SciPost Phys. 5, 040 (2018) ·
published 30 October 2018
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· pdf
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 $\mathfrak{L}$ of renormalizable interactions,
non-perturbative in $\hbar$, and involving arbitrarily high powers of the
gravitational fluctuations. We show that diffeomorphism invariance is violated
for interactions that lie inside $\mathfrak{L}$, 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 $\mathfrak{L}$, 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.
