Quantum gravity, renormalizability and diffeomorphism invariance

Abstract

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 $\hslash$, 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.

arXiv comment64 pages

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Cited by in corpus (10)

• BRST in the Exact RG
• Provable properties of asymptotic safety in $f(R)$ approximation
• QFT with Stubs
• The continuum limit of quantum gravity at first order in perturbation theory
• The continuum limit of quantum gravity at second order in perturbation theory
• The continuum limit of the conformal sector at second order in perturbation theory
• RG and BV-formalism
• Gauge dependence of alternative flow equation for the functional renormalization group
• BRST, Ward identities, gauge dependence, and a functional renormalization group
• Basic properties of an alternative flow equation in gravity theories