What does mathematical consistency in QFT mean?

Question

My question is more naive than Is QFT mathematically self-consistent?

Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? For example (could be in a more general context) , if I refer to https://souravchatterjee.su.domains//qft-lectures-combined.pdf

29.3 Wightman axioms

Axiom 1. There exists a Hilbert space $H$ whose elements represent the possible states of the quantum system under consideration

Unfortunately, no one has been able to construct an interacting quantum field theory in $\mathbb{R}^{1,3}$ that satisfies the Wightman axioms

Do people want to prove the existence of the exact solution in interacting theories? (the exact solution as a possible state in Hilbert space)

If yes, the perturbative expansion is thought to be asymptotic, it may hardly be used to show the existence of the exact solution of interaction system (sum of infinity order of expansion possibly goes to diverge). If I use the lattice approach when the grids approach zero, QED may reach the Landau pole then may become problematic for even closer (not sure what would it be). But, QCD can be defined at the zero grid distance limit, due to the asymptotic freedom. Is QCD then mathematically consistent? Or am I not even wrong?

Answer 1

I have never heard people talk about mathematical "consistency" of QFT in any other sense than that of rigor. Your mention of "Wightman axioms" supports this.

This isn't about the mathematical notion of the consistency of logical system of axioms, it's about having a system of axioms and theorems in the first place and knowing how to exhibit specific examples of things obeying these axioms.

To the extent that we have rigorous axiom systems for QFT (Wightman, Osterwalder-Schrader), we don't know how to use it to e.g. do the Standard Model rigorously. Conversely, we "have" a lot of QFTs at a lower level of rigor - namely that of practicing physicists - where we do not know how to rigorously describe the exact objects and procedures used to produce the physical predictions. That is, while the physicist is happily computing cross sections and expectation values page after page, the mathematician is still stuck on the first line trying to figure out what all these symbols are actually supposed to mean and how to prove the physicist's brazen manipulations are actually meaningful mathematically.

The Yang-Mills millenium problem is precisely about that: We don't know how to treat realistic QFTs like the Standard Model in a completely rigorous way, e.g. give a mathematically flawless description of how to compute n-point functions and show these n-point functions fulfill an axiom system like Osterwalder-Schrader.

This is not about obtaining exact solutions, it's about writing down the equations and algorithms that one would have to solve/perform in the first place to a level of rigor that withstands the typical scrutiny in comparable fields of math. For instance, we don't want to compute the n-point functions, we just want a proper (non-perturbative, i.e. non-approximative) definition of them that doesn't involve nebulous notions like the path integral (which is well-defined only in specific situations and mostly in low dimensions) or the pointwise multiplication of operator-valued distributions (an issue closely related to rigorous formalizations of renormalization). For more on the state of rigor in QFT in general, see this question and its answers. And then we want a proof that these definitions actually constitute a QFT, i.e. obey the Wightman or OS axioms that model what we think a QFT "should be" when formulated rigorously. And lastly this rigorous construction should of course be well-approximated by the already known perturbative and non-rigorous computations for the actual physical theory.

Lattice approaches are not plagued by issues with things like the path integral, but the continuum limit - both whether it exists and whether it converges against the theory we want it to (see, e.g. the triviality of $\phi^4$ theory for why you might be able to take a limit but it's not "the theory you want") - is a thorny issue for rigor as well.