Conformal rotation in perturbative gravity

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Abstract
The classical Euclidean action for general relativity is unbounded below; therefore Euclidean functional integrals weighted by this action are manifestly divergent. However, as a consequence of the positive-energy theorem, physical amplitudes for asymptotically flat spacetimes can indeed be expressed as manifestly convergent Euclidean functional integrals formed in terms of the physical degrees of freedom. From these integrals, we derive expressions for these same physical quantities as Euclidean integrals over the full set of variables for gravity computed as metric perturbations off a flat background. These parametrized Euclidean functional integrals are weighted by manifestly positive actions with rotated conformal factors. They are similar in form to Euclidean functional integrals obtained by the Gibbons-Hawking-Perry prescription of contour rotation.

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