Reformulating QFT to exclude space-time basis This is, I guess, a pretty loose or general question and so I hope it's ok to post in this forum. But I suppose this is what science is all about: asking and seeing what happens. It is just personal interest to see if there's already something that's been done on this or if people are thinking about, or interested in exploring it.
I was reading this article on how time measurement makes time more fuzzy local to the clock, and it got me thinking about the similarities between the space and time dimensions, and why we base all our theories on an assumption that space and time are a given. General relativity being a good example. Space and time are an almost unbreakable fabric through which everything moves, even if mass does bend it. In nearly all forms of physics $(x, y, z, t)$ forms a scaffold on which we perch all our theories, and all our data. What if they were a product of something more fundamental?
My question is: is it possible/useful/interesting/already done to reformulate QFT (and perhaps quantum gravity) in terms of something other than space-time and see if space and time fall out as a natural consequence.
Is it possible that they're a consequence of the interactions of particles, which are determined by other variables. What if, in removing our assumptions of space and time, we find something very similar, but not exactly the same.
What sort of assumptions would need to be made to come back to a system that would resemble QFT? Since that seems to have been a pretty good estimation for reality so far. (with one minor problem at least)
What if the presence of vacuum particles allows for "empty" space? What if as a particle becomes (yeah time, terminology is a bit difficult when talking about space-time) less certain, it defines more space to fill into? What if mass distorting space-time is a stress on a series of particles, each pulling or pushing on each other and therefore distorting their structure and behaviour, and therefore the apparent space-time effect that comes from these particles?
In essence, is it possible to turn our formulation around such that space-time is not fundamental?
I realise these are pretty crazy ideas next to established theory, but I'm just throwing it out there to see what people think. I think the idea is fascinating and maybe someone else does too. 
 A: You are not far off from one of the main research topics nowadays. It is known as quantum gravity (QG), i.e., a marriage of quantum theory's (QT) and General Relativity (GR). This question may already have related questions on this site, but yours merits an answer becuase of some of what you have thought about. 
More of an explanation below, but the end result is that we don't know what happens in the very very small size scale where (or time or very very high energies - numbers below, but far from anything we can currently see or experiment at) spacetime may not make any sense, and at those scales things may look more like an un-formed highly energetic bubbling of 'things', which Wheeler called quantum foam, where spacetime does not yet exist, and neither do any of the particles we know. The thought/hope was  that as the size scale increases that spacetime emerges out of it. There are actually a few theories (really proposed theories) as to what it may really looks like, the idea that it may look like that is that at those sizes and time intervals what might otherwise be spacetime is undergoing quantum fluctuations which is bubbling all over. The foam theory never really got totally formed, the modern version of that is called spin networks. See an article on quantum foam at http://abyss.uoregon.edu/~js/ast123/lectures/lec17.html
which includes a figure used often to depict what it might look like as you look at smaller and smaller scales.
The way spacetime looked like during the early Planckian time is also something that would exist in our universe now if we look small enough, i.e., at the Planck scale, smaller than any elementary particles.  
This epoch in the universe's evolution was thought to be what the universe first looked like, called the Planckian epoch. The scales were from the Big Bang of the universe to about $10^{-34}$ secs. 
See the Wikipedia history of the universe for something on the Planckian epoch at https://en.m.wikipedia.org/wiki/Chronology_of_the_universe
At those times we expect that a Quantum Gravity theory is needed to explain what is there.
The same is true if we try to look NOW at size scales of about $10^{-35}$ m. No atoms, particles, or spacetime, just whatever it is that Quantum Gravity can describe thinGs as. 
Quantum Gravity is described and discussed in the Wikipedia article https://en.m.wikipedia.org/wiki/Quantum_gravity. 
It is understood that in the very very small scale (time scale, size scale, or higher energy scale, roughly 15 orders of magnitude more than the energy we can accelerate particles to in the Large Hadron Collider; it turns out those scales are equivalent and called the Planck scale) QT and GR both apply so they need to work together. But when we try to make them work together the results are either inconsistent or lead to infinities that we can not get away from. This last issue is labeled as that quantization of GR leads to a non-renormalizable theory. 
There are now a number of theories of QG. The best known ones are String Theory (ST, and its cousins, Superstrings and M Theory) and Loop QG (LQG). 
ST says that what exists at those scales are quantum strings, and extensions of those to higher dimensions, like 2-membranes, 3-volumes, etc. ST requires 10 or 11 spacetime dimensions, all spatial except one is time. The idea is that all but 4 are wrapped up in sub-microscopic dimensions, and called Calabi-Yau manifolds. You can see a depiction of one in the QG Wikipedia article. The different vibrations of the strings composing the theory lead to the different possible particles in the universe, and ST dynamics decides what happens. One of those particles is the spin 2 graviton, and in fact at lower energy scales ST recovers General Relativity (i.e., that spacetime variations or curvature is gravity and its equations). The problem with ST is that it has too many arbitrary parameters, and it has not been possible to find experimental evidence for it. The most worked recent part of the theory is one where the Holographic principle may hold, and there have been results found where the String solution has been found based on that Holographic principle (also called the CFT/AdS correspondence if you want to search). ST is also a Theory of Everything. (TOE), it aims to unify/explain all the forces. ST is labeled as background dependent, it posits Lorentzian spacetime in 10-11 dimensions to start with. The Holographic principle makes it weakly background independent. 
The other theory, LQG, is background independent (i.e., it does not assume any spacetime background to start) but it does not aim to be a TOE, just of gravity in the quantum realm. You can Google it. Not as much current research on it as on the different versions of ST, though ST (except for the CFT/AdS versions) is on a lull because it depends on supersymmetry being true (not totally, but the easiest versions), and we keep on not finding a supersymmetric particle although the LHC energies should be in the range of the lightest ones. So things remain at play, without a clear idea of what a true QG theory may look like. 
The bottom line is that nobody knows what spacetime looks like, or is, at the Planckian scales and smaller, and until we do have a working and semi-verified theory spacetime as we know it may not exist at those scales. The Quantum Gravity work aims at finding out what is spacetime at those scales.  
A: Yes, the issue is that you can carry the question a little farther away, into the realm of maths. Any physics we can describe using math has, at least, the same "limitations" than math. And this means that any math theory can be described in terms of sets. So, there only one thing that needs to exist: the empty set. From the empty set you build everything, even the largest infinities. Or, if you limit the universe to the possible computable universes, you can say that there is only a countable string with adequate initial conditions  and running the cellular automata rule 110.
Then you can ask yourself why these specific initial conditions among the other equally justified (i.e., unjustified) ones? Which somehow leads to the idea of strong platonism (AKA the multiverse theory of Max Tegmark). That has many more theories (i.e., universes) than what string theory predicts.
A: To search something with 10 or more dimensions on very small scales may be a fascinating work but at one day there has to be a connection to space and time. Perhaps string theories could explain why space shrinks and expands and time goes slower or faster from the point of observation, far away from a local point of interest.
But time and 3D-space are so fundamental as nothing else could be. Everything is embedded in it, our being as well as every scientific observation. The claimed space-time-continuum is nothing else as the confession to time and 3D-space supplemented by the knowledge that space has a curved topology due to mass accumulations.


