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 Λ. In particular, the momentum space associated to the κ-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 Λ. 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 κ-Poincaré algebra are smoothly recovered in the limit Λ→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.
Highlights
Recent developments in quantum gravity research have revived and given new substance to the longforgotten idea that momentum space should have a nontrivial geometry, an intuition originally due to Max Born [1]
Such an energy scale allows the geometry of momentum space to be nontrivial, and it is a general feature of Deformed Special Relativity (DSR) models that the manifold of momenta has nonzero curvature
In this paper we have shown that the curved momentum space construction can be extended to cases where a nonvanishing spacetime cosmological constant is present
Summary
Recent developments in quantum gravity research have revived and given new substance to the longforgotten idea that momentum space should have a nontrivial geometry, an intuition originally due to Max Born [1]. We are able to explicitly construct the curved momentum space generated by quantum-deformed spacetime symmetries in presence of a nonvanishing cosmological constant We achieve this result by enlarging the momentum space so that it is the manifold of momenta associated to translations on spacetime, but it includes the ‘hyperbolic’ momenta associated to the boost transformations and the angular momenta associated to rotations. In (1+1) dimensions the dual group coordinates are those associated to both the spacetime translations and boosts, and a certain linear action of the dual group on the origin of momentum space generates (half of) a (2+1)-dimensional dS manifold MdS3, spanned by the orbit of the group passing through the origin In this case, the fact that boosts have the same role in the momentum space as translation generators can be understood since their coproducts have the same formal structure. The paper ends with a concluding section in which the applicability of the method here presented to the construction of the κ-AdS momentum space is shown, and the keystones for solving the corresponding (3+1)-dimensional problem are presented
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