Imagine that at some small distance, space itself is quantum, so quantized. I.e. at the bottom, space is a discrete graph with incredibly small edges. Possibly very tangled and not even layed out in obvious dimensions over short distances.
We don’t know if that is the case. That is only one possibility.
But it seems very clear that whatever happens at the Planck distance or lower isn’t simple smooth space as we model it for larger scales.
Unfortunately for this idea, special relativity tells us that Planck length is an observer-dependent phenomenon. That is two objects that are at a distance that is more than the Planck length apart for one observer will be closer than the Planck length for another observer and vice versa. So Planck length can't be a fundamental property of space-time, unless special relativity breaks down at some point even for non-accelerated observers.
I would suspect since general relativity would break down that special relativity certainly would.
The “relativity” aspect will almost certainly still apply in some way, and still form an emergent basis for the special relativity effects you point out.
All of general relativity has to be emergent from the to-be-discovered laws of the underlying small scale structure of space.
(I.e. general relativity isn’t wrong, it is just not complete. Similar relationship as with Newton’s Law of Gravity, which was also correct, but breaks down beyond the conditions it covers well, because it was not complete.)
The smallest scale is also where we expect more “light” shed on the initial conditions of the universe and potentially the insides of black holes. Two other conditions where general relativity already breaks down.
If special relativity breaks down, so does QFT - since QFT is based on special relativity just as much as on QM.
“Breaks down” just means our equations don’t know what to say about a situation.
It doesn’t that any particular phenomena we understand today will break down. Just we will be able to see richer behavior, that has been there all along, than our models cover today. And so be able to understand more conditions than we do today. And perhaps new capabilities to engineer things than we have today.
Obviously quantum field theory, as it exists today, breaks down at the Planck distance too, since it can’t tell us what is happening at smaller distances either.
This isn’t the least bit controversial or surprising. It just means that even with both theories, we can’t explain everything yet.
When we do have an accurate theory of the fine structure of space and time, it will also be a theory of the fine structure of fields. Since space-time is integral to both theories.
It may even be the long sought after unification.
But for distances much larger than the Plank distance, the fine structure theory will still simplify into the GR and QFT we have today.
Just as GR under many conditions we encounter every day, further simplifies to Newton’s Law of Gravity.
I don't think this is the right idea. A new theory can invalidate core assumptions of a successful old theory, revealing it to be a coincidence that it happened to work mathematically in some regimes, it's not always a case that the old theory is simply an approximation of the new one.
For examples of the "good" kind, Newton's laws of motion are indeed just an approximation of special relativity. The Schrodinger equation is just an approximation of QFT.
In contrast, GR came in and showed that Newton's law of universal attraction is completely wrong, at the fundamental level. Sure, it predicts certain phenomena correctly, but so did the epicycles that it replaced. Similarly, QM/QFT showed that Newton's laws of motion are also completely wrong, that objects (or at least particles) don't even move according to some laws of motion, they are only described by a probability wave that moves according to some laws, and between two interactions they have no definite state and are not even localized.
And of course, GR and QFT disagree on these parts - GR generally agrees with Newton's laws of motion (with the SR corrections), and QFT generally agrees with Newtoninan gravity. But you can't use GR's gravity with QFT's laws of motion, so we know one or both are broken. A new theory is very likely to also "overrule" one or both of them and show that they are not just an approximation, but completely wrong, working only "by accident" on the scenarios where they have been tested. Especially if the new theory requires there to exist fixed space distances that all relativistic observers agree on.