Interpretation of the Einstein-Hilbert action

Question

Everyone knows the famous Einstein-Hilbert action $S_{EH} = \int d^4x \sqrt{-g} R$. I'd like to know if, after we first explicit the Ricci scalar in terms of the metric, it could be possible to interpret each one of the terms as a kinetic or potential term, and even more read what could be the free propagator of the graviton, and the graviton-graviton vertexes. If yes, what are the form of those vertexes (how many? are some of them derivatives couplings? how many gravitons per vertex?)

Thanks.

This post imported from StackExchange Physics at 2014-03-22 17:22 (UCT), posted by SE-user toot

Comments

To clarify: are you asking what the Feynman rules for GR (based on the EH action, and known to be perturbatively unrenormalizable) would look like?

This post imported from StackExchange Physics at 2014-03-22 17:22 (UCT), posted by SE-user twistor59

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In some way: yes.

This post imported from StackExchange Physics at 2014-03-22 17:22 (UCT), posted by SE-user toot

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The second derivative terms in R are perfect derivatives, the first derivative terms are a form of pseudo-stress-energy and they have kinetic-potential interpretation. The perturbation theory terms are straightforward to work out but practically hopeless--- there are too many vertices that are too complicated. Here supergravity and string theory are essential--- several graviton scattering amplitudes were computed from string theory first. There is a lot of recent literature on this, look up Dixon's recent papers.

This post imported from StackExchange Physics at 2014-03-22 17:22 (UCT), posted by SE-user Ron Maimon

Answer 1

The only reference I seem to have to this material is a review article by Duff[*] which states some results of calculations performed using the quantities:

$$\tilde{g}^{\mu\nu}\equiv\sqrt{g}g^{\mu\nu}$$ $$\tilde{g}_{\mu\nu}\equiv\frac{1}{\sqrt{g}}g_{\mu\nu}$$

The graviton field $h^{\mu\nu}$ is defined by a perturbation about flat space $$\tilde{g}^{\mu\nu}=\eta^{\mu\nu}+\kappa h^{\mu\nu}$$ together with the corresponding quantity $$\tilde{g}_{\mu\nu}=\eta_{\mu\nu}-\kappa h_{\mu\nu}+\kappa^2h_{\mu\alpha}h_{\alpha\nu}+...$$ (here $\kappa=\sqrt{16\pi G}$). The free graviton momentum space propagator (after some gauge fixing choices) looks like $$D_{\mu\nu\rho\sigma}(p^2)=\frac{1}{p^2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\eta_{\mu\nu}\eta_{\rho\sigma})$$

There are expressions for 3-point, 4-point etc vertices which look rather complicated.

ETA: I found this online reference. The treatment discussing the propagator is around equation (65) onwards. I suspect that there will be much more detail in the original papers of Feynman and de Witt, but I don't have access to them.

[*] M.J.Duff "Covariant Quantization" in Quantum Gravity-an Oxford Symposium. ed Isham, Penrose, Sciama. OUP 1975

This post imported from StackExchange Physics at 2014-03-22 17:22 (UCT), posted by SE-user twistor59