I assume you're talking about perturbative quantum gravity, i.e. trying to give a QFT treatment to the Einstein-Hilbert action by considering fluctuations around, say, a Minkowski background. Then, write out $g_{\mu\nu} = \eta_{\mu\nu} + \kappa\,h_{\mu\nu}$, where $\kappa \propto G_N^{1/2} \sim M_{\rm Pl}^{-1/2}$, for $G_N$ the Newton gravitational constant and $M_{\rm Pl}$ the Planck mass, and $h_{\mu\nu}$ is the metric fluctuation (the graviton). Then, your expansion of the Einstein-Hilbert action generates an infinity of interactions. There is a two graviton term, which is just the kinetic term, and then $n$-point interactions for $n = 3, 4, \ldots$, with a coupling proportional to $\kappa^{n-2} \sim M_{\rm Pl}^{-(n-2)/2}$. Because the coupling $\kappa$ in dimensionful, this generates power dependence (not just logarithmic) on energy in the amplitudes. For example, the two-to-two scattering amplitude scales like $\kappa^2$ and will thus have energy dependence $E/M_{\rm Pl}$ at tree level. Clearly this is irrelevant at particle energy scales, but the point is it grows rapidly with energy. Moreover, the one-loop correction to this term will scale like $(E/M_{\rm Pl})^2$.

So say you fix your scale and want to simply measure these renormalized coefficients using a presumably very sensitive device. Then, you would have to make a measurement for every $n$-point interaction in your expansion of the Einstein-Hilbert action. If you've never seen it written out, take a look at the quantum gravity chapter in Scadron's book "Advanced Quantum Mechanics" to see this process done. There are no diagrams in it, but you can also read the notes from 't Hooft's Erice lectures on perturbative quantum gravity for some of the issues implicit.

This post imported from StackExchange Physics at 2015-04-11 10:41 (UTC), posted by SE-user josh