Advances in High Energy Physics

Volume 2017 (2017), Article ID 7876942, 19 pages

https://doi.org/10.1155/2017/7876942

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

Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain

Correspondence should be addressed to Francisco J. Herranz

Received 12 May 2017; Accepted 6 August 2017; Published 28 November 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 Ángel Ballesteros et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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.

#### 1. Introduction

The connection between quantum groups and Planck scale physics was early suggested in [1]. Quantum deformations of Lie algebras and Lie groups [2–8] have been broadly applied in the construction of deformed symmetries of spacetimes [9–23], especially for the Poincaré and Galilei cases, for which the deformation parameter is known to play the role of a fundamental scale. Among all these quantum kinematical algebras the well known -Poincaré algebra [9, 13, 14, 16, 18] has been frequently considered.

These deformed Poincaré symmetries were later applied in the context of the so-called doubly special relativity (DSR) theories [24–32] which introduced two fundamental scales: the usual observer-independent velocity scale as well as an observer-independent length scale , which was related to the deformation parameter in the algebra. Since from all approaches to quantum gravity [33–37] the Planck scale is thought to play a fundamental role, DSR theories seem to establish a promising link between some Planck scale effects and quantum groups [38, 39].

From a more general viewpoint, we recall that noncommutative spaces have been proposed as a suitable algebraic framework in order to describe the “quantum” structure of the geometry of spacetime at the Planck scale through a noncommutative algebra of quantum spacetime coordinates [40–44]. In this way the deformation parameter characterizes the noncommutativity of the spacetime algebra, thus generating uncertainty relations between noncommuting coordinates that can be thought to model a “fuzzy” or “discrete” nature of the spacetime at very small distances (or high energies) [45, 46]. In particular, the noncommutative spacetime deduced from the -Poincaré algebra is the so-called -Minkowski spacetime [15, 22], which is the algebra defined by the spacetime quantum group coordinates dual to the translation (momenta) generators.

In this framework, spacetime curvature (or nonzero cosmological constant) should play a relevant role concerning the possible cosmological consequences of a quantum spacetime (see, e.g., [38, 47–51] and references therein). Therefore, it seems natural to consider the construction of the -deformation for the (anti-)de Sitter (hereafter (A)dS) groups, and to analyse their possible connections with quantum gravity theories with a nonzero cosmological constant. In this respect, we recall that the Hopf structure for the -deformation of (2 + 1)D (A)dS and Poincaré () algebras were collectively obtained in [19], and their connection between their deformed commutation rules and 2 + 1 quantum gravity has been explored in [38]. The results obtained in [19] correspond to the l.h.s. of the commutative diagram:where vertical arrows indicate a classical deformation [52, 53] that introduces the spacetime curvature (or cosmological constant ) related to the (A)dS radius by , and the horizontal ones show the quantum deformation with parameter (related to the Planck length ); reversed arrows correspond to the spacetime contraction and (classical) nondeformed limit . As a consequence, the construction of noncommutative (A)dS spacetimes in terms of intrinsic and ambient spacetime quantum group coordinates seems worth being explored in detail and, moreover, the same framework could account for new proposals of noncommutative spaces of time-like geodesics (worldlines), which, to the best of our knowledge, have not been considered in the literature yet, even for the Poincaré case.

Here we present an enlarged and updated review version of our unpublished manuscript arXiv:hep-th/0401244, in which the above-mentioned problems are faced for the three relativistic cases simultaneously, that is, by dealing explicitly with the spacetime curvature as a contraction parameter. Hence, we propose to explore the r.h.s. of the diagram (1) (dual to the l.h.s.) by computing the quantum deformation of the (2 + 1)D (A)dS groups (that is, ) which are obtained by quantizing the Poisson–Lie algebra of smooth functions on these groups (namely, Fun() coming from a suitable classical -matrix. In this way, the (2 + 1)D noncommutative spaces (e.g., ) can then be identified as certain subalgebras of the corresponding quantum groups. Moreover, we also construct and study in detail the corresponding 4D noncommutative spaces of worldlines. We stress that in our approach all the -Poincaré relations (including its noncommutative spaces) can be directly recovered from the general (A)dS expressions through the limit . Moreover, all of the resulting noncommutative spaces are covariant under quantum group (co)actions (for the construction of Poisson and quantum homogeneous spaces we refer to [54–58] and references therein).

