# Quantum gravity and relevant/irrelevant operators

I am familiar with the casual dichotomy in QFT between coupling with positive dimensions in energy implies relevant operator on one side and negative dimension implies irrelevant operator on the other side (forgetting about marginal operators) once you have set $\hbar$ and c equal to 1, and then I thought than in a theory of quantum gravity you would have also set G equals to 1 (or $8\pi G = 1$), so it's no more possible to count in energy dimensions, as there are no more dimensions (scales). Does it only mean that our little trick is now over, or is there any conceptual novelty regarding the RG flow?

Thanks

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One doesn't have to set $G=1$ or $8\pi G=1$ in a quantum theory of gravity; it's just one possible choice that may be convenient.

If one wants to study relevant and irrelevant operators in general relativity and its extensions, it's useful not to set $G=1$ or $8\pi G=1$ because by doing so, we would make all quantities dimensionless.

Instead, it is a good idea to reformulate general relativity in the same way as other quantum field theories. Quantum field theories with a weakly coupled classical limit are usually described by Lagrangians $${\mathcal L} = {\mathcal L}_\text{free} + {\mathcal L}_\text{interactions}$$ The fields are normalized and redefined so that the kinetic terms (those with 2 derivatives in the case of bosons, 1 derivative in the case of fermions) have the usual normalization, schematically $(\partial_\mu \phi)^2/2$ for bosonic fields and $\bar\Psi \partial_\mu \gamma^\mu \Psi$ for fermionic fields.

This is possible for general relativity, too. Note that its Lagrangian is the integrand of the Einstein-Hilbert action $${\mathcal L}_{\rm GR} = \frac{1}{16\pi G} R$$ and it is proportionally to Ricci scalar that schematically contains terms $g \partial^2 g$, among others. We may expand the metric around a background, in the simplest case the flat background $$g_{\mu\nu} = \eta_{\mu\nu} + \sqrt{8\pi G} \cdot h_{\mu\nu}$$ The $\eta$ tensor is the flat Minkowski metric; $h$ is the perturbation away from it which carries the "operator character". I have conveniently added the coefficient in front of $h$ because when we insert it to the Einstein-Hilbert action, the leading term will generate $${\mathcal L} \sim (\partial h)^2$$ and the coefficients involving $G$ will cancel; let me be sloppy about the numerical coefficients of order one. However, the nonlinear Einstein-Hilbert action will also produce terms that are of higher order in $h_{\mu\nu}$ but each $h$ will appear together with a factor of $\sqrt{G}$, too.

So the cubic and higher interaction vertices in general relativity are weighted by $\sqrt{G}$ and its higher powers. Because $G$ has a positive dimension of length for $d\gt 2$, the interactions in general relativity are irrelevant (non-renormalizable). That's true even for $d=3$. However, gravity in $d=3$ has no local excitations, it is kind of vacuous, so the non-renormalizability problem may be, to some extent, circumvented.

At any rate, for $d=4$ and higher, even the leading interaction is irrelevant and leads to non-renormalizable divergences already at 2-loop level (and even in ${\mathcal N}=8$ supergravity, which offers as many supersymmetric cancellations as possible, there are new divergences requiring counterterms at the 7-loop level). So the theory breaks down at a cutoff scale that isn't too far from the Planck scale, $m_{\rm Pl} = G^{1/(d-2)}$.

That's where a consistent theory of quantum gravity i.e. string/M-theory has to cure the problems by providing the theory with new states (strings, branes, and – more universally – black hole microstates) and new constraints. Only if we try to study distances shorter than the Planck distance (a sort of a meaningless exercise for many reasons) or energy scales higher than the Planck scale (where the typical "particles" really look like ever larger black holes), we find out that the RG flows break down and become meaningless as a methodology. However, at distances much longer than the Planck length, the RG flows for GR are just fine and behave as they do in any non-renormalizable theory. The naively quantized Einstein's theory isn't predictive at the Planck scale but at much lower energies, one may systematically add new corrections, with a gradually increasing number of derivatives, to make an effective field theory ever more accurate.

Some people, most famously Steven Weinberg, have speculated that there could be a "zero-distance" ultraviolet limit that could run to general relativity at long distances. I think that the research attempting to find evidence for this conjecture remains inconclusive, to say the least. Most folks in quantum gravity are actually convinced that this can't work not only because of the apparent absence of the required scale-invariant field theory describe the UV; but also because such a picture of gravity would contradict holography and black hole thermodynamics.

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