  This paper aims at clarifying the link between Loop Quantum Gravity and
Spin-Foam models in four dimensions. Starting from the canonical framework, we
construct an operator P acting on the space of cylindrical functions
Cyl($\Gamma$), where $\Gamma$ is the 4-simplex graph, such that its ma- trix
elements are, up to some normalization factors, the vertex amplitude of
Spin-Foam models. The Spin-Foam models we are considering are the topological
model, the Barrett-Crane model and the Engle-Pereira-Rovelli model. The
operator P is usually called the "projector" into physical states and its
matrix elements gives the physical scalar product. Therefore, we relate the
physical scalar product of Loop Quantum Gravity to vertex amplitudes of some
Spin-Foam models. We discuss the possibility to extend the action of P to any
cylindrical functions on the space manifold.
