I was thinking about quantum gravity and pre-geometry and wondering this question:
"If space does not exist. How does a particle know how far away it is from another particle?"
i.e. there are no rulers floating about in space measuring everything. And if you want to quantize gravity then you can no longer have a classical metric space as a background on which to measure distances.
Then I thought, imagine if every particle contained within it the distances to all other particles in the Universe. For example if every particle was an abstract sphere with all the other particles in the Univese projected onto it with the brightness being the inverse of the distance like a celestial map. (You could blur it a bit to give a smooth function on the sphere).
Then you could get rid of space all together and just have a collection of these celestial maps. It would even approximate curved space and other topologies. It sounds similar to M-Theory as that also has 2D objects as the fundamental particles but those exist in space where here there is not space.
It is kind of a bootstrap idea in that "the whole is encoded in the part". (Which I think I got from a Douglas Adams book).
If $f(\theta,\sigma)$ was the map on one particle then you could perhaps make it into a field theory by defining functionals $\Phi[f]$ and some action $S[\Phi]$.
Do you know if this idea has been thought of before? (I mean I have heard some wild idea that inside every atom is another Universe. But this is more like every particle contains an (imperfect) map of the Universe).
Also, I'm not sure how collisions would work in this theory when the distance between two particles became small their maps would have to converge into a single sphere. (All the spheres would exist only in abstract space as there would be no space). But this might not be a problem, it just means that as the particles collide and the spheres merge the maps become imperfect for a while.
Like string theory I would imagine that the extra degrees of freedom corresponding to the map would correspond to different types particles.
Instead of a celestial map, another way would be for the sphere to contain the sum of gravitational tidal forces from every other particle which might give a smoother function although with extra ambiguity. And you could decode the information using Fourier analysis of some kind.