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 :
- 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)
- 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)
- 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.