  Covariant loop gravity comes out of the canonical analysis of the Palatini
action and the use of the Dirac brackets arising from dealing with the second
class constraints (``simplicity'' constraints). Within this framework, we
underline a quantization ambiguity due to the existence of a family of possible
Lorentz connections. We show the existence of a Lorentz connection generalizing
the Ashtekar-Barbero connection and we loop-quantize the theory showing that it
leads to the usual SU(2) Loop Quantum Gravity and to the area spectrum given by
the SU(2) Casimir. This covariant point of view allows to analyze closely the
drawbacks of the SU(2) formalism: the quantization based on the (generalized)
Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs
depend non-trivially on the embedding of the canonical hypersurface into the
space-time manifold. On the other hand, there exists a true space-time
connection, transforming properly under all diffeomorphisms. We argue that it
is this connection that should be used in the definition of loop variables.
However, we are still not able to complete the quantization program for this
connection giving a full solution of the second class constraints at the
Hilbert space level. Nevertheless, we show how a canonical quantization of the
Dirac brackets at a finite number of points leads to the kinematical setting of
the Barrett-Crane model, with simple spin networks and an area spectrum given
by the SL(2,C) Casimir.
