Projected spin networks for Lorentz connection: linking spin foams and loop gravity

In the search for a covariant formulation for loop quantum gravity, spin foams have arisen as the corresponding discrete spacetime structure and, among the different models, the Barrett–Crane model seems to be the most promising. Here, we study its boundary states and introduce cylindrical functions on both the Lorentz connection and the time normal to the studied hypersurface. We call them projected cylindrical functions and explain how they would naturally arise in a covariant formulation of loop quantum gravity.