Abstract
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, 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.
Highlights
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
Despite the fact that the κ-Poincaré algebra was initially obtained as the quantum group contraction associated to the flat Λ → 0 limit of the quantum so (3, 2) algebra [3,115], neither the relation among the generators of such so (3, 2) quantum deformation and the kinematical generators { P0, Pa, Ka, Ja } nor the explicit role played by the cosmological constant Λ in the quantum case were explored
A relevant difference between the two limits is that, while in the Carrollian limit the presence of the quantum deformation does not spoil the appearance of an absolute space, in the Galilean limit the mixing between time and space induced by the quantum deformation prevents the emergence of an absolute time, since the commutators between boosts and spatial translations remain non-vanishing in the transition from the κ-Poincaré to the κ-Galilei symmetries
Summary
Deformations of relativistic symmetries have been playing a prominent role in the study of phenomenologically relevant effects of quantum gravity in a “non-quantum” and “non-gravitational” regime, such that both the Planck constant hand the Newton constant q. The novel feature of this deformation, with respect to the classical deformation governed by Λ, is that it can work in two different directions: starting from the Poincaré Lie algebra, one can perform two kinds of contractions, one where c−1 → 0 and one where c → 0, which lead to the Galilei and Carroll Lie algebras and groups, respectively [26,27,28,29] Carroll curved Carroll κ-Newton - Hooke κ-Galilei c κ-(A) de Sitter κ-Poincaré c κ-Carroll curved κ-Carroll In addition to those shown in the previous pictures, here we see a new direction in which the classical deformation governed by the speed of light c can work, linking special-relativistic-like symmetries and Carrollian-like symmetries
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