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EDIT: Better rewording by Chris White:

Is it possible to have a theory that treats both GR and QFT (e.g. QFT on a curved spacetime dynamically influenced by the standard QFT fields)? Is such a theory at least self-consistent (even if it does not apply to nature)? Or is there some fundamental incompatibility we run into without even trying to quantize GR (or perhaps we are somehow forced to quantize GR for consistency)?

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    $\begingroup$ To make this question answerable, you must be more specific about what alternative you want to consider. $\endgroup$
    – user1504
    Sep 26, 2013 at 2:20
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    $\begingroup$ What exactly would it mean to apply GR to QFT without attempting to treat it as a field theory? It's called quantum field theory for a reason. $\endgroup$
    – joshphysics
    Sep 26, 2013 at 2:20
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    $\begingroup$ Possible interpretation/clarification of the question: "Is it possible to have a theory that treats both GR and QFT (e.g. QFT on a curved spacetime dynamically influenced by the standard QFT fields)? Is such a theory at least self-consistent (even if it does not apply to nature)? Or is there some fundamental incompatibility we run into without even trying to quantize GR (or perhaps we are somehow forced to quantize GR for consistency)?" Is this a fair rewording? $\endgroup$
    – user10851
    Sep 26, 2013 at 3:01
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    $\begingroup$ In ordinary circumstances you can certainly treat gravity semi-classically with full quantum field theory for everything else. On the right side of Einstein's equation you have the expectation value of the stress energy tensor $\langle T_{\mu\nu}\rangle$ and the quantum fields evolve in the classical gravity background. In principle you can find a self-consistent solution for the classical metric and QFT, but which fails when you have a superposition of macroscopically different mass distributions. In practice this program is rather difficult and you mostly try ignore backreaction completely. $\endgroup$
    – Michael
    Sep 26, 2013 at 7:05
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    $\begingroup$ Michael Brown is right. Search for QFT in curved spacetime (en.wikipedia.org/wiki/Quantum_field_theory_in_curved_spacetime), and algebraic QFT (en.wikipedia.org/wiki/Local_quantum_field_theory) $\endgroup$
    – Cristi Stoica
    Sep 26, 2013 at 9:34

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This formalism exists, and is called semiclassical gravity. You can calculate effects using it, most notably, the existence of Hawking radiation. As an exact solution, it is somewhat unsatisfactory for these reasons:

  1. First, it is inherently not an exact solution to anything. You treat the quantum field as if the background metric was an exact solution to GR. This invariably causes the field to bend and behave differently. This, obviously, will affect the background metric, which requires that you recalculate the metric, which then requires that you recalculate the field, etc. It's unclear whether this series of successive approximations even will converge to a stable solution. In practical cases, people usually only consider a small number of orders of this backreaction.
  2. There are technical issues. In particular, QFT in a curved spacetime is only well-defined for a special class of background metric. In particular, QFT formalism is heavily dependent on "in" and "out" states. If we're going to rely upon intuition from QFT in Minkowski space, we need a region where the spacetime is flat to compute these states.
  3. Even if we were to relax the previous requirement, we still need a timelike killing vector at infinity in order to define positive and negative frequency states, which is necessary if we are going to normal order our fock space (in non-technical language, consistently define the energy of the vacuum and what states are "particles" and which are "antiparticles")
  4. Finally, there is just an inherent technical issue here, where we can pretty easily violate causality by combining QM effects with GR. For instance, imagine an experiment where a single (very massive) particle is sent through a slit, and after some time, will have a wavefunction whose location is two disjoint regions of space. Is the field a superposition of a mass distribution in both locations? What happens after the wavefucntion is collapsed? Does the metric change non-continuously? How do you resolve the singularity? You could imagine something similar using the EPR experiment, or the like. The main problem is that QM admits non-local effects (albiet non-causal ones), while GR does not. How do you resolve this while naïvely coupling them?

Anyway, if you're interested in the topic, Wald wrote an (extremely technical) book on it: http://press.uchicago.edu/ucp/books/book/chicago/Q/bo3684008.html

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    $\begingroup$ Sorry, points 2 and 3 are quite incorrect, if referring to the achievements of QFT in curved spacetime from, say, 1990 to 2014. QFT in curved spacetime can be defined in every globally hyperbolic spacetime. It is not necessary to require the existence of an (even asymptotic) timlike Killing vector to define physically meaningful states thank to the class of Hadamard states. What you described is, more or less, the state of the art of the end of eighties. $\endgroup$
    – Valter Moretti
    Jan 20, 2014 at 13:16
  • $\begingroup$ Nowadays there exists a complete generally locally covariant QFT in curved spacetime, including generalizations of PCT and spin statistics theorems, perturbative UV renormalization (it is done without fixing a reference state like Minkowski vacuum). $\endgroup$
    – Valter Moretti
    Jan 20, 2014 at 13:17
  • $\begingroup$ @V.Moretti: unless you've also solved the back-reaction problem, which was quite open and intractable when I was in grad school, that's not going to solve the problem of resolving the early big bang singularity. $\endgroup$
    – Zo the Relativist
    Jan 20, 2014 at 14:02
  • $\begingroup$ There are also some relevant progresses in that direction, see for example (in press in Commun. Math. Phys.) arXiv:1309.6303 and the previous Commun. Math. Phys. 305, 563–604 (2011) arXiv:1001.0864 [gr-qc] $\endgroup$
    – Valter Moretti
    Jan 20, 2014 at 14:18
  • $\begingroup$ However was not referring to the the back-reaction problem and the early big bang singularity that remain largely open problems, I was just referring on your statements (2) and (3). $\endgroup$
    – Valter Moretti
    Jan 20, 2014 at 14:20

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