It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?
E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states.
I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest.
There are zero contradictions between quantum mechanics and special relativity; quantum field theory is the framework that unifies them.
General relativity also works perfectly well as a low-energy effective quantum field theory. For questions like the low-energy scattering of photons and gravitons, for instance, the Standard Model coupled to general relativity is a perfectly good theory. It only breaks down when you ask questions involving invariants of order the Planck scale, where it fails to be predictive; this is the problem of "nonrenormalizability."
Nonrenormalizability itself is no big deal; the Fermi theory of weak interactions was nonrenormalizable, but now we know how to complete it into a quantum theory involving W and Z bosons that is consistent at higher energies. So nonrenormalizability doesn't necessarily point to a contradiction in the theory; it merely means the theory is incomplete.
Gravity is more subtle, though: the real problem is not so much nonrenormalizability as high-energy behavior inconsistent with local quantum field theory. In quantum mechanics, if you want to probe physics at short distances, you can scatter particles at high energies. (You can think of this as being due to Heisenberg's uncertainty principle, if you like, or just about properties of Fourier transforms where making localized wave packets requires the use of high frequencies.) By doing ever-higher-energy scattering experiments, you learn about physics at ever-shorter-length scales. (This is why we build the LHC to study physics at the attometer length scale.)
With gravity, this high-energy/short-distance correspondence breaks down. If you could collide two particles with center-of-mass energy much larger than the Planck scale, then when they collide their wave packets would contain more than the Planck energy localized in a Planck-length-sized region. This creates a black hole. If you scatter them at even higher energy, you would make an even bigger black hole, because the Schwarzschild radius grows with mass. So the harder you try to study shorter distances, the worse off you are: you make black holes that are bigger and bigger and swallow up ever-larger distances. No matter what completes general relativity to solve the renormalizability problem, the physics of large black holes will be dominated by the Einstein action, so we can make this statement even without knowing the full details of quantum gravity.
This tells us that quantum gravity, at very high energies, is not a quantum field theory in the traditional sense. It's a stranger theory, which probably involves a subtle sort of nonlocality that is relevant for situations like black hole horizons.
None of this is really a contradiction between general relativity and quantum mechanics. For instance, string theory is a quantum mechanical theory that includes general relativity as a low-energy limit. What it does mean is that quantum field theory, the framework we use to understand all non-gravitational forces, is not sufficient for understanding gravity. Black holes lead to subtle issues that are still not fully understood.
Matt Reece gives a good answer, but one additional area of tension that seems worth mentioning is the problem of time. The role of time in quantum theory is quite different from general relativity.
For a review of some of the issues involved, see
Canonical Quantum Gravity and the Problem of Time. C. J. Isham. "Recent Problems in Mathematical Physics", NATO Advanced Study Institute, Salamanca, June 15-27, 1992. arXiv:gr-qc/9210011.
A superposition of causal structures. More precisely, given two events A and B, they could be in a superposition of being spacelike, null and timelike separated. Quantum field theory is built upon a sharp distinction between localized operators which are spacelike separated from those which aren't. With a superposition of causal structures, such distinctions break down.
I myself overlooked it too but wikipedia actually happens to have a great such list at https://en.wikipedia.org/wiki/Quantum_gravity#Points_of_tension
There are other points of tension between quantum mechanics and general relativity.
First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in the neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place).
Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. The resolution of these points may come from a better understanding of general relativity.
Third, there is the Problem of Time in quantum gravity. Time has a different meaning in quantum mechanics and general relativity and hence there are subtle issues to resolve when trying to formulate a theory which combines the two.
I'm not sure this should be an answer - it is really an anti-answer.
In Quantum Relativity, David Finkelstein has a list of analogies between QM and relativity, detailing "an extended parallel between the structures and developments of relativity and quantum theory". (Section 1.4.2)
Yes, he does have GR in mind when he speaks of relativity.
While he points out some deep similarities, the rest of the book explores in depth, in ways that would thrill only a theorist, the underlying nature of each, analogies and differences.
Anyone interested in the relation of QM and GR would benefit from a browse through this book, though it is hardly the only one that should be read.
All of the other answers regard how to quantise the background spacetime. Linear perturbations were quantised by Matvei https://doi.org/10.1007%2Fs10714-011-1285-4.
However, before asking about how we quantise spacetime, we can also ask what quantum field theory looks like in a classical background spacetime, which does not behave quantum mechanically. This semiclassical approach is justified by the use of semiclassical methods in electrodynamics.
The problem with Quantum Field Theory in Curved Spacetime, is that the vacuum state is not unique, which means that different (inertial) observers can see a different particle spectrum. This is a naive interpretation though, as it is physically unreasonable that the motion of an observer should determine the vacuum state/Fock space/physical particle spectrum of reality.
This kind of problem is also related to the Unruh effect.
The typical text for these kinds of issues is the book by Birrell & Davies. But much progress has been made regarding the interpretation, and consulting the literature is also necessary.
Main reason of incompatibility between QM and GR is quantum foam.
Quantum foam are fluctuations on scales below planck length that are so strong that space and time lose its ordinary sence. There the uncertainty principle which causes quantum fluctuations is in straight conflict with smooth spacetime geometry that general relativity requires.
This problem can be solved as strings in string theory can weaken those fluctuations by spreading them.
