Dominant Topologies in Euclidean Quantum Gravity

Details Dominant Topologies in Euclidean Quantum Gravity Dominant Topologies in Euclidean Quantum Gravity

1997-10-27

arxiv.org/abs/gr-qc/9710114v2

Class.Quant.Grav.15:2629-2638,1998

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

14 pages, LaTeX. Minor additions (computability, relation to ``minimal volume'' in topology); error in eqn (3.5) corrected; re

Scientific paper

10.1088/0264-9381/15/9/010

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $\Lambda>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $\Lambda<0$, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ``density of topologies'' grows fast enough to overwhelm this suppression. The value $\Lambda=0$ is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wave function.

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