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 Arnowitt-Deser-Misner 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.