A note on the spin connection representation of gravity

Abstract

The formulation of gravity in 3+1 dimensions in which the spin connection is the basic field ($ω$-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 to space-time 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.