Semiclassical path measure and factor ordering in quantum cosmology

The euclidean path integral over compact four-metrics proposed by Hartle and Hawking to define the quantum state of the Universe is examined in the full semiclassical approximation Psi = P exp(-I/sup c/), where I/sup c/ is the action of the extremum and the prefactor P is given by a gaussian integral over small fluctuations around the extremum. We consider spatially homogeneous minisuperspace models of Bianchi types I and III and of the Kantowski--Sachs type, and infinite dimensional models containing the perturbative gravitational field and a massless scalar field on a k = +1 Friedmann--Robertson--Walker minisuperspace background. The prefactor is evaluated using zeta-function regularisation and adopting a scale invariant measure, and it is then related to the factor ordering dependent first derivative terms in the Wheeler--DeWitt equation HPsi = 0 through the semiclassical expansion. In the minisuperspace models, the prefactor obtained is consistent with ordering the kinetic term in H as the natural Laplacian as proposed by Hawking and Page. In the infinite dimensional models, a preferred factor ordering may be found by regularising the divergent semiclassical expansion of the Wheeler--DeWitt equation naively in terms of the Riemann zeta-function; however, no contact with covariant regularisation methods or an infinite dimensional Laplacian is attepted. copyright 1988 Academic Press, Inc