We show that the Wilsonian renormalization group (RG) provides a
natural regularisation of the Quantum Master Equation such that to first
order the BRST algebra closes on local functionals spanned by the
eigenoperators with constant couplings. We then apply this to quantum
gravity. Around the Gaussian fixed point, RG properties of the conformal
factor of the metric allow the construction of a Hilbert space
\Ll of
renormalizable interactions, non-perturbative in
\hbarℏ,
and involving arbitrarily high powers of the gravitational fluctuations.
We show that diffeomorphism invariance is violated for interactions that
lie inside
\Ll,
in the sense that only a trivial quantum BRST cohomology exists for
interactions at first order in the couplings. However by taking a limit
to the boundary of
\Ll,
the couplings can be constrained to recover Newton’s constant, and
standard realisations of diffeomorphism invariance, whilst retaining
renormalizability. The limits are sufficiently flexible to allow this
also at higher orders. This leaves open a number of questions that
should find their answer at second order. We develop much of the
framework that will allow these calculations to be performed.
Fixed point structure of the conformal factor field in quantum gravity