The Einstein Field Equations emerge when applying the principle of least action to the Einstein-Hilbert action, and from what I understand the path integral formulation generalizes the principle of least action. What happens when you apply the path integral instead of the action principle to the Einstein-Hilbert action?
[Physics] What happens when you apply the path integral to the Einstein-Hilbert action
general-relativitypath-integralquantum-field-theoryquantum-gravityrenormalization
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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
In some sense, yes it is, and in others it is certainly not.
The fact that in the Einstein field equations: $G_{\mu \nu} = 8 \pi T_{\mu \nu}$, you have complete 'freedom' to define $T_{\mu \nu}$ however you would like (within some kind of exceptions), means you can allow for highly non-trivial curvature dependent terms to be coupled to the stress-energy. See this really interesting paper by Capozziello et al (http://arxiv.org/abs/grqc/0703067) that shows how you can 'bunch' the additional curvature terms for an $f(R)$ gravity into the 'effective' stress-energy tensor. In this way, the uniqueness of what you may call the 'Einstein' field equations does not hold.
However, once one moves to the vacuum the answer is suddently a definitive yes. The Einstein equations are generated by having an action that contains a geometric invariant quantity. If it did not then the principles of general covariance would no longer hold since arbitrary coordinate transformations would necessarily destroy the gauge symmetries. Now, that being said, one can consider the most general possible geometric invariant Lagrangian in this sense, it would look something like.. $\mathcal{L} = f(g_{\mu \nu} R^{\mu \nu}, R^{\mu \nu} R_{\mu \nu}, R^{\alpha \beta \gamma \delta} R_{\alpha \beta \gamma \delta}, C^{\alpha \beta \gamma \delta} C_{\alpha \beta \gamma \delta}, g^{\mu \nu} R^{\alpha \beta} R_{\alpha \beta \mu \nu}...)$ and the list of possibilities goes on. Since you want to replicate the Einstein equations in themselves you can immediatly throw away all but the linear terms in the above. Any of the other variants when you perform the calculus of variations gives you higher than second order derivatives.
Now, keeping only the linear terms you see that the action will necessarily be the Einstein-Hilbert one plus (maybe) some other curvature invariants. Since these will not vanish once you perform the variation, unless they either vanish identically or vanish on the boundary as a divergence term, your field equations will necessarily still be the Einstein-Hilbert ones provided you only keep $\mathcal{L} = R^{\sigma}_{\sigma}$.
A rigorous proof of this would be interesting. I am unaware of any in the literature.
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Your questions essentially amounts to ask
which is the starting point of quantum gravity (QG). GR is a non-renormalizable theory, at least from the traditional perspective of perturbation theory in QFT. So the path integral with the (exponentiated) Einstein-Hilbert action as weight factor cannot easily be used to make meaningful physical predictions. New approaches to QG are needed, such as e.g. string theory (ST).