LQG vertex with finite Immirzi parameter

We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter gamma. We cover both the Euclidean and Lorentzian cases. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on gamma, and, remarkably, holds in the Lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite gamma. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions. (c) 2008 Elsevier B.V. All rights reserved.