The path integral measure, conformal factor problem and stability of the ground state of quantum gravity

Abstract

The functional measure for the Feynman path integral over four-geometries is constructed by the same generally covariant methods used by Polyakov for string theory. Ultralocality and general covariance determine the gaussian measure almost uniquely, and lead to a jacobian factor in the path integral which is just that required to render the linearized conformal perturbations (σ) of any Ricci-flat background into non-propagating, constrained modes. The non-trivial jacobian in the measure is equivalent to the non-local field redefinition χ = √−▽ 2 σ. It is the euclidean continuation of χ (rather than σ) which leads to a completely convergent euclidean path integral, consistent with unitarity. Thus, the conformal factor problem in one-loop euclidean quantum gravity is understood to be an artifact of the naive analytic continuation of the Einstein-Hilbert action. We reconsider the issue of the ground state of quantum gravity in light of this result, and show that flat space-time is absolutely stable to gaussian quantum fluctuations in the infrared. The extension of the method to non-Ricci-flat backgrounds is discussed as well.