What I am interested to see are specific examples/reasons why gravity should be quantized. Something more than "well, everything else is, so why not gravity too". For example, isn't it possible that a quantum field theory on curved space-time would be the way treat QFT and gravity in questions where the effects of neither can be ignored?
[Physics] What are the reasons to expect that gravity should be quantized
gravityquantization
Best Answer
Gravity has to be subject to quantum mechanics because everything else is quantum, too. The question seems to prohibit this answer but that can't change the fact that it is the only correct answer. This proposition is no vague speculation but a logically indisputable proof of the quantumness.
Consider a simple thought experiment. Install a detector of a decaying nucleus, connected to a Schrödinger cat. The cat is connected to a bomb that divides the Earth into two rocks when it explodes. The gravitational field of the two half-Earths differs from the gravitational field of the single planet we know and love.
The nucleus is evolving into a superposition of several states, inevitably doing the same thing with the cat and with the Earth, too. Consequently, the value of the gravitational field of the place previously occupied by the Earth will also be found in a superposition of several states corresponding to several values - because there is some probability amplitude for the Earth to have exploded and some probability amplitude for it to have survived.
If it were possible to "objectively" say whether the gravitational field is that of one Earth or two half-Earths, it would also be possible to "objectively" say whether the nucleus has decayed or not. More generally, one could make "objective" or classical statements about any quantum system, so the microscopic systems would have to follow the logic of classical physics, too. Clearly, they don't, so it must be impossible for the gravitational field to be "just classical".
This is just an explicit proof. However, one may present thousands of related inconsistencies that would follow from any attempt to combine quantum objects with the classical ones in a single theory. Such a combination is simply logically impossible - it is mathematically inconsistent.
In particular, it would be impossible for the "classical objects" in the hybrid theory to evolve according to expectation values of some quantum operators. If this were the case, the "collapse of the wave function" would become a physical process - because it changes the expectation values, and that would be reflected in the classical quantities describing the classical sector of the would-be world (e.g. if the gravitational field depended on expectation values of the energy density only).
Such a physicality of the collapse would lead to violations of locality, Lorentz invariance, and therefore causality as well. One could superluminally transmit the information about the collapse of a wave function, and so on. It is totally essential for the consistency of quantum mechanics - and its compatibility with relativity - to keep the "collapse" of a wave function as an unphysical process. That prohibits observable quantities to depend on expectation values of others. In particular, it prohibits classical dynamical observables mutually interacting with quantum observables.