Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via a wick rotation, and the theories were constructed in euclidean space rather than Minkowski space.
Now, I have two questions... first, were the euclidean theories then wick rotated back into Minkowski space? Otherwise, how else were they converted to Minkowski spacetime?
Also, because of the introduction of a complex coordinate, an n dimensional Minkowski qft would have to be formulated in an n+1 dimensional euclidean space, right? If so, is it the addition of a coordinate that makes it difficult to formulate 4 dimensional qfts using constructive methods? Or does the addition of a coordinate actually help in some convoluted way?
Sorry if the questions are vague; I am still introducing myself to mathematical physics.
Also, any level of answer is fine, although I would prefer technical to layman level.



How does Constructive Quantum Field Theory work?
An euclidean theory which fulfills the Osterwalder-Schrader axioms gives a Wightman theory on Minkowski space via analytical continuation of the correlation function by the <http://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem>


E.g. a 4 dim theory gives a Wightman theory on 3+1 dim Minkowski space, so there is no additional dimension...


