GR and quantum mechanics



Hi, I am just learning GR (a gap in my physics education). I have a
few elementary questions:
1. Does the Feynman path integral formulation of quantum geometry lead
to the Schrodinger equation, or to a set of 4 equations of which the
Schrodinger equation is but the time component?
2. Where do the famous divergences occur, specifically? In the
computation of some Green's function through a perturbative approach
(which Green's function)? Or in trying to repeat some previous
perturbative computations in QED with a quantum spacetime (which
computation)? The former does not seem a compelling reason to say that
there is an incompatibility.
3. Can someone explain roughly, without technical details, how adding
more dimensions and making the basic objects strings (does that mean
real valued functions having their domain on a string in spacetime
instead of all of spacetime?) instead of fields, fixes either of these
problems, or whatever the problem is?

Thanks,
-Iuval

.



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