I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always felt a bit stupid to ask this in real:
Many parts of physics look mainly analytic to me, i.e. electrodynamics and fluid dynamics look like an application of vector analysis and PDEs, quantum mechanics seems to rely heavily on functional analysis. I also understand that there is a lot of structure hidden in those theories: The best example would be symplectic geometry and classical mechanics or topological phases in quantum mechanics.
Despite, it seems to me that this is a more modern development in the physics community and the core of those theories as they would be taught to undergraduates would still be mainly analytic. (Sorry if I am wrong about this assumptions but it is my feeling).
However, when it comes to Quantum Field Theory, I feel that very much revolves (especially from the math-community side) around topological and algebraic questions. There is for example a visible math-community with analysis background working on mathematical quantum mechanics, but I never noticed this community in Quantum Field Theory.
Please consider these thoughts of mine, to justify my question: Is there no analysis present in QFT or why do mathematicians concentrate so heavily on those other aspects?
Best Answer
As Robert Israel indicates, if you really wanted to define a QFT, it would certainly be very analytic, say, to define the path integral. So analytic, in fact, that no one can do it for even semicomplicated theories.
However, one of the big math/physics developments in the last few decades is a class of QFTs where the observables are topological in nature. These are the topological QFTs or TQFTs. In these theories, you can ignore all or most of the (too) hard analysis and deal with much more well-defined spaces. Of course, from the mathematics point of view, these theories still involve a path integral that isn't defined (due to all that hard analysis we're ignoring), but enough structure can be found and has been developed to lead to all sorts of cool mathematics (see the mathematical definition of TQFTs, most recently axiomatized by Lurie building on lots of prior work). And even without this structure, physical intuition about these not mathematically well-defined theories has led to countless conjectures, theorems and the like, for example in mirror symmetry and various invariants like Donaldson and Seiberg-Witten.