Revised Canonical Quantum Gravity via the Frame Fixing

Abstract

We present a new reformulation of the canonical quantum geometrodynamics, which allows to overcome the fundamental problem of the frozen formalism and, therefore, to construct an appropriate Hilbert space associate to the solution of the restated dynamics. More precisely, to remove the ambiguity contained in the Wheeler-DeWitt approach, with respect to the possibility of a (3 + 1)-splitting when the space-time is in a quantum regime, we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so-called kinematical action; as a consequence the new super-Hamiltonian constraint becomes a parabolic one and we arrive to a Schroedinger-like approach for the quantum dynamics. In the semiclassical limit our theory provides General Relativity in the presence of an additional energy-momentum density contribution coming from no longer zero eigenvalues of the Hamiltonian constraints; the interpretation of these new contributions comes out in natural way as soon as it is recognized that the kinematical action can be recasted in such a way it describes a pressureless, but, in general, non geodesic perfect fluid.