SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter

An so(4, C)-covariant Hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so-called Immirzi parameter) is developed It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study the dependence of the quantum system on the Immirzi parameter. The path-integral quantization shows no dependence on it. A method of modifying the loop approach in accordance with the formalism developed here is briefly outlined.