A note on thespin connection representation of gravity

Abstract

The formulation of gravity in 3 + 1 dimensions in which the spin connection is the basic field (-frame) leads to a theory with first- and second-class constraints. Here, the Dirac brackets for the second-class constraints are evaluated and the Dirac algebra of first-class constraints is found to be the usual algebra associated with spacetime reparametrizations and tangent space rotations. This establishes the classical equivalence with the vierbein approach (e-frame). The explicit form of the path integral for this theory is given and the quantum equivalence with the e-frame is also established.