(E0)-(E4) plus the linear growth condition (E0') appears to be stronger than the Wightman axioms. History At first, Osterwalder and Schrader claimed a stronger theorem that the axioms (E0)-(E4) by themselves imply the Wightman axioms, 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. The proof in is based on a complicated inductive argument (proposed also by Vladimir Glaser in ), 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. Ref. also contains another theorem replacing (E0') by yet another assumption called . This other theorem is rarely used, since is hard to check in practice. See e.g. for a review of these facts. Other axioms for Schwinger functions Axioms by Glimm and Jaffe An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book. In this approach one assumes that one is given a measure on the space of distributions . One then considers a generating functional which is assumed to satisfy properties OS0-OS4: (OS0) Analyticity. This asserts that is an entire-analytic function of for any collection of compactly supported test functions . Intuitively, this means that the measure decays faster than any exponential. (OS1) Regularity. This demands a growth bound for in terms of , such as. See for the precise condition. (OS2) Euclidean invariance. This says that the functional is invariant under Euclidean transformations . (OS3) Reflection positivity. Take a finite sequence of test functions which are all supported in the upper half-space i.e. at . Denote by where is a reflection operation defined above. This axioms says that the matrix has to be positive semidefinite. (OS4) Ergodicity. The time translation semigroup acts ergodically on the measure space . See for the precise condition. Relation to Osterwalder–Schrader axioms Although the above axioms were named by Glimm and Jaffe (OS0)-(OS4) in honor of Osterwalder and Schrader, they are not equivalent to the Osterwalder–Schrader axioms. Given (OS0)-(OS4), one can define Schwinger functions of as moments of the measure , and show that these moments satisfy Osterwalder–Schrader axioms (E0)-(E4) and also the linear