QFT and its non-rigorous assumptions

Question

I have been trying to figure out all the non-rigorous assumptions of QFT (as performed in an operator theory) that allow it to function as it currently is. So far, the three big candidates I found are these :

1. The product of distributions is allowed in some sense (this runs afoul of Schwartz's impossibility theorem, although it is possible to define it rigorously but it is rarely done in QFT)
2. The interacting field operator is related to the free field operator by some unitary transformation (Wrong in most cases, as shown by Haag's theorem and other such things)
3. The vacuum of the interacting theory is related to the vacuum of the free theory. In particular, we have that $\langle 0 \vert \Omega \rangle \neq 0$. According to Haag (p. 71), this is also wrong. The usual formula for relating the vacuas fail to converge.

Are those the only common wrong assumptions? I'm also thinking that derivatives of the field operators may be ill defined as derivatives on operators are usually well defined for bounded operators, but I'm not quite sure.

Answer 1

I'm not sure there is a definite answer, since nobody seems to agree as to what QFT really is (at least so says Nate Seiberg in his 2015 Breakthrough Prize talk), but the problems you mention are well known and accounted for.

Quantum Mechanics gets loads of issues with infinite degrees of freedom. So in reality what we actually do is to regularize the theory, both infrared (e.g. putting in finite volume) and ultraviolet (e.g. putting in a lattice). When properly regularized QFT has finite degrees of freedom, and therefore all Haag's objections go away. You have a perfectly defined interaction picture and a mapping between free and interacting ground states, so everything goes smoothly.

This has the drawback of breaking Lorentz Symmetry, but the hope is that once we renormalize, by taking appropriate limits, we are left with sensible answers and recover Poincaré invariance.

Since the couplings in QFT are very singular and ill-defined we should have done the regularization in the first place just to know what the hamiltonian is, so this is a key part of QFT.

The problem with Haag's approach, and other axiomatic ones, is that they work directly with already renormalized fields, and therefore run on multiple problems. This is why more recent textbooks, starting with Weinberg (I think), emphasize that regularization and the renormalization group are key concepts one cannot do QFT without, even in the absence of interactions.