  In our model of quantum gravity the quantum development of a Cauchy
hypersurface is governed by a wave equation derived as the result of a
canonical quantization process. To find physically interesting solutions of the
wave equation we employ the separation of variables by considering a temporal
eigenvalue problem which has a complete countable set of eigenfunctions with
positive eigenvalues and also a spatial eigenvalue problem which has a complete
set of eigendistributions. Assuming that the Cauchy hypersurface is
asymtotically Euclidean we prove that the temporal eigenvalues are also spatial
eigenvalues and the product of corresponding eigenfunctions and
eigendistributions, which will be smooth functions with polynomial growth, are
the physically interesting solutions of the wave equation. We consider these
solutions to describe the quantum development of the Cauchy hypersurface.
