I mean open up a book on QFT and I know things like path integrals still don't have a fully worked formalism but then you see Fourier transforms and residues from complex analysis being applied to integrals of "field operators". I don't mean the functional analysis of quantum mechanics, I mean mathematically speaking are field operators as studied under functional analysis? Or do they need another formalism. Maybe there is no formalism and it is just hoped there is some way that the math works right and that it makes sense.
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1There are some partial axiomatizations, I think. For conformal field theories specifically there are things like vertex operator algebras (https://en.wikipedia.org/wiki/Vertex_operator_algebra) and in general there is stuff like the Haag-Kastler axioms (https://ncatlab.org/nlab/show/Haag-Kastler+axioms). – Qiaochu Yuan Sep 26 '23 at 20:50
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2Ok. That is answer. Thanks for the answer. Don't know why everybody always comments instead of asnwering – Lina Jane Sep 26 '23 at 23:47