Paul Colby said:
The graviton (at least initially) is an idea that stems from applying local quantum field theory techniques to linearized GR. The resulting attempt only provides answers to first order questions.
This was thought to be the case when Feynman's book was published, but it turns out it's not. See below.
Paul Colby said:
Feynman wrote a book on this subject you might find interesting.
If you're referring to the Feynman Lectures on Gravitation, while it's a good book, a lot of the material in it is out of date now, since it is based on lectures he gave at Caltech just at the start of a period of development of the quantum field theory of a massless spin-2 field. When that development finished in the early 1970s (and Feynman was an important figure in that), we knew some things that Feynman didn't know when those lectures were done and published:
There
is a quantum field theory of a massless spin-2 field, which can be formulated with as much mathematical rigor as any other quantum field theory. (How much rigor that actually is depends on which theorist you talk to.) This massless spin-2 field (or its particle aspect, if you like) is called the "graviton".
That QFT can be developed on similar lines to QED, which is the QFT of a massless, spin-1 field, with appropriate differences for spin-2 (for example, "like charges" attract in the spin-2 theory, which is what we want since "like charges" for gravity means mass, and we want gravity between masses to be attractive).
The massless, spin-2 field QFT is
not limited to linear order: it is a full, nonlinear QFT, correct to all orders. Figuring out how to show this was the main thing that took until the early 1970s.
The classical limit of the massless spin-2 QFT is classical General Relativity, i.e., the classical field equation is the Einstein Field Equation. (This is similar to the way the field equation of the classical limit of QED is Maxwell's Equations.) So classical GR can be viewed as a classical approximation to this QFT.
However, the spin-2 QFT, unlike the spin-1 QFT, is not renormalizable. That means the spin-2 QFT cannot be a final theory of quantum gravity. It can only be an "effective theory", which leaves out physics at high enough energy scales. In the early 1970s, that was thought to be a big difference as compared to QED; but since then, with the final development of the Standard Model (and along with it the understanding that QED itself is only one piece of the puzzle), physicists have come to believe that
all of our current QFTs are effective theories.
Given all that, the big open question regarding the spin-2 QFT is whether there is actually any physical regime in which it's relevant--i.e., in which classical GR isn't enough to make good predictions, but we're still in a regime where whatever new physics lies underneath the spin-2 QFT hasn't come into play yet. For QED and the Standard Model, we know there
is such a regime--we know there are plenty of physical situations where the QFT is necessary to make accurate predictions, but it's also sufficient--we don't need to make use of any new physics beyond the Standard Model. We don't know if that's the case for the graviton: it might be that, by the time you reach a regime where classical GR no longer works, the massless spin-2 QFT no longer works either--you need whatever new physics lies beyond it.