# Is it known whether Wightman's axiomatic QFT is logically equivalent to Osterwalder–Schrader's axiomatic QFT?

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?

• 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 Feb 4 at 3:28