Schwinger-Dyson equations for Quantum Gravity

[mhill]

using the Schwinger Dyson equations that gives us a differential expression for the functional Z[J] so Z[0] is just the path integral over 4-dimensional spaces .then for Einstein equation (no matter) they read (system of 10 functional equations)

$$R _{a,b}( -i \frac{ \delta Z (J)}{\delta J})+ J(x)Z(J)=0$$

then let's suppose we had a super-powerfull computer so we could solve these S-D equations Numerically could it be a solution to the problem of QG ?? , in fact could someone say me what methods are used to solve these kind of equations with functional derivatives ?? .. if possible in the perturbative and Non-perturbative expansions, thanks

 

[lbrits]

Well the functional derivative expansion maps directly into an expansion onto Feynman diagrams. But the expression you have written will probably blow up due in an uncontrollable way due to non-renormalizability. Also, $$R_{ab}$$ is a function of position on that manifold, but position is only meaningful for a particular metric. If the metric itself is fluctuating, what exactly is the meaning of your equation then? I don't think this is the way to go about quantum gravity. What is Z[0]?