Curved momentum spaces from quantum groups with cosmological constant

Abstract

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincar\'e algebra are smoothly recovered in the limit $\Lambda\to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.Comment: 13 page

Curved momentum spaces from quantum (anti–)de Sitter groups in ( 3+1 ) dimensions

“…We follow the presentation of [7], and make use of the group-theoretical framework recently introduced in [22]. We show that the κ-Poincaré curved momentum space can be obtained as a specific orbit of the action of the dual κ-Poisson-Lie group on a (4 þ 1)-dimensional ambient Minkowski space, and it turns out to be (half of) the (3 þ 1) dS space.…”

“…The aim of this paper is the generalization of the previous construction to the case with nonvanishing cosmological constant, by following the approach presented in [22] for the (1 þ 1)-and (2 þ 1)-dimensional dS cases. In this way, a global picture of the interplay between the cosmological constant and the Planck-scale deformation parameter z ¼ 1=κ can be presented.…”

“…Therefore, when the cosmological constant does not vanish, translations and boosts have similar algebraic properties and play complementary geometric roles (see [45] for a more detailed discussion in the context of homogeneous spaces of worldlines). As a consequence, the main idea introduced in [22] for the construction of curved momentum spaces for the κ-(A)dS algebras in (2 þ 1) dimensions becomes fully applicable: when ω ≠ 0 the momentum space has to be enlarged by including the angular momenta associated to the rotation symmetries and the hyperbolic momenta associated to boosts. This means that the momentum space arises as the orbit of an appropriate action of the dual Poisson-Lie group G à ω , whose Lie algebra g à ω is obtained by dualizing δ. Namely,…”

“…3 The physical interpretation of such a dispersion relation requires giving a physical meaning to the hyperbolic momenta χ a . This can be done thanks to the identification (22), which states that the local group coordinates have the same Poisson brackets as the generators of the κ-(A)dS algebra, as discussed in detail in [22]. It is then possible to represent the local coordinates of the dual group in terms of the usual phase space coordinates π μ and x ν with fπ μ ; x ν g ¼ δ ν μ , following a procedure similar to that found in [8,49].…”

“…In recent work we demonstrated that in fact it is possible to define momentum space in the presence of a cosmological constant [22]. The nontrivial interplay between curvature of spacetime and of momentum space is resolved by generalizing the momentum space so that it contains the hyperbolic momenta associated to boosts besides the momenta associated to spacetime translations.…”