Question

What's the mathematical interest in QFTs?

Answer 1

I’m a mathematical physicist working in QFT; I’m not going to give much meat in this answer since I’ll probably go on a tangent and because other answer here are pretty good.

QFT is a large, less rigorously mathematically established field compared to say QM or GR, with many approaches being prodded at (topological QFT, for example). We don’t know why certain heuristics employed by physicists in this space work as well as they do.

I’d like to add that finding QFT uninteresting or non-mathematical is probably because you simply don’t know enough about it to accurately make a judgement, it’s quite raw and abstract in its mathematical nature than many other areas of theoretical physics (again compare it to something like GR). And this is precisely why it is of interest to us mathematical physicists and mathematicians, even some theoretical physicists take interest in the attempted rigorous underpinning of QFT.

So I suppose the answer would be “Learn more math, learn more physics, to find out how much more there is to learn and why it is interesting”.

Answer 2

The math in GR is easier to appreciate at your level because it‘s more concretely physical. QFT by contrast is actually a lot more abstract; if you feel like it’s not mathematical enough then that’s just because you haven’t learned much about it yet.

My guess is that, at your level, the things you’ve been told about QFT are largely “lies we tell to students”; things that are sort of true, but they give you the illusion of understanding rather than the actuality of it. In a lot of cases this consists of simple recipes and formulas that seem to make sense on their own, but once you learn more you’ll realize that they’re just simplifications to make the subject approachable for someone who doesn’t know very much yet.

QFT is interesting from a mathematical perspective because it’s not actually a clear, mathematically consistent theory. In a lot of ways its foundations are a collection of heuristics and tricks that happen to correctly predict the outcomes of experiments. That means that there’s still work left to be done in terms of explaining how or why the math works (or why it doesn’t).

A big issue in QFT is dealing with the fact that the calculations in it, when done naively, result in infinities. Physicists have tricks for getting rid of the infinities, but it’s still not clear if those tricks are mathematically sound or not. The trouble with combining QFT and GR is that even the physicists haven’t come up with any good tricks for getting rid of the infinities in that case, mathematically sound or otherwise. There’s a lot of math involved in trying to figure out solutions to these problems.

Nonrelativistic quantum mechanics, by contrast, is a clear and consistent theory that can be derived from a collection of well-understood axioms.

Answer 3

I will add that a lot of work is also being done in topological QFT's since it may be easier actually to find an exact solution in a geometric sense.

Answer 4

QFT is interesting because it's less well-trodden territory, and there are many aspects of the theory that have room for interesting development.  Also, while some of the tools developed lack rigor, that's an invitation to shore up the foundations.  Lastly, the tools themselves have much wider applicability than QFT itself.

Answer 5

First of all, QFT is directly related to Algebra, Analysis, Combinatorics, Geometry, and even more. It's a huge subject at the intersection of a lot of beautiful math. I think that's cool.

Second of all, it works, but it lacks mathematical rigor. That's precisely what I'd like to figure out. In GR they basically can turn any physical question into a differential equation. This is fine, they don't need my help. But QFT is a mess, it's now mathematicians' job to figure these stuff out.

Answer 6

Something I've not seen talked about already but is interesting to me as a homotopy theorist is that TQFTs are a good way of studying the cobordism (∞-)category, which is a useful tool for studying the homotopy groups of spheres, for instance. The idea is that a TQFT is a functor (think "map of theories") from cobordism to vector spaces, so we can view a TQFT as a way of encoding homotopy data as vector spaces, which are much better for calculations:)

Answer 7

Phycisists working on QFT care about representations of certain Lie groups (e.g. SU(2n), SO(2n+1), E8, etc.) as they describe physical symmetries. It turns out that fully understanding the representation theory of these groups is a horrifyingly difficult task: in fact, as most of these fall under the class of so-called (quasi-)split reductive groups, they are objects of interest for people in the Langlands Programme, which is a very large research programme within pure mathematics.

There are also many objects in the theory of Lie algebras, such as Kac-Moody algebras, Yangians,the Heisenberg algebra, or vertex operator algebras, which originated from QFT. Mathematicians care about these things because they exhibit interesting combinatorial behaviours, and also because they too pertain to the Langlands Programme (although a bit more tangentially, via what's known as the Quantum Local Langlands Programme).

People working in PDEs obviously care about the many equations coming from physics.

Algebraic geometers also care a bit about QFT stuff, thanks to the emergence of topics like mirror symmetry and algebro-geometric objects such as Calabi-Yau manifolds.

Answer 8

Physics major, never went to grad school. My take on this is that physics and mathematics are two totally different approaches to explaining observations, but they get easily conflated since physics on paper contains a lot of math, and proof-like argumentative structure. However mathematics always starts from foundational postulates, whereas physics revolves around conjectures then extrapolates to look for a fit to observations. This is the instance of QFTs and why there are so many of them. To limit physics to the container of mathematics would be beside the point

Answer 9

I read this thread too long thinking of the Quantum Fourier Transform rather than Quantum Field Theory.

Answer 10

I'm not really into Quite Fungible Tokens, sorry