Dominant Topologies in Euclidean Quantum Gravity

Abstract

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.

14 pages, LaTeX. Minor additions (computability, relation to ``minimal volume'' in topology); error in eqn (3.5) corrected; references added. To appear in Class. Quant. Grav

References in corpus (2)

• On the role of power expansions in quantum field theory
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