  Two sets of spatially diffeomorphism invariant operators are constructed in
the loop representation formulation of quantum gravity. This is done by
coupling general relativity to an anti- symmetric tensor gauge field and using
that field to pick out sets of surfaces, with boundaries, in the spatial three
manifold. The two sets of observables then measure the areas of these surfaces
and the Wilson loops for the self-dual connection around their boundaries. The
operators that represent these observables are finite and background
independent when constructed through a proper regularization procedure.
Furthermore, the spectra of the area operators are discrete so that the
possible values that one can obtain by a measurement of the area of a physical
surface in quantum gravity are valued in a discrete set that includes integral
multiples of half the Planck area. These results make possible the construction
of a correspondence between any three geometry whose curvature is small in
Planck units and a diffeomorphism invariant state of the gravitational and
matter fields. This correspondence relies on the approximation of the classical
geometry by a piecewise flat Regge manifold, which is then put in
correspondence with a diffeomorphism invariant state of the gravity-matter
system in which the matter fields specify the faces of the triangulation and
the gravitational field is in an eigenstate of the operators that measure their
areas.
