Lorentzian LQG vertex amplitude

We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian Barrett-Crane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit. Introduction Loop quantum gravity (LQG) [1] provides a well defined, background independent, construction of the kinematical Hilbert space of quantum general relativity. Spin foam techniques [2] have been developed as a possible framework to study the quantum dynamics. A spin foam is a two complex (union of edges, faces and vertices) colored by quantum numbers (faces are labelled by representations of a given group and edges by intertwiners). It can be interpreted as the history of a spin network (more precisely, the boundary of a spin foam is a spin network). A spin foam model is given by the assignment of amplitudes to faces, edges and vertices. The most studied model so far is the Barrett-Crane (BC) model for both Lorentzian [3] and Euclidean [4] signatures. It is obtained as a modification of a topological BF quantum field theory by imposing the discrete analogues of the constraints called simplicity constraints that, in the continuum limit, reduce BF theory to general relativity [5]. Much work has been carried out in recent years to extract the low energy behavior of this model [6] and it turns out that some components of the two-point functions are in disagreement with the expected behavior determined by standard perturbative quantum gravity [7]. As argued in [8, 9] the problem can be traced back to the way some of the constraints are imposed in the Barrett-Crane model. In fact, the simplicity constraints form a second class system [10] and in the BC model these are imposed as strong operator equations [11], killing then physical degrees of freedom. In [8, 9] a reformulation of these constraints has been proposed and this allows for a new sector of solutions. This can be obtained from the BC model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit. In [8, 9] only the Euclidean signature case was considered. Here we extend the construction to the Lorentzian case. The main features of the Euclidean model are preserved in the Lorentzian case. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. ∗Unité mixte de recherche (UMR 6207) du CNRS et des Universités de Provence (Aix-Marseille I), de la Meditarranée (Aix-Marseille II) et du Sud (Toulon-Var); laboratoire affilié à la FRUMAM (FR 2291).