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

The quantum development of an asymptotically Euclidean Cauchy hypersurface