Probabilistic spacetime curvature as the union of general relativity and quantum mechanics? General relativity says that an electron exists at some position from where it curves the spacetime around it. Quantum mechanics says that an electron does not exist at any position, but is just a bunch of probability values across space. So to combine these ideas, we make spacetime curvature probabilistic? Like, the probability of existence of a particular spacetime curvature corresponding to a particular electron position is that same as the probability of the existence of the electron at that position. What's wrong with this method?
 A: There is a nice brief discussion of this kind of thing in Wald, General Relativity, sec. 14.1. A longer discussion is given in the book Feynman Lectures on Gravitation (not the same thing as the Feynman lectures). Wald writes with 20-20 hindsight about why such attempts failed. Feynman writes a first-hand account of how you would go about constructing such a theory, and what problems you run into -- he gave the lectures as a description of his own ongoing attempt to treat gravity this way, as just another quantum field theory.
There is a problem with causality. In normal QFT, the way you ensure causality is by making sure that field operators at spacelike-separated points commute. But if you try to do this with the metric as your field, you don't even have a way of defining what points are spacelike-separated.
More generally, the analogy between gravity and other field theories breaks down somewhat, because when you formulate other field theories classically, you make use of the metric. Without a metric, you can't define basic things like how to differentiate a vector field, or how to find the magnitude of a field vector. In the path integral formulation, you can't set up the integration without having a measure already defined on spacetime.
Even if you lower your sights a little and just try to do perturbation theory on a flat background, the theory is not renormalizable.
