For clarity let me say that I am not asking for the general definition of time-ordered product in QFT, nor the definition of Wightman functions.
Precision: In the first displayed equation on pg. 394 (section 2.5) of the book Quantum fields and strings: a course for mathematicians, volume 1, the author defines the time-ordered Wightman function at $v$ as a distributional limit for $\epsilon \to 0+$ of the Schwinger function evaluated at $ v+ i\epsilon v = (1+i\epsilon)v$, where $v$ is a point in a cartesian power of the Minkowski space:
$$ \mathcal{W}^T_n(v) = \lim\limits_{\epsilon \to 0+} \mathfrak{S}_n(v + i \epsilon v) $$
(https://www.math.ias.edu/QFT/fall/lect2.tex)
My problem: I do not understand it.
Examples of problems I have with the definition:
- In general, the point $(1+i\epsilon)v$ doesn’t belong to the domain of the Schwinger functions (given at the bottom of pg. 389), since, to start, $i \epsilon v$ has imaginary values at the spacial coordinates if $v$ is an usual element of the cartesian power of the Minkowski space.
- The point $v$ could in principle lie in one of the diagonals, but that is forbidden for the elements of the domain of the Schwinger function.
- I’m not even sure that the point he is evaluating the Schwinger function at even belongs to the full domain of the holomorphic function whose boundary value gives the usual Wightman function. For example, one could pick $v$ to be zero, which need not be in the full domain of the holomorphic function.
I am very likely missing something quite basic here. Thanks for the help!
PS: This definition was mentioned in one of the answers to a previous post (How do I define time-ordering for Wightman functions?), but the contributors didn’t discuss the points I am raising, probably because it was clear to them.
