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We show that the Wilsonian renormalization group (RG) provides a naturalregularisation of the Quantum Master Equation such that to first order the BRSTalgebra closes on local functionals spanned by the eigenoperators with constantcouplings. We then apply this to quantum gravity. Around the Gaussian fixedpoint, RG properties of the conformal factor of the metric allow theconstruction of a Hilbert space $\mathfrak{L}$ of renormalizable interactions,non-perturbative in $\hbar$, and involving arbitrarily high powers of thegravitational fluctuations. We show that diffeomorphism invariance is violatedfor interactions that lie inside $\mathfrak{L}$, in the sense that only atrivial quantum BRST cohomology exists for interactions at first order in thecouplings. However by taking a limit to the boundary of $\mathfrak{L}$, thecouplings can be constrained to recover Newton's constant, and standardrealisations of diffeomorphism invariance, whilst retaining renormalizability.The limits are sufficiently flexible to allow this also at higher orders. Thisleaves open a number of questions that should find their answer at secondorder. We develop much of the framework that will allow these calculations tobe performed.

Quantum gravity, renormalizability and diffeomorphism invariance