Mathematics – Logic
Scientific paper
Dec 1989
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989phrvd..40.4011h&link_type=abstract
Physical Review D (Particles and Fields), Volume 40, Issue 12, 15 December 1989, pp.4011-4022
Mathematics
Logic
25
Origin And Formation Of The Universe, Particle-Theory And Field-Theory Models Of The Early Universe
Scientific paper
We consider the minisuperspace path integral for a closed homogeneous isotropic model in 2+1 dimensions using Einstein gravity with a cosmological constant and an antisymmetric tensor field matter source. We apply Hartle-Hawking boundary conditions. We therefore sum over three-geometries connecting a two-sphere of given radius to zero. The minisuperspace path integral reduces to a single ordinary integration over the lapse, and we study this integral by finding the saddle points-representing classical solutions-and the steepest-descent contours connecting them. We find that the integral has an infinite number of saddle points. Each represents a three-geometry interpolating between zero and the final two-sphere with an arbitrary number of wormhole-connected three-spheres in between. The value of the integral depends crucially on how the contour of integration is chosen, and the extent to which certain multiple-sphere configurations contribute depends on whether or not a chosen contour may be distorted into a steepest-descent path passing through the corresponding saddle points. Although steepest-descent contours passing through an infinite number of saddle points may be found, we argue, using standard steepest-descent arguments, that it is misleading to approximate the value of the path integral by summing exp(-I) for each of the saddle points. The reason is that the polynomial corrections about the dominant saddle point are much greater than any of the contributions from the subdominant saddle points. This means that although the path integral has the formal appearance of a sum over multiple-sphere configurations, the only significant contribution comes from the one or two configurations with least action. Possible implications for the sum over spheres in the Coleman mechanism are discussed.
Halliwell Jonathan J.
Myers Robert C.
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