Is quantum mechanics compatible with general relativity?
The difficulty to reconcile quantum mechanics and general relativity counts as one of the big problems of fundamental physics. There appears to be a problem because canonical quantum gravity based on quantizing the Hilbert action is nonrenormalizable. (See the section on Renormalization in quantum gravity in this FAQ about how nevertheless to renormalize a nonrenormalizable field theory.)
The difference between renormalizable and unrenormalizable theories is that the former are specified by a (small) finite number of parameters while the latter are specified by an infinite number of parameters. In both cases, it is possible to extract approximate results from computations, and the parameters can be tuned to fit the experimental results. This gives a consistent procedure for predictions. Indeed, many nonrenormalizable theories are in use as effective field theories. (See hep-ph/0308266 for a recent survey on effective field theories.)
People who dislike nonrenormalizable theories do this on the basis of a claim that their predictive value is nil because of the infinitely many constants. But this is as unfounded as saying that thermodynamics is not predictive because it depends on a function (the expression for the free energy, say) that requires an infinite number of degrees of freedom for its complete specification. Clearly, in the latter case, the widespread use of finitely parameterized imperfect free energies does not hamper the usefulness of thermodynamics. The same can be said about nonrenormalizable field theories. It only implies that to extract arbitrarily precise predictions one needs correspondingly much information as input. We know that this is the case already for many simpler phenomena in physics. (For indications that canonical quantum gravity is nonperturbatively renormalizable see, e.g., hep-th/0110021, hep-th/0312114, hep-th/0304222.)
A different matter is the dream of a fundamental theory without any free parameters, which of course conflicts with a theory in which infinitely parameters are needed for its complete specification. But there is no theorem that says that nature is governed by unique principles. It is quite likely that the designer of the universe had some choices besides the constraints imposed by logical consistency. Thus I think this dream (which also fuels string theory) is misguided, and the correct quantum version of general relativity is standard, nonrenormalizable canonical quantum gravity.
This means that, quite likely, general relativity is fully compatible with quantum mechanics.
Of course this conflicts with the view of powerful groups within theoretical physics, who maintain that their approach to quantum gravity (either string theory, or loop quantum gravity) is the road to suuccess. But from what I have seen (at a somewhat superficial level of understanding) I trust neither string theory nor loop quantum gravity to be close to the truth. In any case, both are completely separated from experiemental verification.
If experiments in the near future can probe some features of quantum gravity, it will be for small quantum systems interacting with external electromagnetic and gravitational fields. See gr-qc/0408010.
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