Editing Schwinger function (section)

==Osterwalder–Schrader theorem<!--'Osterwalder–Schrader theorem' redirects here-->==
The '''Osterwalder–Schrader theorem'''<ref name="Osterwalder Schrader 1975">{{cite journal | last1=Osterwalder | first1=Konrad | last2=Schrader | first2=Robert | title=Axioms for Euclidean Green's functions II | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=42 | issue=3 | year=1975 | issn=0010-3616 | doi=10.1007/bf01608978 | pages=281–305| s2cid=119389461 | url=http://projecteuclid.org/euclid.cmp/1103899050 }}</ref> states that Euclidean Schwinger functions which satisfy the above axioms (E0)-(E4) and an additional property (E0') called '''linear growth condition''' can be analytically continued to Lorentzian Wightman distributions which satisfy [[Wightman axioms]] and thus define a [[quantum field theory]].

=== Linear growth condition ===

This condition, called (E0') in,<ref name="Osterwalder Schrader 1975" /> asserts that when the Schwinger function of order <math> n</math> is paired with an arbitrary [[Schwartz space|Schwartz]] test function <math>f</math> which vanishes at coincident points, we have the following bound:

:<math>|S_{n}(f)|\leq \sigma_n |f|_{C\cdot n},</math>

where <math> C\in \mathbb{N}</math> is an integer constant, <math>|f|_{C\cdot n}</math> is the Schwartz-space seminorm of order <math>N=C\cdot n</math>, i.e.

:<math>|f|_{N} = \sup_{|\alpha|\leq N, x\in \mathbb{R}^d} |(1+|x|)^N D^\alpha f(x)|,</math>

and <math>\sigma_n </math> a sequence of constants of '''factorial growth''', i.e. <math>\sigma_n \leq A (n!)^B </math> with some constants <math>A,B</math>.

Linear growth condition is subtle as it has to be satisfied for all Schwinger functions simultaneously. It also has not been derived from the [[Wightman axioms]], so that the system of OS axioms (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]],<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" />