I know there are two ways to do quantum gravity.
One can pick a background space-time (usually Minkowski flat space-time) and then at any time slice one can define the state of the universe as the probability of gravitons being at positions: $x_1, x_2, x_3, x_4, x_5,...$
$$\psi(x_1,x_2,x_3,x_4,x_5....)$$
Or one could specify the geometry given by the 3D metric tensor g. Then the wave function is
$$\psi[g]$$
Then one either sums over all the Feynman diagrams for the graviton paths or alternatively one sums over all the different geometries.
What is the connection between the two ways of doing it? Is there a way to show that the same degrees of freedom are in both? How are the above wave functions related? Is one right and one wrong?
(As an extra, can one do a similar thing with the electromagnetic field. Is there a wave function $\psi(A)$ where A(x) is the electromagnetic field? Summing over fields instead of photons?)
What I don't get is that in QED there is no one electric field. Each electron has it's own electric field (minus it's own one). But in quantum gravity people talk as if there is one geometry. Surely each particle will feel a different geometry (minus its own self interaction?)
Edit:
I don't think they are compatible since the first one is calculated by:
$$ Amp(In,Out) = \int e^{ i S[g,\phi] } In(g,\phi)Out(g,\phi) D[g]$$
where for example: $$ In(g) = a + \int g(x)\psi(x) dx^4 + \int g(x)g(y)\psi(x,y)dx^4 dy^4 +...$$
Where the second one is calculated as something like:
$$ Amp(g_{in},g_{out}) = \int_{g_{in}}^{g^{out}} e^{i S[g,\phi] } D[g]D[\phi] $$
Which seems to only be an amplitude for a single graviton field. Whereas it is believed that gravitons are quantised.
However, maybe just because gravity is also to do with space-time these two ways are somehow equivalent? Or that gravitons don't exist?
(Let's just assume for a moment that Supergravity is finite. Just to aid the discussion).
Edit 2
I think in the first description it is easy to describe entangled gravitons (which surely must exist), while in the second geometric description it is not possible to do this. So it is not a description of more than one graviton. Unless I am mistaken and entangled gravitons cannot exist. (Or entangled closed strings which amount to the same thing). So summing over geometries cannot be correct as it can't describe entangled gravitons. Or is something wrong with this argument? On the other hand if we believe the brane description of the Universe a single brane can describe multiple particles, so maybe a single geometry represents multiple gravitons?
Edit 3
I think in loop quantum gravity instead of expanding $\psi[g]$ in terms of positions of gravitons they are expanding it in terms of loops or spin networks. So an individual spin network would not correspond to one particular geometry but rather the superposition of various numbers of entangled gravitons. So I think this is fine. And probably also why it is difficult to reconstruct GR from LQG.
On the other hand I think the Hawking-Hartle type of summing over geometries cannot be true or otherwise only true in approximation.