  The Wilsonian renormalization group (RG) requires Euclidean signature. The
conformal factor of the metric then has a wrong-sign kinetic term, which has a
profound effect on its RG properties. In particular around the Gaussian fixed
point, it supports a Hilbert space of renormalizable interactions involving
arbitrarily high powers of the gravitational fluctuations. These interactions
are characterised by being exponentially suppressed for large field amplitude,
perturbative in Newton's constant but non-perturbative in Planck's constant. By
taking a limit to the boundary of the Hilbert space, diffeomorphism invariance
is recovered whilst retaining renormalizability. Thus the so-called conformal
factor instability points the way to constructing a perturbatively
renormalizable theory of quantum gravity.
