Editing Schwinger function (section)

=== History ===

At first, Osterwalder and Schrader claimed a stronger theorem that the axioms (E0)-(E4) by themselves imply the [[Wightman axioms]],<ref name=":0" /> however their proof contained an error which could not be corrected without adding extra assumptions. Two years later they published a new theorem, with the linear growth condition added as an assumption, and a correct proof.<ref name="Osterwalder Schrader 1975" /> The new proof is based on a complicated inductive argument (proposed also by [[Vladimir Glaser]]),<ref name="Glaser 1974">{{cite journal | last=Glaser | first=V. | title=On the equivalence of the Euclidean and Wightman formulation of field theory | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=37 | issue=4 | year=1974 | issn=0010-3616 | doi=10.1007/bf01645941 | pages=257–272| s2cid=121257568 | url=https://cds.cern.ch/record/873612 }}</ref> by which the region of analyticity of Schwinger functions is gradually extended towards the Minkowski space, and Wightman distributions are recovered as a limit. The linear growth condition (E0') is crucially used to show that the limit exists and is a tempered distribution.

Osterwalder's and Schrader's paper also contains another theorem replacing (E0') by yet another assumption called <math>\check{\text{(E0)}}</math>.<ref name="Osterwalder Schrader 1975" /> This other theorem is rarely used, since <math>\check{\text{(E0)}}</math> is hard to check in practice.<ref name="Kravchuk Qiao Rychkov 2021" />