arXiv:gr-qc/0401102 v1 26 Jan 2004 A New Approach in Quantum Gravity and its Cosmological Implications Simone Mercuri Giovanni Montani ICRA—International Center for Relativistic Astrophysics Dipartimento di Fisica (G9), Universit` a di Roma, “La Sapienza”, Piazzale Aldo Moro 5, 00185 Rome, Italy. e-mail: email@example.com, firstname.lastname@example.org PACS: 04.60.Ds – 98.80.Qc Abstract This work concerns a new reformulation of quantum geometrodynamics, which allows to overcome a fundamental ambiguity contained in the canonical approach to quantum gravity: the possibility of performing a (3+1)-slicing of space-time, when the metric tensor is in a quantum regime. Our formulation provides also a procedure to solve the problems connectedto the so called frozen formalism. In particular 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 hamiltonian constraints become parabolic, so arrivingto evolutive (Schr¨odinger-like) equations for the quantum dynamics. The kinematical action can be interpreted as the action of a pressure less, but, in general, non geodesic perfect fluid, so in the semi classical limit our theory leads to the dynamics of the gravitational field coupled to a dust which represents the material reference frame we have introduced fixing the slicing. We also investigate thecosmological implications ofthe presence ofthe dust, which, in the WKB limit of a cosmological problem, makes account for a dark matter component and could play, at present time, a dynamical role.