Steepest-descent contours in the path-integral approach to quantum cosmology. III. A general method with applications to anisotropic minisuperspace models

This paper is the third of a series concerned with the contour of integration in the path-integral approach to quantum cosmology. We describe a general method for the approximate evaluation of the path integral for spatially homogeneous minisuperspace models. In this method the path integral reduces, after some trivial functional integrals, to a single ordinary integration over the lapse. The lapse integration contours can then be studied in detail by finding the steepest-descent paths. By choosing different complex contours, different solutions to the Wheeler-DeWitt equation may be generated. The method also proves to be useful for finding and studying the complex solutions to the Einstein equations that inevitably arise as saddle points. We apply our method to a class of anisotropic minisuperspace models, namely, Bianchi types I and III, and the Kantowski-Sachs model. After a general discussion of convergent contours in these models, we attempt to implement two particular boundary-condition proposals: the no-boundary proposal of Hartle and Hawking, and the path-integral version of the tunneling proposal of Linde and Vilenkin.

In the 3+1 formalism we use, the issue of finding initial data corresponding to the no-boundary proposal turns out to be rather subtle, and we discuss it in detail. Although the no-boundary proposal fixes the initial data, the contour is not obviously fixed, leaving considerable ambiguity in the wave function. We find two explicit contour choices for the no-boundary proposal such that the resulting wave functions lead to interesting physical predictions. The path integral for the no-boundary wave function has complex saddle points, corresponding to complex solutions of the full Einstein equations with real boundary data. These solutions are genuinely complex and are in no sense equivalent to a combination of real Euclidean and real Lorentzian solutions. The path-integral version of the tunneling proposal is found to fix the integration contour, but leaves ambiguity in the initial data. We investigate the relation of the path-integral proposals to Vilenkin’s outgoing-flux tunneling proposal. The amplitude generated by the path-integral tunneling proposal can be made to satisfy the outgoing-flux tunneling proposal provided the initial data in the path integral are chosen suitably. However, neither version of the tunneling proposals fixes the initial data completely, and moreover, the two versions are found not to be equivalent in our models. We discuss the relationship of our approach to the ‘‘microsuperspace’’ approach of paper II of this series. We discuss Lorentzian proposals for the wave function of the universe. We discuss the possible difficulty in our anisotropic models of recovering quantum field theory in the semiclassical approximation, in that the complex saddle points have Re(√g )<0. In the case of the Kantowski-Sachs model, the path integral for the no-boundary wave function turns out to be closely related to that for the partition function of a black hole in a box, and this connection is briefly discussed.