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The semi-classical approximation $G_{ab}=\langle T_{ab}\rangle_{\omega}$ is the approximation to the theory of quantum gravity in which one treats matter fields as being quantum in a state $\omega$ and the gravitational field as being classical.

For concreteness, I am interested in the derivation of the following statement that I found in Wald's Quantum Field Theory in Curved Spacetimes and Black Hole Thermodynamics Ch. 4.6, p. 98:

Indeed, eq. (4.6.12) [$G_{ab}=\langle T_{ab}\rangle_{\omega}$] can be formally derived from a full theory of quantum gravity in the "$\frac{1}{N}$" approximation, wherein one assume the presence of $N$ scalar fields (each coupled to gravity with coupling proportional to $\frac{1}{N}$) and then takes the limit $N\rightarrow\infty.$

I am also interested to know if there are any known solutions for this system. Any additional information will be welcomed.

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    $\begingroup$ That seems an awfully broad question ... $\endgroup$ Apr 11, 2016 at 17:48
  • $\begingroup$ I agree. I asked the question because I think it may be useful to have a canonical answer who expands the information beyond the wikipedia article and make some more technical comments. $\endgroup$
    – yess
    Apr 11, 2016 at 18:09
  • $\begingroup$ I have tried to narrow the questions. $\endgroup$
    – yess
    Apr 12, 2016 at 17:49
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    $\begingroup$ I can't really tell what the question is. You want derivations of three separate statements, or you want to know whether or not such derivations exist? If the former, I'd say cut it down to one statement instead of three. $\endgroup$
    – DanielSank
    Apr 13, 2016 at 7:46

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