1) quantum gravity contexts (like combinatorial quantisation, state sum models or spin foams); and quantum kinematical groups and their noncommutative spacetimes in connection with deformed special relativity and quantum gravity phenomenology.
Prof. Angel Ballesteros Dr. Giulia Gubitosi Prof. Francisco J. Herranz Guest Editors