Steepest-descent contours in the path-integral approach to quantum cosmology. II. Microsuperspace

This paper is the second of a series concerned with the contour of integration in the path-integral approach to quantum cosmology. We study the Hartle-Hawking no-boundary proposal for the wave function of the universe in a simple very restrictive model. It consists of a three-sphere closed off by four-geometries which are precisely sections of a four-sphere. The path integral for the model therefore reduces to a single ordinary integration over the radius r of the four-sphere. Models of this type, in which the path integral reduces to a finite number of ordinary integrations, we christen ‘‘microsuperspace’’ models. They have the attractive feature that, in contrast with minisuperspace, one deals with the four-geometry directly. We argue that to integrate r over the real geometrically obvious range, i.e., to integrate over real Euclidean four-geometries, is not satisfactory, because (i) convergence depends on an unknown measure, (ii) it does not work in more general cases, and (iii) it does not lead to the oscillatory wave function needed to predict that spacetime is classical when the Universe is large. We therefore look for a complex contour. Satisfactory contours may be found, but we find that there is no unique convergent contour, reinforcing the conclusion of paper I that the Hartle-Hawking proposal does not fix the wave function uniquely. We also argue that certain complex saddle points which dominate the integral are most correctly thought of as genuinely complex four-metrics and not, as previously assumed, as combinations of real Euclidean and real Lorentzian four-metrics. The general utility of microsuperspace models is discussed.