Can you have quantum gravity without gravitons? I was thinking about whether quantum gravity needs gravitons. One can do a sum over histories of curved spaces perhaps without having to have gravitons.
But the term "quantization" implies that you are taking classical gravity and writing it in terms of quanta of the fields. Yet a sum over histories is part of quantum field theory and yet doesn't necessarily have to include quanta.
Yes, the electromagnetic field has a description in terms of fields or particles.
On the other hand, one can formally write down a 'fermion field' but this is merely a shorthand for the collection of fermion particles. (ie. there is no such thing a fermion field in the sense that it has a value for every point in space).
So is there anything wrong with treating gravity purely in terms of sum over histories of curved surfaces (as Hawking was inclinded to do), or is there some necessity to write it in terms of 'quanta' like gravitons as in string theory. After all, gravitons might not exist, or be unable to be detected by any means.
The only thing I can see that might be inconsistent is to decide how to treat feynman graphs of fermions entering a black hole and then the black hole evaporating. But Feynman tells us we must add up all paths including those travelling faster than light, so there would be no problem including fermion lines that entered a black hole and came out again.
So you would have feynman graphs that were overlayed on curved space which included black holes. And summed over them. That seems to cover most of what can happen in the Universe...
What am I missing?
 A: I think you are missing that the desire by most theoretical physicists to have quantization of gravity is not in order just to describe gravitational interactions, which after all Newtonian and General Relativity tools do well. It is the desire to have a Theory of Everything (TOE). In the same way that we say $SU(3)\times SU(2)\times U(1)$ are the quantum field theory of all three forces, to come to a mathematical expression that would include all four forces. That is the holy grail.
In order to include the first three, the theory must be quantized. For a TOE the $SU(3)\times SU(2)\times U(1)$ must be embedded because it is mainly a fit to the plethora of data. One extra reason is that gravitational waves are analogous to electromagnetic waves, and the graviton is assumed to be for gravitation the analogue of the photon in the TOE. That is why people are struggling with string theories, because in mathematical principle they can do this, except there are too many of them.
A: First of all, gravitons would still arise at the level of effective field theory even if they are not fundamental. In Standard Model pions still exist even if they are not fundamental objects, and we can still consider effective field theories with pions as mediators of nuclear force.
But otherwise, OP's idea reminds me of Sakharov's “induced gravity” hypothesis.
For a (technical) review and links to earlier literature see:

*

*Visser, M. (2002). Sakharov's induced gravity: a modern perspective. Modern Physics Letters A, 17(15n17), 977-991, doi:10.1142/S0217732302006886, arXiv:gr-qc/0204062.

In the text of the paper we find this overview:

To set the general framework:

*

*Assume you are given a Lorentzian manifold.

*Make no assumptions about the dynamics of this geometry; leave it free to flap in the breeze. Do not attempt to quantize geometry/gravity itself, but quantize everything else. (The geometry is considered as a classical background.)

*Consider one-loop quantum field theory on this manifold.

*Then at one loop the effective action is guaranteed to contain terms of the form:
$$∫d^4x\sqrt{−g}\{c_0+c_1\,R(g) +c_2\,(“R^2”)\}.\tag{1}$$
<…>
That is: the one loop effective action automatically contains terms proportional to the cosmological constant, the Einstein–Hilbert action, plus “curvature-squared” terms.


