Polarization mode symmetries of massless particles

[lomidrevo]

I am just reading Carroll's textbook on GR, where at the end of chapter 7.4 Gravitational Wave Solutions he discuss how rotational symmetries in polarization modes are related to spin of massless particles. He then explains that we could expect associated spin-2 particles to gravity - gravitons - followed from the two polarization modes $+$ and $\times$ that are invariant under $180^{\circ}$. In next paragraph he writes:

Imagine starting with the lagrangian for the symmetric tensor $h_{\mu\nu}$, but now imagining that this "really is" a physical field propagating in Minkowski spacetime rather than a perturbation to a dynamical metric... Now make the additional demand that $h_{\mu\nu}$ couple to its own energy-momentum tensor, as well as to the matter energy-momentum tensor.
...
we end up with fully nonlinear glory of general relativity.

I think I got a sense of this, but let me double-check by question:
So if we start with flat Minkowski spacetime and allow existence of a spin-2 field with such properties, we basically "reconstruct" classical general relativity, $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$?

If we demand that this field couples to its own energy-momentum tensor, does it mean that graviton can interact with other gravitons, similarly as we can say in GR "gravity gravitates"?

If the above is correct, is this reasoning one of the main building blocks of all quantum theories of gravity?

Notes:

• I understand that graviton is only hypothetical particle for the time being, until there exist complete theory of quantum gravity
• this was my first encounter with gravitons (and quantum theory of gravity) in a serious textbook (I am not counting all the popsci book I read before)
• I haven't yet studied QFT, so my further understanding will be surely limited

Thanks

 

[Paul Colby]

expect associated spin-2 particles to gravity - gravitons - followed from the two polarization modes + and × that are invariant under 180∘.

While this is true I believe $+$ and $\times$ polarizations are interchanged under a 45 degree rotation. For EM (spin 1) linear polarizations are interchanged with a 90 degree rotation.