Before posting my question, let me make some remarks:

**[MS]** Salmhofer's book on renormalization begins with a nice discussion on Feynman's path integral. At some point, the author states the following:

In quantum field theory, one is not dealing with a single particle, but with infinitely many particles, because one has to account for the creation and annihilation of particles. One can formally write down a Hamiltonian, but it becomes very difficult to give a mathematical definition of it.

We shall simply define the theory by the functional integral.

**[AA]** I think in the same spirit of the above statement, Abdelmalek's paper on QFT for mathematicians states that, from the mathematical point of view, the fundamental problem is to give meaning to and study the properties of integrals of the form:
\begin{eqnarray}
\mathbb{E}[\mathcal{O}_{A_{1}}(x_{1})\cdots \mathcal{O}_{A_{n}}(x_{n})] = \frac{\int_{\mathcal{F}}\mathcal{O}_{A_{1}}(x_{1})\cdots \mathcal{O}_{A_{n}}(x_{n})e^{-S(\phi)}D\phi}{\int_{\mathcal{F}}e^{-S(\phi)}D\phi}\tag{1}\label{1}.
\end{eqnarray}

Now, I'd like to understand both these statements, in particular the one in bold typed in [MS].

**[EW; BS]** As discussed in Edson de Faria and Wellington de Melo's book and Reed & Simon's book, a first mathematical description of QFT was given by Garding and Wightman. They proposed a set of axioms, known today as Wightman axioms, which defines mathematically what we mean by a quantum field theory. This is called Axiomatic QFT. Also, there is a famous result called the Wightman reconstruction theorem which states that one can completely recover a QFT from its Wightman correlations functions.

QFT and Euclidean QFT are related by a Wick rotation to imaginary time. As a consequence, Wightman correlation functions become Schwinger functions and a set of axioms for Schwinger functions can also be defined. These axioms are called Osterwalder-Schrader axioms. As before, there is a reconstruction theorem which states that the Euclidean QFT can be fully recovered from its Schwinger functions.

Concerning the above discussion, I have two questions:

**Q1:** Are these reconstruction theorems the reason for both statements [MS] and [AA]? In other words, is Euclidean QFT's mathematical description basically a study of Schwinger functions (and, thus, functional integrals) because the underlying QFT can be recovered from them?

**Q2:** I've been told once that, although axiomatic QFT is very precise mathematically, it is still very limited in its ability to produce results in terms of the physics behind it. I was even told that axiomatic QFT is "more like a mathematical theory than a physics theory". I'm very inexperienced and I don't know if this is accurate or not, but with the above discussion, it seems to me that axiomatic QFT is not necessarily trying to produce results in terms of physics, but rather it is trying to produce a solid mathematical ground which will certainly contribute to finding results in physics at some point. Is this accurate? Moreover, is axiomatic QFT even limited?

Lorentzsignature QFT can be recovered from the Euclidean signature Schwinger functions. $\endgroup$ – user1504 Oct 6 '20 at 0:06