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High Energy Physics - Theory, hep-th,Astrophysics, Cosmology and Extragalactic Astrophysics, astro-ph.CO,General Relativity and Quantum Cosmology, gr-qc
Abstract:
Complex metrics are a double-edged sword: they allow one to replace singular
spacetimes, such as those containing a big bang, with regular metrics, yet they
can also describe unphysical solutions in which quantum transitions may be more
probable than ordinary classical evolution. In the cosmological context, we
investigate a criterion proposed by Witten (based on works of Kontsevich &
Segal and of Louko & Sorkin) to decide whether a complex metric is allowable or
not. Because of the freedom to deform complex metrics using Cauchy's theorem,
deciding whether a metric is allowable in general requires solving a
complicated optimisation problem. We describe a method that allows one to
quickly determine the allowability of minisuperspace metrics. This enables us
to study the off-shell structure of minisuperspace path integrals, which we
investigate for various boundary conditions. Classical transitions always
reside on the boundary of the domain of allowable metrics, and care must be
taken in defining appropriate integration contours for the corresponding
gravitational path integral. Perhaps more surprisingly, we find that proposed
quantum (`tunnelling') transitions from a contracting to an expanding universe
violate the allowability criterion and may thus be unphysical. No-boundary
solutions, by contrast, are found to be allowable, and moreover we demonstrate
that with an initial momentum condition an integration contour over allowable
metrics may be explicitly described in arbitrary spacetime dimensions.
