Eric Hannah and Kenneth Eppley in 1977 presented a thought experiment that illuminated nicely why coupling a quantized to an unquantized field inevitably spells trouble, published in their article "The necessity of quantizing the gravitational field." The experiment is deceptively simple. You prepare a quantum particle in a state with a well-known momentum (in some direction). It doesn't necessarily have to be a momentum eigenstate, but something with a small momentum uncertainty. From Heisenberg's uncertainty principle, we know then that its position uncertainty will be large. Now you measure the position of the particle with a classical gravitational wave.
If gravity wasn't quantized, gravitational waves wouldn't have to fulfill the relation p = ℏk, which was famously shown to hold for photons by Einstein, using the photoelectric effect. It would then be possible to prepare a gravitational wave with a small wavelength (high frequency) but small momentum. If you use this gravitational wave to measure the position of the quantum particle, there are, so argue Hannah and Eppley, three different possible outcomes:
- You collapse the wavefunction of the quantum particle and measure its position to a precision determined by the short wavelength of the gravitational wave yet without transferring a large momentum. It is then possible to violate Heisenberg's uncertainty principle, thus the quantum part of the theory doesn't survive.
- You collapse the wavefunction of the quantum particle without violating Heisenberg's uncertainty principle, then you will violate energy conservation because your wave can't provide the necessary spread in momentum.
- You don't collapse the wavefunction, in which case you can use your measurement for superluminal communication. You then had two types of measurements, one that does and one that doesn't collapse the wavefunction. By spatially separating an entangled state and monitoring one part of it without collapsing it, you can find out, instantaneously, when a collapse was induced in the other part.
Since gravity is an extremely weak interaction, this experiment is far beyond experimental possibility; the detector's mass for example would have to exceed that of our galaxy. Hannah and Eppley claimed that their experiment would at least in principle be possible to construct with the matter content of our universe. It was however later shown by James Mattingly, in his paper Why Eppley and Hannah's Experiment Isn't (the title evidently did not make it through peer review), that Hannah and Eppley underestimated the experimental challenges. Mattingly crunched the numbers and showed that the cosmic background radiation spoils the sensitivity of the detectors and, worse, that the detector would have to be so massive it would sit inside a black hole.
Thus, Hannah and Eppley's experiment isn't even in principle possible. While their reasoning is physically plausible, this puts one into a philosophically difficult spot. There clearly is a theoretical problem with coupling a classical to a quantum field, but if we can show there are no practical consequences in our universe, is it a problem we should worry about?
I like Hannah and Eppley's thought experiment. It is not the best motivation one can have for quantizing gravity, but it is a lean way to illuminate the problem.