The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes

Ballesteros, Ángel; Gubitosi, Giulia; Gutiérrez-Sagredo, Iván; Herranz, Francisco J.

doi:10.1016/j.physletb.2020.135461

Cited by 24 publications

(54 citation statements)

References 55 publications

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“…Most papers however consider a nonrelativistic version of the model, where only spatial coordinates present a Snyder structure, while time is unaffected. This case is of course easier to treat and is expected to correspond to the correct low-energy limit (see however [26]). In this context, some standard problems have been discussed, both in classical and quantum settings.…”

Section: Phenomenological Implicationsmentioning

confidence: 99%

Progress in Snyder model

Mignemi¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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Section: Phenomenological Implicationsmentioning

confidence: 99%

Progress in Snyder model

Mignemi¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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“…Therefore, no higher-order terms in the classical and quantum coordinates arise. By contrast, when the κ-deformation is applied to a curved manifold instead of (24), higher-order terms in the coordinates appear in the Poisson homogeneous spacetime, so that the corresponding quantization is not straightfoward at all, as the recent constructions of the κ-noncommutative (anti-)de Sitter [70], Newtonian and Carrollian [71] spacetimes explicitly show.…”

Section: Quantum Groups and Noncommutative Spacesmentioning

confidence: 99%

“…This is just the same result coming from a particular solution of the Z ⊗4 2 -graded contraction equations for so(5) [17] (see [20] for the general solution). Explicitly, the non-vanishing commutation relations of so ω (5) read (71) We remark that, although the factor √ ω ab = 0 in the map ( 68) can be an imaginary number, enabling to change the real form of the algebra, the resulting commutation relations (70) of so ω (5) only comprise real Lie algebras. Moreover, the zero value for ω ab is consistently allowed in (70), which is equivalent to apply an Inönü-Wigner contraction [13,31], leading to a more abelian (contracted) Lie algebra.…”

Section: The Drinfel'd-jimbo Lie Bialgebra For the Cayley-klein Algebra So ω (5)mentioning

confidence: 99%

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Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Gutiérrez-Sagredo

Herranz

2021

Symmetry

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The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

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“…There are different motivations to consider a deformation of the classical-quantum transition [1][2][3][4][5][6][7][8][9][10][11][12]. The difficulties to find a consistent quantum theory incorporating the gravitational interaction could be due to the present formulation of quantum theories [13].…”

Section: Introductionmentioning

confidence: 99%

Deformed classical-quantum mechanics transition

Cortés

Gamboa

2020

Phys. Rev. D

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An approach to study a generalization of the classical-quantum transition for general systems is proposed. In order to develop the idea, a deformation of the ladder operators algebra is proposed that contains a realization of the quantum group SUð2Þ q as a particular case. In this deformation Planck's constant becomes an operator whose eigenvalues approach ℏ for small values of n (the eigenvalue of the number operator) and zero for large values of n (the system is classicalized).

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The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes

Cited by 24 publications

References 55 publications

Progress in Snyder model

Progress in Snyder model

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Deformed classical-quantum mechanics transition

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