Hamiltonian gravity in tetrad-connection variables

A systematic Hamiltonian formulation of the Einstein--Cartan system, based on the Hilbert--Palatini action with the Barbero--Immirzi and cosmological constants, is performed using the traditional ADM decomposition and without fixing the time gauge. This procedure results in a larger phase space compared to that of the Ashtekar--Barbero approach as well as a larger set of first-class constraints generating gauge transformations that are on-shell equivalent to spacetime diffeomorphisms and SO(1,3) transformations. The imbalance in the number of components between the tetrad and the connection is resolved by the identification of second-class constraints implied by the action, which can be implemented by use of Dirac brackets or by solving them directly. The Hamiltonian system remains well-defined off the second-class constraint surface in an extended phase space with additional degrees of freedom, implying a more general geometric theory. Implications for canonical quantum gravity are discussed.