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Covariant loop gravity comes out of the canonical analysis of the Palatiniaction and the use of the Dirac brackets arising from dealing with the secondclass constraints (``simplicity'' constraints). Within this framework, weunderline a quantization ambiguity due to the existence of a family of possibleLorentz connections. We show the existence of a Lorentz connection generalizingthe Ashtekar-Barbero connection and we loop-quantize the theory showing that itleads to the usual SU(2) Loop Quantum Gravity and to the area spectrum given bythe SU(2) Casimir. This covariant point of view allows to analyze closely thedrawbacks of the SU(2) formalism: the quantization based on the (generalized)Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputsdepend non-trivially on the embedding of the canonical hypersurface into thespace-time manifold. On the other hand, there exists a true space-timeconnection, transforming properly under all diffeomorphisms. We argue that itis this connection that should be used in the definition of loop variables.However, we are still not able to complete the quantization program for thisconnection giving a full solution of the second class constraints at theHilbert space level. Nevertheless, we show how a canonical quantization of theDirac brackets at a finite number of points leads to the kinematical setting ofthe Barrett-Crane model, with simple spin networks and an area spectrum givenby the SL(2,C) Casimir.Comment: LaTeX, 21 pages; a sign mistake correcte

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>SU</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo><mml:mn /></mml:math>loop quantum gravity seen from covariant theory

SU(2)loop quantum gravity seen from covariant theory