Steepest-descent contours in the path-integral approach to quantum cosmology. I. The de Sitter minisuperspace model

We consider the issue of finding a convergent contour of integration in the path-integral representation of the wave function for a simple exactly soluble model, the de Sitter minisuperspace model. Following a suggestion of Hartle, we look for the steepest-descent contour in the space of complex four-metrics. We determine all the possible contours which give solutions to the Wheeler-DeWitt equation or Green’s functions of the Wheeler-DeWitt operator. We attempt to apply the boundary-condition proposal of Hartle and Hawking. We find that the proposal does not fix the solution uniquely because, although the initial point of the paths is fixed, the contour is not. We find a contour which represents the Vilenkin wave function and discuss the differences between the Hartle-Hawking and Vilenkin wave functions. We also discuss some of the implications of integrating over complex metrics. One consequence is that the signature of the metric is not respected, even at the semiclassical level.