# Glimm and Jaffe's paper on the construction of 2D QFTs

I am presently interested in the construction of low-dimensional (2D/3D) QFTs where all the Wightman axioms have been proved and to this end, I started reading this article, which is recommended in this discussion.

The article looks interesting but there are some points that I cannot get.

For instance, concerning the definition of $y_0$ (the time of the annihilators). In the paragraph below eq. (2.2.6), it is claimed to belong to $\mathbf{Z} + \frac{1}{2}$ (i.e it is a half-integer). But in the second paragraph below (2.2.7), it is said that $y_0 = 0$ for the so-called initial terms. Also, if you look at the details of the contour expansion algorithm in section 2.3 (more precisely, page 610), you'll see $y_0$ is incremented by units, so there seems to be no way to reconcile both statements.

Also, in the proof of lemma 2.4.3 as I understand it, we want to lower bound the separation between $y_0$ and the time of the fields appearing in the Wick monomial $B$. They claim that $\sigma_i \geq 1$ (I inferred that $\sigma_i$ meant this separation by comparing with the proof of the previous lemma, though the notation is never defined AFAIK). But if you merely know $y_0 \in [T - 1, T]$ and $B$ is localized at times greater than $T$, then this separation could be arbitrarily small from my understanding...

Could someone please shed light on these points? Or maybe just give general feedback on the article (as I understood, it is a classic in Axiomatic QFT, the authors are renowned, etc. but it would be great to hear from people who read it carefully).

• Abdelmalek Abdesselam mentions improvements and simplifications and gives some more references here. Commented May 6, 2018 at 3:52
• Thanks for your suggestion! The topic and references look quite interesting. Yet, I would still like to understand the paper to begin with (mostly because it looks relatively self-contained and is widely cited, so that many people must have read it). If you think, however, that the proof is full of typos and therefore not worth reading, or want to point me to corrections/comments relative to this article, please do not hesitate! Commented May 7, 2018 at 13:42