The κ-(A)dS noncommutative spacetime

Abstract

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed by quantizing its semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. This turns out to be the only possible generalization of the well-known κ-Minkowski spacetime to the case of non-vanishing cosmological constant, under the condition that the time translation generator of the corresponding quantum (A)dS algebra is primitive. Moreover, the κ-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the cosmological constant parameter define a quantum sphere, while the commutators between time and space coordinates preserve the same structure of the κ-Minkowski spacetime. When expressed in ambient coordinates, the quantum κ-(A)dS spacetime is shown to be defined as a noncommutative pseudosphere.

Asymptotic freedom for $$\lambda \phi ^4_{\star }$$ QFT in Snyder–de Sitter space

“…For this reason, we extend our previous study of the Snyder scalar QFT to the case of a de Sitter background. Some other approaches to noncommutative curved spaces can be found in [13][14][15], where the authors motivate their mathematical construction from Poisson-Lie algebras, and [16,17], where a grouprepresentation approach is used and some astrophysical consequences are discussed.…”

Effects of Planck-scale-modified dispersion relations on the thermodynamics of charged black holes

“…As a matter of fact, this procedure presents some similarities with some of the most appealing approaches that demonstrated corrections on the black hole thermodynamics driven by modified dispersion relations [50] (besides generalized uncertainty principles). In that case, a generalized Stephan-Boltzmann law is calculated considering approximated MDRs, similarly to our calculations of the energy density in the denominator of the equation of state parameter (14) or (15). And the temperature of this radiation fluid is also identified with the temperature assigned to the back hole.…”

“…An appealing way to connect this approach to actual astrophysical observations consists in promoting such deformed flat metrics to deformed curved ones, in order to manifest the gravitational field degrees of freedom. There exist some proposals that promote the κ-Poincaré algebra to a curved setup (see [15] and references therein), some that explore curved Finsler and Hamilton geometries [16][17][18], disformal transformations on a metric [19], among others. In this paper we shall focus on the simplest and most fruitful of these proposals, called rainbow gravity (RG) [21].…”

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

“…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.…”

“…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. Consequently, each graded contraction parameter ω m can take a positive, negative or zero value in (70), and, when ω m = 0, it can be reduced to ±1 through scaling of the Lie generators.…”