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Noncommutative spaces of worldlines
https://repo.scoap3.org/record/31774
The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the κ -Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of κ -Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the parameter κ−1 plays the very same role as the Planck constant ħ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.Ballesteros, AngelSat, 23 Mar 2019 09:09:24 GMThttps://repo.scoap3.org/record/317742019-05-10Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups
https://repo.scoap3.org/record/22520
The κ -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 κ -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 κ -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.Herranz, Francisco J.Tue, 28 Nov 2017 21:16:11 GMThttps://repo.scoap3.org/record/22520urn:ISSN:1687-7365Hindawi2017-11-28The κ -(A)dS quantum algebra in (3 + 1) dimensions
https://repo.scoap3.org/record/18620
The quantum duality principle is used to obtain explicitly the Poisson analogue of the κ -(A)dS quantum algebra in ( 3+1 ) dimensions as the corresponding Poisson–Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant Λ is included as a Poisson–Lie group contraction parameter, and the limit Λ→0 leads to the well-known κ -Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this κ -(A)dS deformation is sketched.Ballesteros, ÁngelMon, 16 Jan 2017 16:13:45 GMThttps://repo.scoap3.org/record/18620urn:ISSN:0370-2693Elsevier2017-02-04Towards ( 3+1 ) gravity through Drinfel'd doubles with cosmological constant
https://repo.scoap3.org/record/9999
We present the generalisation to ( 3+1 ) dimensions of a quantum deformation of the ( 2+1 ) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern–Simons formulation of ( 2+1 ) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the ( 3+1 ) scenario as a possible guiding principle towards the description of ( 3+1 ) gravity. To this aim, a canonical classical r -matrix arising from a Drinfel'd double structure for the three ( 3+1 ) Lorentzian algebras is obtained. This r -matrix turns out to be a twisted version of the one corresponding to the ( 3+1 ) κ -deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the κ -Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.Ballesteros, AngelThu, 23 Apr 2015 16:17:26 GMThttps://repo.scoap3.org/record/9999urn:ISSN:0370-2693Elsevier2015-06-30A ( <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>1</mn></math> ) non-commutative Drinfel'd double spacetime with cosmological constant
https://repo.scoap3.org/record/1722
We show that the Drinfel'd double associated to the standard quantum deformation slη(2,R) is isomorphic to the (2+1) -dimensional AdS algebra with the initial deformation parameter η related to the cosmological constant Λ=−η2 . This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case Λ≠0 are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear Λ -deformation of the Minkowskian one.Ballesteros, AngelFri, 21 Mar 2014 23:48:10 GMThttps://repo.scoap3.org/record/1722urn:ISSN:0370-2693Elsevier2014-05-01