Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Abstract

The -deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel' d-double and the Poisson-Lie structure underlying the -deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize the -Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of "duality" between the cosmological constant and the Planck scale is also envisaged.

Curved momentum spaces from quantum (anti–)de Sitter groups in ( 3+1 ) dimensions

“…Notice that this coproduct is written in a "bicrossproducttype" basis that generalizes the one corresponding to the (2 þ 1) κ-AdS ω algebra [44,45]. As it can be easily checked, the κ-Poincaré coproduct (2) is obtained from the above expressions in the limit ω → 0.…”

“…(iii) The Minkowski spacetime M 3þ1 ≡ ISOð3; 1Þ= SOð3; 1Þ arises when ω ¼ Λ ¼ 0. Explicit ambient space coordinates for these three maximally symmetric Lorentzian spacetimes can be obtained by making use of a suitable realization of the Lie groups obtained by exponentiation of a faithful representation of the AdS ω Lie algebra [see [43][44][45] for details in the (2 þ 1)-dimensional case].…”

“…Therefore, when the cosmological constant does not vanish, translations and boosts have similar algebraic properties and play complementary geometric roles (see [45] for a more detailed discussion in the context of homogeneous spaces of worldlines). As a consequence, the main idea introduced in [22] for the construction of curved momentum spaces for the κ-(A)dS algebras in (2 þ 1) dimensions becomes fully applicable: when ω ≠ 0 the momentum space has to be enlarged by including the angular momenta associated to the rotation symmetries and the hyperbolic momenta associated to boosts.…”

Noncommutative spaces of worldlines

“…Moreover, it is well-known that most of the structures that can be defined on classical homogeneous spaces of geodesics can be inherited from their associated motion groups. For instance, in [43] all symplectic, complex and metric structures were described, and we recall that in [39] all homogeneous spaces of worldlines corresponding to kinematical groups were studied in detail, including the pseudo-Riemannian metrics defined on them, and in [44] the (2+1) Lorentzian spaces of worldlines were considered. In particular, for the Poincaré case it was found that an invariant foliation exists in the space of worldlines and that the resulting homogeneous space is of negative curvature (see [39] for details).…”

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

“…At this point it could seem surprising that the noncommutative (A)dS spacetimes (64) and (66) do not depend on the cosmological constant and, therefore, coincide with the corresponding noncommutative Minkowski spacetimes. Indeed, this is true only at first order, and higher order contributions depending on are expected to appear when the full quantum coproduct z is constructed and the all-orders noncommutative spacetime is obtained by applying the full Hopf algebra duality or, alternatively, by quantizing the allorders Poisson-Lie group whose linearization corresponds to the extended noncommutative spacetimes here presented (see [50][51][52] for explicit examples, including the noncommutative κ-(A)dS spacetime in (2 + 1) dimensions, which turns out to be a nonlinear deformation of the κ-Minkowski spacetime).…”