FINITE DIFFEOMORPHISM-INVARIANT OBSERVABLES IN QUANTUM-GRAVITY

Two sets of spatially diffeomorphism-invariant operators are construct ed in the loop representation formulation of quantum gravity. This is done by coupling general relativity to an antisymmetric tensor gauge f ield and using that field to pick out sets of surfaces, with boundarie s, in the spatial three-manifold. The two sets of observables then mea sure the areas of these surfaces and the Wilson loops for the self-dua l connection around their boundaries. The operators that represent the se 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 quant um gravity are valued in a discrete set that includes integral multipl es of half the Planck area. These results make possible the constructi on of a correspondence between any three-geometry whose curvature is s mall in Planck units and a diffeomorphism-invariant state of the gravi tational and matter fields. This correspondence relies on the approxim ation of the classical geometry by a piecewise flat Regge manifold, wh ich is then put in correspondence with a diffeomorphism-invariant stat e of the gravity-matter system in which the matter fields specify the faces of the triangulation and the gravitational field is in an eigens tate of the operators that measure their areas.