There is no "exact result" from QM or QFT that says this. It is inherent in the axioms (postulates) of the theory. In non-relativistic QM the mystery being solved was how to describe a "particle" moving subject to the apparent paradoxes of the day, e.g. particle-wave duality, uncertainty, etc. NR-QM provides a prescription whereby we assign operators to the classical degrees of freedom of a system, in this case the coordinates and momentum of a particle. The full machinery of QM then provides one with a probability amplitude function that allows you to calculate the probability to observe a particular "classical state" for the particle.
In field theory there are no particles, we start with a field configuration as the classical degree of freedom and apply the same postulates to the classical field. This is not a result but rather a correct application of the prescription to non-particle systems.
One place where this works out completely is quantum electrodynamics (QED). We have complete descriptions of the state of EM waves in terms of quantum operators and states.
In the development of relativistic QM one ran into problems with the prescription that, for a while, led to the idea of second quantization. Quantizing the wave function. I still hear people using this terminology but I think it is outdated and give the wrong impression of what is actually being done.
In GR the situation is more interesting as we do not have fields propagating on a flat space-time but the geometry of space-time itself being subjected to quantization.
In my opinion a full theory of QGR without inherent issues will not arise form applying QM to GR. Keep in mind that historically QM was something you did to a classical system to reveal its quantum nature. This has been extended to say QM is fundamental and we need to extract classical reality from the large N limit of QM results. Hence our very expectations regarding the topological nature of time and spatial dimensions is a classical observation and may need to be thrown out as a starting point, hopefully to emerge in the large N limit.