Quantum mechanics ... is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles.Wikipedia


I quote that because sometimes there are tries to adopt QM to macroscopic processes. The problem for unification of QM and GR is our knowledge about how electromagnetic radiation follows the geodesic path initiated by gravitational potential from mass accumulations.

But I suppose this is what science is all about: asking and seeing what happens.

Here is one of such a try: Is energy change in gravitational field an electromagnetic phenomenon?
A: Space and time are an inevitable property of the Universe, as Bob Bee said you are very close to one of the main Research Topics of Physics today the unification of Quantum Physics and General Relativity to form a Quantum Theory of Gravity, this theory will allow us to formulate - "The Theory of Everything", which will finally unify all the fundamental forces of Universe into one. We have formulated Quantum Theories for all the other forces - Electromagnetic , Weak Nuclear and Strong Nuclear, Gravity is the only force that is left out, it is so because of the inevitable property of the ultimate presence of space and time. I am going to explain this without any mathematical calculations or references. The other three fundamental forces did not require the presence of the space-time fabric to explain their existence and working, but Gravity itself is built in the threshold of this property. In order to formulate a Quantum Theory of Gravity we must either exclude the space-time property for gravity or include it into the other three fundamental forces.

General Relativity describes Gravity as a consequence of the warping of space and time fabric caused due to presence of mass or energy. Newtonian theory of gravity simply explained Gravity as a force that exists without explaining its reason of existence. What you are saying has been already done in a theory known as the Quantum Loop Gravity Theory, in which space and time are not quantities woven into a single fabric, but the fabric itself is made up of small chunks of space and time, and that there are such indivisible chunks present. Therefore, the indivisible chunks of space and time states that a body can occupy least possible space and that there can be a least simplest event that can take place, there have been other attempts all of which makes sense mathematically but cannot be verified experimentally. We will sooner or later be able to formulate a magnificent theory of Quantum Gravity which will help us unfold the marvellous mysteries of the Universe.

