Quantizing Canonical Gravity in the Real Domain

There are several possibilities of formulating the classical Hamiltonian theory of pure Einstein gravity. The traditional one, proposed by Arnowitt-DeserMisner, is in terms of a canonical pair (gab, π ) of a Riemannian three-metric and its conjugate momentum. Introducing local, rotational SO(3)-degrees of freedom, one obtains a closely related formulation, based on a variable pair (E i ,K i a), where the (inverse, densitized) three-metric is expressible as a function of the triad E i , g ab = E i E , and its conjugate momentum is the extrinsic curvature of the three-manifold Σ, with one of its spatial indices converted to an internal one. Starting from this latter formulation, one may perform a canonical transformation and end up with yet another canonical pair of variables (Aa, E a i ), where Aa is now an SO(3)-valued connection variable. To be precise, we will be interested in two variants of this approach. In the first case, the generator of the canonical transformation is i ∫