# Axiomatic QFT, the reconstruction theorem and functional integrals

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?

• I think for [AA] the key is simply that predicted correlation functions are the only sorts of things we know how to test with experiments. Oct 5, 2020 at 15:39
• Minor quibble: The reconstruction theorem says that a Lorentz signature QFT can be recovered from the Euclidean signature Schwinger functions. Oct 6, 2020 at 0:06

I think what you say in Q2 is correct. It is certainly desirable to have a mathematically rigorous axiomatic formulation of QFT, but it seems unlikely that this would lead to any new physics results. This is my opinion but one which is shared by others including some of the leading field theorists like Banks and Weinberg.

As you say, it will probably contribute at some point in future to some new physics just because it provides solid mathematical ground to stand on, but I cannot see it leading directly to any interesting physics results in QFT.

Q1: This is basically correct. For a discussion, cf. the discussion in the first chapter of the book by Montvay and Münster and the references given therein.

Q2: This is quite correct. Axiomatic QFT can rigorously prove results like the CPT theorem or the spin-statistics theorem, but it is of very limited (not to say no) use in calculating physical observables. For a discussion cf. the book by Streater and Wightman, from whence I lift the (somewhat tongue-in-cheek) quotation about axiomatic QFT that

[c]ynical observers have compared them to the Shakers, a religious sect of New England who built solid barns and led celibate lives, a non-scientific equivalent of proving rigorous theorems and calculating no cross sections.

• Wunderful quote! Though we should not underestimate the advantages of a solid barn over a non-solid one, in particular when the weather sometimes becomes stormy... :-) Oct 5, 2020 at 11:14
• Indeed, the CPT theorem is a very solid refuge for field theory.
– gmvh
Oct 5, 2020 at 18:15
• Yes, the CPT theorem is essential and it is a genuine theorem. Dec 16, 2020 at 23:05