Musings

Wading through all these papers is not an easy task. But you can get a pretty good idea of where things are headed by looking at the first one, Nonperturbative Evolution Equation For Quantum Gravity. It’s a very pretty paper, which generalizes the so-called Exact Renormalization Group of Polchinski to (pure) gravity.

There are a bunch of technical innovations required for the gravity case: one has to work in background field gauge, deal with gauge fixing, etc. But the basic idea of the Exact RGE can be understood by looking at Polchinski’s construction for a scalar field.

Define

We assume that the source has no short-wavelength Fourier modes, $$J(p) =0\, \text{unless} p^2\ll \Lambda^2$$ The UV cutoff function, $$K(s) =\left\{ \array{1 & s\lt 1\\ 0& s\gg 1}\right.$$ and is assumed to be smooth. $m^2$ is an IR regulator and is not the location of the pole in the inverse 2-point function (or whatever). The latter is controlled, in part, by the quadratic terms in $L(\phi,\Lambda)$, which we are not being explicit about. $L(\phi,\Lambda)$ is a completely general effective Lagrangian, which varies according to

Plugging into (1), one formally obtains $$\array{\arrayopts{\colalign{right left}} \Lambda \frac{d Z}{d\Lambda}=&\int d^4 p \Lambda \frac{\partial K}{\partial\Lambda}\cdot\int D\phi \frac{\partial}{\partial\phi(p)}\left[\left(\phi(p)K^{-1}\left(p^2/\Lambda^2\right) + \frac{(2\pi)^4}{2} (p^2+m^2)^{-1} \frac{\partial}{\partial\phi(-p)}\right)\exp S(\phi,J,\Lambda)\right]\\ =&0}$$ These formal manipulations are justified because we have imposed both a UV and an IR cutoff.

(2) is the Exact RG equation. It is a coupled set of differential equations for the infinite number of couplings in $L(\phi,\Lambda)$. Graphically, the first term on the RHS of (2) (for the $\beta$-function of the $n$-point coupling $g_n(p_1,p_2,\dots,p_{n-1},-p_1-p_2-\dots-p_{n-1};\Lambda)$) is given by a tree diagram, where we sew together $g_{k+1}$ and $g_{n-k+1}$ using the “propagator” $$\frac{1}{p^2+m^2}\Lambda\frac{\partial K\left(p^2/\Lambda^2\right)}{\partial\Lambda}$$ The second term corresponds to a 1-loop diagram, where we tie together two of the legs of $g_{n+2}$. The $\beta$-function for each coupling in $L(\phi,\Lambda)$, is exactly given by a quadratic expression in the other couplings.

I’ve somewhat belaboured the derivation of this “Exact” RGE to make two points very clear

1. This really is a coupled set of differential equations in an infinite number of variables. Except for the Gaussian fixed point, $$g_2(p,-p)\neq 0,\quad g_n=0\,\, \forall n\gt 0$$ the fixed-point sets (though they are finite dimensional) do not line up nicely, so that only a finite number of coupling in $L_*(\phi)$ are nonzero.
2. Despite the adjective “Exact”, (2) is very much a perturbative equation. The formal manipulations, by which we arrived at it, were justified to the extent that introducing the cutoff function, $K\left(p^2/\Lambda^2\right)$ really does provide a UV regulator. This is true in perturbation theory, where it provides a cutoff on the loop-momentum integrals for all Feynman diagrams¹. It is, most certainly, not true nonperturbatively².

What’s “exact” about (2) is not that we have somehow magically learned something about the nonperturbative behaviour of the theory by summing some tree and 1-loop Feynman diagrams. What’s exact is that, whereas in the usual formulations of the perturbative RGE the $\beta$-function receives contributions at all loop-orders, here the perturbative $\beta$-function is given exactly by 1-loop. The cost of this “simplification” is that we must consider, simultaneously, the $\beta$-functions for the full infinite set of couplings in the effective Lagrangian.

So, anyone reading the title of Reuter’s first paper should have been very nervous.

Einstein gravity in $d=2+\epsilon$ dimensions has a fixed-point at weak coupling. As Reuter reviews, this is successfully reproduced by the “exact” RGE. This is not surprising, as perturbation theory should be good for small $\epsilon$.

His subsequent papers go on to study a purported strong coupling fixed point in $d=4$ dimensions.

Technically, of course, no one can do anything with the full set of coupled equations, (2). So Reuter truncates to a finite number of couplings (cosmological constant + Einstein-Hilbert term), and studies the truncated set of equations. As I emphasized in my previous post, there’s no reason to expect that the infinite set of higher couplings are negligible in the UV. Reuter and collaborators perform various tests (throwing in an $R^2$ coupling, and seeing its effect) to attempt to justify neglecting these other couplings. I could discuss the details of that. I could re-emphasize that including matter will, almost certainly, destroy whatever fixed point you found in the pure gravity theory. I could …

But this post has gone on long enough already. Most fundamentally, there’s really no reason to trust the perturbatively “exact” RGE for strong coupling (where Reuter et al wish to use it). So I won’t detain you, Dear Reader, any further.