Reparametrization-invariant path integral in gr and "Big Bang" of quantum universe

The reparametrization-invariant generating functional for the unitary and c ausal perturbation theory in general relativity in a finite space-time is o btained. The region of validity of the Faddeev-Popov-DeWitt functional is s tudied. It is shown that the invariant content of general relativity as a c onstrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale fact or) and "geometric" (given by the Levi-Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang," the Hubble evolution, and creation of matter fields by the "geo metric" vacuum are described by the inverted Levi-Civita transformation of the geomeric system into the dynamic one. The particular case of the Levi-C ivita transformations are the Bogoliubov ones of the particle variables (di agonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing t he equations of motion). The choice of initial conditions for the "Big Bang " in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann-Robertson-Walker universe in their conformal (Hoyle-Narlikar) versions.