I was just pondering whether this result would eventually be useful to physics or science generally in say ways like Galois' ideas and Group Theory eventually became useful. Just a thought.
Someone once told me that mathematicians are frequently doing "the science of 100 years from now." I hope we don't have to wait that long to find out if design theory has a place in the Theory of Everything.
https://physics.stackexchange.com/questions/1603/application...
(Emphasis mine)
You make it sound as if the Standard Model of particle physics were mathematically well-defined but it's anything but, see
https://en.wikipedia.org/wiki/Haag%27s_theorem
https://en.wikipedia.org/wiki/Wightman_axioms#Existence_of_t...
https://mathoverflow.net/questions/19495/mathematics-of-path...
---
Ok, so, when you say "mathematically well-defined," you do have a bit of a point there. But, it's a misleading phrase to say the least, because the actual problems that you need to worry about more or less amount to:
> "We have a real world over here, and a mathematical model over here. The model has had a lot of success predicting and explaining things, but we're not exactly sure if they match up exactly."
That is a very valid concern, if you're a physicist. However, what this is explicitly saying is that the concerns about the model are not mathematical per se. I would, of course, put a rather large asterisk on that statement, because the unitary gauge field theory of SU(3) × SU(2) × U(1), while it is relativistic in the sense of incorporating special relativity, it does not incorporate general relativity at all.
Again, if you're a physicist, that's pretty bad, at least in a principled, theoretical sense. But, going back to this notion that the SM has been a very practically successful theory, I would bet that one could have a nice career as a theoretical physicist without stepping too hard on those types of concerns. Not being a physicist, I may very well be wrong about that, but it seems like a truthy statement to me, at least.
This "relative" disparity (pun intended) becomes an issue at the point where gravitons are expected to appear. That carries its own extra special bag of problems, because GR is not normalizable, so you're going to get divergence to infinity in places you would really rather not have them. So, that's kinda bad, but, the good news is that gravity is the weakest force, which means that until the point where all four forces are unified, we're pretty okay from a physical PoV. That is, unless you happen to be very near the event horizon of a black hole, in which case you have weightier problems (pun again intended lol).
The overall expectation seems to be that a true quantum theory of gravity must almost by necessity solve all or most of these issues. So, again, not so much of a problem mathematically, not much of a big deal until you're getting up close and personal at distances where the other 3 forces really reign supreme.
All the issues you've mentioned are valid issues but they're about the physics. They are relevant and important but, as you say, the Standard Model and QFT have been very successful at describing a large number of physical phenomena at a certain scale, so the model is fine. Its applicability might be limited to certain scales or situations but that's not really different from any other theory in physics.
The problem with quantum field theory from a mathematical point of view, however, is that many of the mathematical objects that the symbols in the equations in physics text books represent (e.g. the "quantum fields", operators, inner products, integrals etc.) don't even exist in any (mathematically) meaningful way. Physicists pretend they do, manipulate them following very questionable rules (The limit N → ∞ doesn't exist? Doesn't matter, we'll apply it anyway and commute it with the integral!) and only the fact that, somehow, when the dust settles, those equations produce numbers that we actually see in experiments, saves the theories.
Take, e.g. the path integral for any of the field theories in the standard model: Physicists derive predictions from it. Yet it doesn't exist[0]: The integral doesn't exist. The measure doesn't exist. The sigma algebra doesn't exist. Etc. etc.
[0]: Or at least the question of its existence has eluded entire armies of mathematicians. So at the very least it is undefined.