Seminarium Teorii Względności i Grawitacji

I will present two Hilbert spaces constructed over the set of all metrics of arbitrary but fixed signature, defined on a manifold. Every state in one of the Hilbert spaces is built of an uncountable number of wave functions representing some elementary quantum degrees of freedom, while every state in the other space is built of a countable numberof them. Each Hilbert space is unique up to natural isomorphisms and carries a unitary representation of the diffeomorphism group of the underlying manifold. The spaces obtained for signature (3,0) may possibly serve as a kinematical Hilbert space for canonical quantization of the ADM formalism.