GENERALIZED WEYL SYSTEMS AND Κ-Minkowski SPACE

Agostini, Alessandra; Lizzi, Fedele; Zampini, Alessandro

doi:10.1142/s021773230200871x

Cited by 51 publications

(61 citation statements)

References 35 publications

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“…For c = 1 2 , we have a symmetric ordering, which is completely different from the totally symmetric Weyl ordering [63]. Using △N = N ⊗1+1⊗N , △A = A⊗1+1⊗A, and [N, A] = 0, it is easy to verify that the above class of twist operators F c satisfies the cocycle condition…”

Section: Star Productmentioning

confidence: 99%

Quantum statistics and noncommutative black holes

2012

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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...

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Section: Star Productmentioning

confidence: 99%

Quantum statistics and noncommutative black holes

2012

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“…In this paper we would like to focus on the simplest case, that of spinless particles; then, since any purely internal degrees of freedom (commuting with P κ ) can be safely ignored, we may drop the index σ from now on. 5 One also often sees the "modified" bicrossproduct basis [38,35,36,8], which is defined by P…”

Section: Intertwiners For the κ-Poincaré Hopf Algebramentioning

confidence: 99%

Covariant particle statistics and intertwiners of the κ-deformed Poincaré algebra

Young

Zegers

2008

Nuclear Physics B

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To speak about identical particles -bosons or fermions -in quantum field theories with κ-deformed Poincaré symmetry, one must have a κ-covariant notion of particle exchange. This means constructing intertwiners of the relevant representations of κ-Poincaré. We show, in the simple case of spinless particles, that intertwiners exist, and, supported by a perturbative calculation to third order in 1 κ , make a conjecture about the existence and uniqueness of a certain preferred intertwiner defining particle exchange in κ-deformed theories.

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“…It is instead a property of the star-product scheme as it appears in the unifying form described in [27]. The symmetry could yield to interesting results in other relevant cases as those considered in [22,25], which we plan to investigate later.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, [22] a general method is proposed, which produces various new non-formal star products on R 3 using a variation of the Jordan Schwinger map [23]. The method includes previous results of [24] and it is easily extendable to higher dimensions [25].…”

Section: Introductionmentioning

confidence: 99%

Phase space distributions and a duality symmetry for star products

Man’ko

Marmo²,

Vitale³

2005

Physics Letters A

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A duality property for star products is exhibited. In view of it, known starproduct schemes, like the Weyl-Wigner-Moyal formalism, the Husimi and the GlauberSudarshan maps are revisited and their dual partners elucidated. The tomographic map, which has been recently described as yet another star product scheme, is considered. It yields a noncommutative algebra of operator symbols which are positive definite probability distributions. Through the duality symmetry a new noncommutative algebra of operator symbols is found, equipped with a new star product. The kernel of the new star product is established in explicit form and examples are considered.

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GENERALIZED WEYL SYSTEMS AND Κ-Minkowski SPACE

Abstract: We introduce the notion of generalized Weyl system, and use it to define * -products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various * -products which generalize the κ-Minkowski commutation relation.

Cited by 51 publications

References 35 publications

Quantum statistics and noncommutative black holes

Quantum statistics and noncommutative black holes

Covariant particle statistics and intertwiners of the κ-deformed Poincaré algebra

Phase space distributions and a duality symmetry for star products

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