We develop the quantization of spherically symmetric gravitational systems using the Dirac formalism and the WKB approximation. The particular application is the nucleation of false-vacuum bubbles, which may have been the origin of the inflationary universe and matter. Consistency between our results and those obtained by the Euclidean method requires that the latter include metrics with degeneracies. Our work also raises several other questions of principle, including the meaning of the wave function and the necessity of topology change. A tentative application of this work to quantum cosmology indicates that the "Hartle-Hawking wave function" is inconsistent.

• Received 7 May 1990

DOI:https://doi.org/10.1103/PhysRevD.42.4042