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

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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 the classical paper by Glimm and Jaffe: http://www.jstor.org/stable/1970959?seq=1#page_scan_tab_contents.

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 actually read it carefully).

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