Abstract
Kontsevich and Segal (K-S) have proposed a criterion to determine which complex metrics should be allowed, based on the requirement that quantum field theories may consistently be defined on these metrics, and Witten has recently suggested that their proposal should also apply to gravity. We explore this criterion in the context of gravitational path integrals, in simple minisuperspace models, specifically considering de Sitter (dS), no-boundary and anti--de Sitter (AdS) examples. These simple examples allow us to gain some understanding of the off-shell structure of gravitational path integrals. In all cases, we find that the saddle points of the integral lie right at the edge of the allowable domain of metrics, even when the saddle points are complex or Euclidean. Moreover the Lefschetz thimbles, in particular the steepest descent contours for the lapse integral, are cut off as they intrude into the domain of nonallowable metrics. In the AdS case, the implied restriction on the integration contour is found to have a simple physical interpretation. In the dS case, the lapse integral is forced to become asymptotically Euclidean. We also point out that the K-S criterion provides a reason, in the context of the no-boundary proposal, for why scalar fields would start their evolution at local extrema of their potential.
Highlights
Despite the fact that we live in a Lorentzian universe, Euclidean and complex metrics are often used in theoretical physics
A prominent example is provided by black hole metrics in imaginary time, which offer the quickest way of deriving the thermodynamic properties of black holes [1]
Supporting evidence came from the fact that for such metrics they found that the Gauss-Bonnet integral
Summary
Despite the fact that we live in a Lorentzian universe, Euclidean and complex metrics are often used in theoretical physics. Louko and Sorkin [2] looked at this question for topology changing transitions in two dimensions, and found that only certain kinds of complex deformations make sense The condition that they employed was to require a scalar field theory to be well defined on the complexified background in question, i.e., that the path integral for a (real) scalar field on a given complex manifold should be convergent. The setting we choose consists of the simplest minisuperspace models of quantum gravity This is because these models offer rather good analytic control, yet they are examples where off-shell configurations play an important role, as one needs to understand the off-shell structure in order to define gravitational path integrals. Our results certainly provide support for the idea that the Kontsevich-Segal approach has relevance when extended to quantum gravity
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