Abstract
The non-perturbative renormalisation of quantum gravity is investigated allowing for the metric to be reparameterised along the RG flow, such that only the essential couplings constants are renormalised. This allows us to identify a universality class of quantum gravity which is guaranteed to be unitary, since the physical degrees of freedom are those of general relativity without matter and with a vanishing cosmological constant. Considering all diffeomorphism invariant operators with up to four derivatives, only Newton’s constant is essential at the Gaussian infrared fixed point associated to the linearised Einstein–Hilbert action. The other inessential couplings can then be fixed to the values they take at the Gaussian fixed point along the RG flow within this universality class. In the ultraviolet, the corresponding beta function for Newton’s constant vanishes at the interacting Reuter fixed point. The properties of the Reuter fixed point are stable between the Einstein–Hilbert approximation and the approximation including all diffeomorphism invariant four derivative terms in the flow equation. Our results suggest that Newton’s constant is the only relevant essential coupling at the Reuter fixed point. Therefore, we conjecture that quantum Einstein gravity, the ultraviolet completion of Einstein’s theory of general relativity in the asymptotic safety scenario, has no free parameters in the absence of matter and in particular predicts a vanishing cosmological constant.
Highlights
Wilson’s exact renormalisation group (RG) [1] provides a framework to construct a consistent quantum field theory (QFT) that describes gravity
Attempts to find a suitable fixed point have required fixed points for all gravitational couplings, included in a given approximation, instead of incorporating field reparameterisations into the RG equations and checking which of the couplings are inessential. We shall remedy this oversight by incorporating field reparameterisations in the gravitational RG equations which allow us to eliminate the inessential couplings from the flow equations
This conclusion could change by including higher-order terms, this seems unlikely since all higher-order terms are canonically irrelevant and, the quantum correction to their scaling dimensions would have to be large
Summary
Wilson’s exact renormalisation group (RG) [1] provides a framework to construct a consistent quantum field theory (QFT) that describes gravity. Our strategy will be to adapt the minimal essential scheme devised in [26], in the context of scalar field theories, to remove all inessential couplings in pure gravity This can be carried out order by order in the derivative expansion, where only terms with up to s-derivatives of the fields are included in the effective action. P p this implies that higher derivative terms coming from the operators det g R2 and det g Rμν Rμν are inessential in the universality class of the GFP This means that, contrary to the expectations based on perturbative renormalisability, the existence of the Reuter fixed point in the minimal essential scheme suggests that a possible UV-completion of the gravitational theory does not require additional degrees of freedom. The derivation of RG equations in the minimal essential scheme for QEG at fourth order in the derivative expansion for a generic dimension and a generic regulator cut-off are presented in Appendix A.1
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