Constructive QFT has provided some interesting models for dimension $d < 4$ of space-time, satisfying specific axiomatic versions of QFT. On the other hand, it is a well known fact that an axiomatic theory can admit more than one model, the question here would be: is it known whether there are models that satisfy Wightman's set of axioms but not those of Osterwalder and Schrader, or vice versa?

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    $\begingroup$ Anything that satisfies OS must satisfy W, by the original OS reconstruction theorem. OS also originally had a theorem published that proved equivalence in the other direction too, but it ended up containing a mistake in the proof, afaik it is still not proved that any wightman QFT has an OS interpretation. I will write a more detailed answer with references later, have to run now unfortunately $\endgroup$ Feb 4 at 3:28


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