>>11322775That's not the point; you want your fields to form reps of , regardless if Streater-Wightman axioms are able to reproduce the full power of QFT. No one said that those are complete, but they (as well as the additional completeness assumption on the Hilbert space) for sure form a lower bound.
>extremely basic scenariosIf you consider a full theory of free fields, which forms the basis of literally every scatter computation, to be "extremely basic", then I really don't know what to tell you.
>Weren't those uncapable of describing anythingNo axiomatic approach to general QFT (i.e. not TQFT/CFT) has yet been completely able to capture what QFTists are doing. The closest we have is Haag-Kastler's local operator nets and Strocchi-Swieca/Osterwalder-Schrader's analytic constructions of Wightman/Schwinger functions, and these in some scenarios the regularity requirements are too stringent to produce any useful -matrix.
Aside from obvious obstructions like Haag's theorem, we can't find the Poisson structures needed for Kostant-Souriau-Sardanashvily geometric quantization of jets, and Zhang-Baez's AQFT struggles to get even basic classical limits like WKB. One needs to strike the perfect balance between formalism and applicability in order to produce useful axiomatization of QFT.
The gap is much less significant than you think, however, as CFTs (IR fixed points of QFTs) can in fact be fully captured by the mathematical theory of Friedan-Shenkar's gauge system. Not to mention Atiyah-Lurie's axiomatization of TQFTs has produced much progress in both physics and mathematics.
