# Wightman's theorem, in terms of time-ordered functions$.$

According to Wightman's theorem, given a set of distributions $\{W_n\}$ satisfying a set of axioms, we may conclude the existence of a set of operators $\{\phi\}$ which satisfy some properties, and such that \begin{equation} W_n(x_1,\dots,x_n)=\langle 0|\phi(x_1)\cdots\phi(x_n)|0\rangle\tag1 \end{equation}

For example, if we assume $W_n(x_1+a,\dots,x_n+a)=W_n(x_1,\dots,x_n)$, we may conclude that the operators satisfy \begin{equation} [P_\mu,\phi(x)]=\partial_\mu\phi(x)\tag2 \end{equation} where $P_\mu$ is the generator of translations. Given other properties of $W_n$ we may deduce other properties for the $\phi$'s.

So far so good. My concern is that these matters are always stated in terms of Wightman functions instead of time-ordered correlation functions, \begin{equation} G_n(x_1,\dots,x_n)=\langle 0|{\color{red}{\mathrm T}}\ \phi(x_1)\cdots\phi(x_n)|0\rangle\tag3 \end{equation}

Of course, given $W_n$ we can find $G_n$, but not the other way around. My question is about a possible reconstruction theorem using time-ordered functions instead of Wightman functions. The theorems must clearly be weaker, but I'd like to know how far we may push the programme. Given a certain characterisation of the $\{G_n\}$, such as

1. They are tempered distributions.
2. They satisfy clustering properties.
3. They are Poincaré covariant.
4. Etc.

can we conclude something similar to Wightman's theorem? can we deduce the existence of operators $\{\phi\}$ satisfying some properties, such as $(2)$?

You can't get the $G_n$ from the $W_n$ as easily as you say because this involves multiplying distributions by non-smooth Heaviside functions of time differences. That's mathematically illegal. The axiomatic setting with time-ordered distributions is also called the LSZ theory. I am not an expert of it but I know it is discussed at length in Ch. 13 of the book "General Principles of Quantum Field Theory" by Bogolubov et al. A perhaps even better reference for your needs is the article "Time-ordered products and Schwinger functions" by Eckmann and Epstein (they give the axioms of T-products in the second page of this article). I would say that provided you have the regularity hypotheses to do the Osterwalder-Schrader back and forth between Euclidean and Minkowski, you can actually get the $W_n$ from the $G_n$ because in the Euclidean setting that is just restricting a distribution on $(\mathbb{R}^d)^n$ to the open set given by the complement of the big diagonal.