Quantum gravity is a challenge for field theories from "summary" of The Quantum Theory of Fields by Steven Weinberg,Weinberg S
The problem of quantum gravity is that it is not just a small modification of familiar quantum field theory. Gravity is not even a field theory, in the usual sense. The gravitational field is the spacetime metric itself, not a set of fields defined at each point in spacetime. This means that the metric is subject to quantum fluctuations, which are not well understood. One might try to treat the metric as a field on a fixed flat background spacetime, but this approach is inherently inconsistent because the metric determines the geometry of spacetime itself. One might also try to quantize gravity by treating the metric as an operator on a Hilbert space of wave functions, but this leads to severe mathematical and conceptual problems. The problem is not just technical. The very nature of spacetime is called into question by quantum gravity. In the framework of general relativity, spacetime is a smooth manifold that can be curved by matter and energy. In quantum theory, however, spacetime is expected to be fundamentally uncertain and fluctuating. These two pictures of spacetime are difficult to reconcile, and resolving this conflict is at t...
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