The structure of the paper is the following. In the next section we recall the basics on the (A)dS groups in (2 + 1) dimensions and their associated homogeneous (2 + 1)D spacetimes and 4D spaces of worldlines (time-like lines). Both types of spaces are described in terms of intrinsic quantities (related to group parameters) as well as in ambient coordinates with one and two extra dimensions, respectively, which will be further used in their noncommutative versions. By starting from the classical -matrix that generates the -deformation, we construct in Section 3 the corresponding Drinfel’d-double and obtain some preliminary information on the first-order quantum deformation, from which first-order noncommutative spaces arise. On one hand, we find that at first-order in the deformation parameter the three noncommutative relativistic spacetimes are given by the same -Minkowski algebra. Moreover, we show that the deformation parameter can be interpreted as a curvature on a classical dS spacetime for the three cases, thus generalizing the results obtained in [29, 30] for -Poincaré. On the other hand, we obtain that the first-order noncommutative spaces of worldlines are in fact nondeformed ones, and a relationship with the nonrelativistic (Newtonian) kinematical groups is thus established.

As an intermediate stage in the search of the quantum (A)dS groups, we compute in Section 4 the invariant (A)dS vector fields and next the Poisson–Lie structures coming from the classical -matrix generating the -deformation. These results enable us to propose in Section 5 the noncommutative (A)dS spaces, which are written in both intrinsic and ambient coordinates. The resulting noncommutative spacetimes show how the curvature modifies the underlying first-order -Minkowski space, while for the noncommutative spaces of worldlines we find that 2D velocity/rapidity space (spanned by the dual coordinates to the boost generators) remains nondeformed for -Poincaré but becomes deformed for the (A)dS cases. Hence Lorentz invariance seems to be lost (or somewhat “deformed”) when a nonzero curvature/cosmological constant is considered.

Section 6 is devoted to study the (dual) quantum (A)dS algebras and their deformed Casimirs in two different bases. In particular, starting from the expressions given in [19], a nonlinear transformation involving the generators of the stabilizer subgroup of a time-like line allows us to obtain these quantum algebras in a new basis that generalizes for any the bicrossproduct basis of -Poincaré [16]. These results are analysed in connection with 2 + 1 quantum gravity [38] and a “duality” between curvature/cosmological constant and deformation parameter/Planck length is suggested along the same lines of the so-called “semidualization” approach for Hopf algebras in 2 + 1 quantum gravity [59] associated with the exchange of the cosmological length scale and the Planck mass (see also [60, 61]). Finally, some remarks and comments concerning recent findings in this framework close the paper.

#### 2. (Anti-)de Sitter Lie Groups and Their Homogeneous Spaces

The Lie algebras of the three (2 + 1)D relativistic spacetimes of constant curvature can collectively be described by means of a real (graded) contraction parameter [19], and we denote them by . If are, in this order, the generators of rotations, time translations, space translations, and boosts, the commutation relations of readwhere from now on we assume that Latin indices , Greek ones , , and is a skew symmetric tensor such that . For a positive, zero, and negative value of , provides a Lie algebra isomorphic to , , and , respectively. The case can also be understood as an Inönü–Wigner contraction [62]: .

Parity Π and time-reversal are involutive automorphisms of defined by [63]which together with the composition,and the identity determine a Abelian group of involutions [64]. The automorphisms and give rise, in this order, to the Cartan decompositions:where is the Lorentz subalgebra and covers the Lie subalgebras , , and for , correspondingly. In fact, the contraction parameter is related to the -grading associated to .

When the Lie group is considered, two relevant families of symmetric homogeneous spaces [65] can be constructed as follows.

(i) The (2 + 1)D spacetime: this is a rank 1 space, associated with the automorphism (4) and Cartan decomposition (5), which is defined through the quotientwhere is the Lorentz subgroup spanned by and . Thus momenta characterize the tangent space at the origin. This space turns out to have constant curvature equal to the contraction parameter: for (A)dS and () for Minkowski, where is the universe radius.

(ii) The 4D space of time-like lines (or wordlines): this is a rank 2 space, related to the automorphism (3) and Cartan decomposition (6), which is given bywhere and . This space is of hyperbolic type as this has constant curvature equal to (i.e., in terms of the speed of light). The tangent space is determined by spatial momenta and boosts . In fact, can also be interpreted as a D relativistic phase space [66] in which position and momentum coordinates are related to the group parameters dual to and , respectively.

We display in Table 1 each of the above symmetrical homogeneous spaces for each of the three Lorentzian Lie groups.