Semiclassical path measure and factor ordering in quantum cosmology

Abstract

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 Ψ = P exp(− I c ), where I 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 backgroud. 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 ĤΨ = 0 through the semiclassical expansion. In the minisuperspace models, the prefactor obtained is consistent with ordering the kinetic term in Ĥ 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 attempted.