Editing Schwinger function (section)

===Axioms by Glimm and Jaffe===

An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book.<ref name="Glimm 1987">{{cite book | last1=Glimm | first1=James | last2=Jaffe | first2 = Arthur| title=Quantum Physics : a Functional Integral Point of View | publisher=Springer New York | publication-place=New York, NY | year=1987 | isbn=978-1-4612-4728-9 | oclc=852790676 | page=}}</ref> In this approach one assumes that one is given a measure <math>d\mu </math> on the space of distributions <math> \phi \in D'(\mathbb{R}^d)</math>. One then considers a generating functional

:<math> S(f) =\int e^{\phi(f)} d\mu,\quad f\in D(\mathbb{R}^d)</math>

which is assumed to satisfy properties OS0-OS4:
* '''(OS0) Analyticity.''' This asserts that
:<math>z=(z_1,\ldots,z_n)\mapsto S\left(\sum_{i=1}^n z_i f_i\right)</math>
is an entire-analytic function of <math>z\in \mathbb{R}^n</math> for any collection of <math>n</math> compactly supported test functions <math>f_i\in D(\mathbb{R}^d)</math>. Intuitively, this means that the measure <math>d\mu</math> decays faster than any exponential.

* '''(OS1) Regularity'''. This demands a growth bound for <math>S(f)</math> in terms of <math>f</math>, such as<math>|S(f)|\leq \exp\left(C \int d^dx |f(x)|\right)</math>.  See <ref name="Glimm 1987" /> for the precise condition.
* '''(OS2) Euclidean invariance.''' This says that the functional <math>S(f)</math> is invariant under Euclidean transformations <math>f(x)\mapsto f(R x+b)</math>. 
* '''(OS3) Reflection positivity.''' Take a finite sequence of test functions <math>f_i\in D(\mathbb{R}^d)</math> which are all supported in the upper half-space i.e. at <math>x^0>0</math>. Denote by <math>\theta f_i(x)=f_i(\theta x)</math> where <math>\theta</math> is a reflection operation defined above. This axioms says that the matrix <math>M_{ij}=S(f_i+\theta f_j)</math> has to be positive semidefinite.
* '''(OS4) Ergodicity.''' The time translation semigroup acts ergodically on the measure space <math> (D'(\mathbb{R}^d),d\mu)</math>. See <ref name="Glimm 1987" /> 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 <math>\phi</math> as moments of the measure <math>d\mu </math>, and show that these moments satisfy Osterwalder–Schrader axioms (E0)-(E4) and also the linear growth conditions (E0'). Then one can appeal to the Osterwalder–Schrader theorem to show that [[Wightman functions]] are tempered distributions. Alternatively, and much easier, one can derive [[Wightman axioms]] directly from (OS0)-(OS4).<ref name="Glimm 1987" />

Note however that the full [[quantum field theory]] will contain infinitely many other local operators apart from <math>\phi</math>, such as <math>\phi^2</math>, <math>\phi^4</math> and other composite operators built from <math>\phi</math> and its derivatives. It's not easy to extract these Schwinger functions from the measure  <math>d\mu </math> and show that they satisfy OS axioms, as it should be the case.

To summarize, the axioms called (OS0)-(OS4) by Glimm and Jaffe are stronger than the OS axioms as far as the correlators of the field  <math>\phi</math>  are concerned, but weaker than then the full set of OS axioms since they don't say much about correlators of composite operators.