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

Abstract

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.