Hilbert spaces built over metrics of fixed signature

We construct two Hilbert spaces 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 number of them. Each Hilbert space is unique up to natural isomorphisms and carries a unitary representation of the diffeomorphism group of the underlying manifold. The Hilbert spaces constructed in the case of signature (3, 0) may be possibly used for canonical quantization of the ADM formulation of general relativity.

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