DIFFERENTIAL STRUCTURE ON Κ-Minkowski SPACE, AND Κ-Poincaré ALGEBRA

We study the behaviour of a scalar field coupled to a noncommutative black hole which is described by a κ-cylinder Hopf algebra. We introduce a new class of realizations of this algebra which has a smooth limit as the deformation parameter vanishes. The twisted flip operator is independent of the choice of realization within this class. We demonstrate that the R-matrix is quasi-triangular up to the first order in the deformation parameter. Our results indicate how a scalar field might behave in the vicinity of a black hole at the Planck scale.Keywords: κ deformed space, noncommutative black holes, twisted statistics PACS numbers: 11.10.Nx, 11.30.Cp INTRODUCTIONNoncommutative geometry offers a framework for describing the quantum structure of space-time at the Planck scale [1]. Einstein's theory of general relativity together with the uncertainty principle of quantum mechanics leads to a class of models with space-time noncommutativity [2,3]. The smooth space-time geometry of classical general relativity is thus replaced with a Hopf algebra at the Planck scale. There are many examples of such Hopf algebras including the Moyal plane, κ-space and Snyder space. The analysis of [2, 3] does not suggest any preferred choice among these models.Further insight about the possible features of the space-time algebra at the Planck scale comes from the analysis of noncommutative black holes. The algebraic structure associated with a noncommutative black hole can be revealed by studying a simple toy model, such as the noncommutative deformation of the BTZ black hole [4,5]. The resulting space-time algebra resembles a noncommutative cylinder [6,7], belonging to the general class of κ-deformed space-time [8][9][10][11]. The appearance of the κ-cylinder algebra is not restricted to the deformation of the BTZ black hole alone. Such an algebra describes noncommutative Kerr black holes [12] within the framework of twisted gravity theories [13][14][15]. It also appears in the context of noncommutative FRW cosmologies [16]. In addition, the κ-Minkowski algebra is relevant in models of doubly-special relativity and in the analysis of astrophysical data from the GRB's [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. This wide-ranging appearance of the κ-cylinder algebra * kumars.gupta@saha.ac.in † meljanac@irb.hr ‡ asamsarov@irb.hr suggests that it captures certain generic features of noncommutative gravity and black holes and is therefore an interesting toy model to explore Planck scale physics. In this Letter we shall investigate certain features of the κ-cylinder algebra using a scalar field as a simple probe. In order to study quantum field theory in any space-time, it is essential to specify the statistics of the quantum field. It has been known for a long time that quantum gravity can admit exotic statistics [31][32][33]. More recently, the idea of twisted statistics and the associated R-matrices have appeared in the context of quantum field theories in noncommutative space-time [34][35][36][37][38][39], including the κ-d...

show abstract

“…The algebra (12) admits the realization of the type (11), which also includes the class of noncovariant realizations given by [62]x…”

Section: Generalitiesmentioning

confidence: 99%

Quantum statistics and noncommutative black holes

2012

View full text Add to dashboard Cite

show abstract

“…In this paper we are establishing numerous results valuable from both mathematical and physical point of view, generalizing previous results on realizations [17][18][19][20]. Our motivation is to unify κ-Minkowski spacetime and Lorentz algebra in the unique Lie algebra [18].…”

Section: Introductionmentioning

confidence: 53%

Kappa-Minkowski Spacetime, Kappa-Poincaré Hopf Algebra and Realizations

Kovačević

Meljanac

2012

Int. J. Geom. Methods Mod. Phys.

Self Cite

View full text Add to dashboard Cite

The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.

show abstract

“…where m is the multiplication map m : a ⊗ b → ab of the Heisenberg algebra. These realizations are generalizations of those discussed in [26,27,28,29].…”

Section: Coordinate Realizations Of the κ-Minkowski Spacetime And Stamentioning

confidence: 66%

Interpolations between Jordanian twists, the Poincaré–Weyl algebra and dispersion relations

Meljanac

Škoda

et al. 2020

Int. J. Mod. Phys. A

Self Cite

View full text Add to dashboard Cite

We consider a two parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the κ-Minkowski noncommutative space time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincaré-Weyl Hopf algebra and two types of one parameter families of dispersion relations are constructed. Mathematically equivalent deformations may lead to differences in physical phenomena. * Daniel.Meljanac@irb.hr † meljanac@irb.hr ‡ zoran.skoda@uhk.cz § rina.strajn@unidu.hr

show abstract

DIFFERENTIAL STRUCTURE ON Κ-Minkowski SPACE, AND Κ-Poincaré ALGEBRA

Cited by 33 publications

References 50 publications

Quantum statistics and noncommutative black holes

Quantum statistics and noncommutative black holes

Kappa-Minkowski Spacetime, Kappa-Poincaré Hopf Algebra and Realizations

Interpolations between Jordanian twists, the Poincaré–Weyl algebra and dispersion relations

Contact Info

Product

Resources

About