Loop-Quantum-Gravity Vertex Amplitude

Spin foam models are hoped to provide the dynamics of loop-quantum gravity. However, the most popular of these, the Barrett-Crane model, does not have the good boundary state space and there are indications that it fails to yield good low-energy $n$-point functions. We present an alternative dynamics that can be derived as a quantization of a Regge discretization of Euclidean general relativity, where second class constraints are imposed weakly. Its state space matches the $SO(3)$ loop gravity one and it yields an $SO(4)$-covariant vertex amplitude for Euclidean loop gravity.