Triad approach to the Hamiltonian of general relativity

The Sparling-Thirring forms are constructed using a connection over a frame bundle defined in terms of a spacelike triad and lapse and shift functions. From the forms, one can identify a first-order Lagrangian from which to construct the Hamiltonian using triad vector densities as basic configuration-space variables. If one uses the self-dual part of the forms, the resulting first-order Lagrangian (its imaginary part is a total divergence) leads to momenta which are the dual of the Sen connection which was introduced by Ashtekar in a spinor representation. The resulting formalism is equivalent to that of Ashtekar.