REPRESENTATIONS OF QUANTUM GENERAL-RELATIVITY USING ASHTEKARS VARIABLES

On the basis of the link class solutions for quantum general relativity found by Rovelli and Smolin in the loop representation, we describe a possible procedure for assigning a Hilbert space structure to this set of solutions. On this Hilbert space operators can be defined by going back to the connection representation; this is possible consistently since the 'gauge choice' one makes by picking out specific loops in the link classes only corresponds to general coordinate transformations on the observables connected to the operators. In this way the loop transform of Rovelli and Smolin is restricted to the physical states of the theory. However, the results still contain products of delta functions at one point, and thus the description is not complete until a satisfying regularization procedure has been found. Nevertheless this formalism shows how physical observables of general relativity can be defined in a natural way as operators on the space of smooth link classes.