Hoku said:
I'm looking for the reasons we believe(d) that gravity is incompatable with QM.
Let me correct this question as follows: "You looking for the reasons we believe(d) that gravity is incompatible with the well-known framework of QFT". It has become clear over the years that all known interactions (w/o gravity) can be described by (relativistic) quantum field theories. There are many hints that these QFTs are incomplete or even "not fully consistent", but nevertheless QFT is the only known framework able to describe nature (w/o gravity).
Now as this is clear one can ask why QFT does not work for gravity. There are a couple of technical reasons. The basic one is related to the above mentioned fact of incompleteness. As QFT has been understood over a couple of decades, perturbation theory is not only a calculational tool but in many cases a
definition of QFT. The starting point of QFTs (e.g. a path integral) is formal; in many cases there is no rigorous mathematical definition.
One problem is that this perturbative quantization fails for gravity. But it should be clear that this is not only a problem for gravity but for the framework of QFT framework.
Even if one is able to overcome the difficulty of the perturbative approach, there is the general problem that the definition of a QFT relies on classical, pre-existing spacetime structures. That means that spacetime serves as the stage for QFTs. As long as this holds, the
QFT approach forbids spacetime becoming quantized. But on the other hand if one looks at the Einstein equations (formalyy G = T) the left side (spacetime) staying classical and the right side (matter) being quantized seems to be inconsistent.
Then there are rather general arguments, that the Planck scale is a limit for the applicability of QM/QFT methods: from gravity we know that objects which are compressed to fit into their Schwarzschild horizons will form a black hole. As in QFT there is no intrinsic limit for quantum fluctuations one argues that
fluctuations will eventually collaps into micro-black holes as soon as the Compton wave length [energy] of a lump of energy becomes smaller [larger] than the Planck length [energy].
My conclusion is that gravity is not necessarily incompatible with the quantum nature, but with the well-know framework of QFT. So it could very well be that it's rather a lack of appropriate tools than a full incompatibility. Most approaches to quantum gravity do exactly this: they try to enlarge the scope of QFT, try to improve the tools, try to overcome the above mentioned technical difficulties. In parallel they find that spacetime as we know it from GR fails to exist at the quantum level; it is replaced by something more fundamental from which classical spacetime will emergy at lower energies as an effictive theory (unfortunately different theories do agree on this "something").
Examples:
1) string theory uses standard QFT methods, but they are not applied to spacetime but to the string world sheet
2) supergravity adds local supersymmetry in order to improve the behavior of the perturbation expansion; in addition the attempts to show finiteness of the perturbation expansion rely on on-shell methods (instead of standard off-shell Greens functions)
3) loop quantum gravity uses an approach known as canonical quantization, but it goes through a formalism which is manifestly inequivalent to the standard fock space construction
4) causal dynamical triangulation (and related approaches) uses a kind of discretization of spacetime which sounds familiar if one knows lattice gauge theories, but again differs in some fundamental aspects.