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The most recent discussion of what axioms one might drop from the Wightman axioms to allow the construction of realistic models that I'm aware of is Streater, Rep. Prog. Phys. 1975 38 771-846, "Outline of axiomatic relativistic quantum field theory". I'm looking for any more recent review that seriously discusses the axioms.

A critique of the Haag-Kastler axioms would also be welcome, but I would prefer to stay close enough to Lagrangian QFT to allow relatively immediate characterization of the difference between models of the reduced axiomatic system and the standard models that are relatively empirically successful.

I'm specially interested in any reviews that include a discussion of what models are possible if one relinquishes the existence of a lowest energy vacuum state (we know that, at least, this weakening allows the construction of thermal sectors of the free field, and that such a sector contains a thermal state that is thermodynamically stable even though it is not minimum energy, and that a Poincaré invariant "extra quantum fluctuations" sector is also possible—I'd like to know what is the full gamut of such models?).

[Added: This question was partly inspired by a Cosmic Variance post on the subject of QFT, particularly the link to John Norton, obviously with my research interests added.]

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In the last years there has been good progress on incorporating standard methods and results of perturbative Lagrangian QFT into AQFT. A commented list of references is here.

The basic observation is that the Stückelberg-Bogoliubov-Epstein-Glaser perturbative local S-matrices yield a local net of observables.

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  • $\begingroup$ Thanks for the reminder to go to the nCatLab. I'm gradually becoming able to read some of the content and the cited papers there with profit. It'll take me a few days to go through the papers on the particular webpage you linked to. BTW, the citation for the paper "V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978)" is slightly wrong, the page numbers are 578-585. FWIW, the links to arXiv papers are also pretty messy. $\endgroup$
    – Peter Morgan
    Feb 22, 2012 at 20:33
  • $\begingroup$ I have added the page number. Anything else that is "wrong"? What do you find messy about the links? $\endgroup$
    – Urs Schreiber
    Feb 23, 2012 at 6:28
  • $\begingroup$ Sorry, I should have been explicit. Two arXiv links are direct to ps instead of to abs. The papers are definitely useful, thanks, but having read them my feeling is that they don't much answer the question as I think of it. The question mentions Lagrangian QFT, and the John Norton link is preoccupied with the relation between Lagrangian and axiomatic methods, but I'm more interested in concentrated critiques of the Wightman or Haag-Kastler axioms, and in what models can be constructed if we drop individual axioms or parts of axioms, by someone like Streater, from the inside. $\endgroup$
    – Peter Morgan
    Feb 24, 2012 at 0:01
  • $\begingroup$ @UrsSchreiber do you know any reference about the nature of the operators in supersymmetric field theories? For example, physicists say "Coulomb branch operators". Is there a formal definition of those in terms, say $C^*$ algebras or whatever equivalent? $\endgroup$
    – Marion
    May 23, 2017 at 22:55
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Nonperturbative constructive results in 1+3 dimensisons are still completely missing. (Are you interested in lower dimensions?)

In particular, all realistic field theories are either nonrenormalizable (so that not even perturbation theory defined them well), or gauge theories (for which the Wightman axioms are inappropriate as they don't allow gauge-dependent charged states).

It is unknown how to modify the Wightman axioms so as to account for gauge invariance; but there has been interesting work, especially by Strocchi, on structural requirements in this direction. [author:strocchi gauge entered at http://scholar.google.at/scholar provides reading material]

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  • $\begingroup$ I dislike lower dimensions! Definitely bad of me, but there it is. I'm also a little down on gauge theories, which is much worse, given they're everything in empirically useful QFT. I'd like to present the same interacting quantum fields using only observables; likely that's hopeless! Good to be reminded of Strocchi, thanks, he's a little different. $\endgroup$
    – Peter Morgan
    Mar 12, 2012 at 19:48

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