Cosmological Deformation of Lorentzian Spin Foam Models

TL;DR

This paper explores the quantum deformation of the Lorentzian spin foam model, incorporating a positive cosmological constant, which leads to finite integrals and models quantum gravity in de-Sitter space.

Contribution

It introduces a quantum Lorentzian spin foam model with a cosmological constant, demonstrating divergence removal and finite evaluations using harmonic analysis on quantum groups.

Findings

Infrared divergences are removed with cosmological constant.

Evaluation of spin networks becomes finite due to quantum deformation.

Model describes quantum gravity in de-Sitter space.

Abstract

We study the quantum deformation of the Barrett-Crane Lorentzian spin foam model which is conjectured to be the discretization of Lorentzian Plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de-Sitter space. This spin foam model is constructed using harmonic analysis on the quantum Lorentz group. The evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that the introduction of the cosmological constant removes all the infrared divergences: for any fixed triangulation, the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces.