  We apply the ADM approach to obtain a Hamiltonian description of the
Einstein-Hilbert action. In doing so we add four new ingredients: (i) We
eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt
g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $\chi$ such
that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider
the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers
F(x) which can be treated as Lorentzian manifolds equipped with the
Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic.
(iv) The Hamiltonian operator $H$ is a normally hyperbolic operator in the
bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu=0$ is a
hyperbolic equation in the bundle. Since the corresponding Cauchy problem can
be solved for arbitrary smooth data with compact support, we then apply the
standard techniques of QFT which can be naturally modified to work in the
bundle.
