$\mathrm{SU}\left(2\right)\mathrm{}$ loop quantum gravity seen from covariant theory

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 $\mathrm{SU}\left(2\right)\mathrm{}$ loop quantum gravity and to the area spectrum given by the $\mathrm{SU}\left(2\right)\mathrm{}$ Casimir operator. This covariant point of view allows us to analyze closely the drawbacks of the $\mathrm{SU}\left(2\right)\mathrm{}$ formalism: the quantization based on the (generalized) Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs depend nontrivially 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 $\mathrm{SL}(2,C)\mathrm{}$ Casimir operator.