Deformed oscillator algebras and QFT in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>κ</mml:mi></mml:math>-Minkowski spacetime

Govindarajan, T. R.; Gupta, Kumar S.; Harikumar, E.; Meljanac, Stjepan; Meljanac, Daniel

doi:10.1103/physrevd.80.025014

Cited by 101 publications

(119 citation statements)

References 77 publications

(217 reference statements)

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“…This particular realization corresponds to symmetric ordering of coordinates and it gives rise to a generalized Klein-Gordon equation [49],…”

Section: Twisted Oscillator Algebra In the Noncommutative Black Hole mentioning

confidence: 99%

“…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-deformed spaces [40][41][42][43][44][45][46][47][48][49]. To this end, it is useful to work with realizations of the κ-space and the associated star products [51][52][53][54][55].…”

Section: Introductionmentioning

confidence: 99%

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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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“…This particular realization corresponds to symmetric ordering of coordinates and it gives rise to a generalized Klein-Gordon equation [49],…”

Section: Twisted Oscillator Algebra In the Noncommutative Black Hole mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum statistics and noncommutative black holes

2012

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“…Using the solution for U 0 obtained in eqn. (19) in the above , and after straight forward simplification, we get the equation satisfied by U 2 as…”

Section: Kappa-deformed Laplacian In Euclidean Space-timementioning

confidence: 99%

“…(10) also as the κ-deformed Laplacian. Both D µ D µ and operators have been analysed as possible generalisations of Klein-Gordon operator in κ-space-time [18,19].…”

Section: Kappa-deformed Laplacian In Euclidean Space-timementioning

confidence: 99%

Spectral dimension of kappa-deformed spacetime

Anjana

Harikumar

2015

Phys. Rev. D

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We investigate the spectral dimension of κ-space-time using the κ-deformed diffusion equation. The deformed equation is constructed for two different choices of Laplacians in n-dimensional, κ-deformed Euclidean space-time. We use an approach where the deformed Laplacians are expressed in the commutative space-time itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimension of κ-deformed space-time and show that it decreases as the probe length decreases. By introducing a bound on the deformation parameter, spectral dimension is guaranteed to be positive definite. We find that, for one of the choices of the Laplacian, the non-commutative correction to the spectral dimension depends on the topological dimension of the space-time whereas for the other, it is independent of the topological dimension. We have also analysed the dimensional flow for the case where the probe particle has a finite extension, unlike a point particle.1

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“…The exact mathematical formulation of classical field theories on geometries, like κ-Minkowski [26][27][28][29][30][31] and Snyder [32], with respect to deformation parameters κ and β, respectively, is also very important. However, quantum properties of these sibling theories are not as easy to characterize as the Moyal theories [29,31].…”

Section: Introductionmentioning

confidence: 99%

UV-IR mixing in nonassociative Snyder ϕ4 theory

et al. 2018

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Using a quantization of the nonassociative and noncommutative Snyder ϕ 4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter β. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative ϕ 4 products can generate different momentum-conserving integrals. Finally, most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore, infrared divergences should exist, in general, in the Snyder ϕ 4 theory. We consider infrared divergences at the limit p → 0 as UV/IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative ϕ 4 products. We also discuss the extrapolation of the Snyder deformation parameter β to negative values as well as certain general properties of one-loop quantum corrections in Snyder ϕ 4 theory at the zero-momentum limit.

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Deformed oscillator algebras and QFT inκ-Minkowski spacetime

Cited by 101 publications

References 77 publications

Quantum statistics and noncommutative black holes

Quantum statistics and noncommutative black holes

Spectral dimension of kappa-deformed spacetime

UV-IR mixing in nonassociative Snyder ϕ4 theory

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