Editing Schwinger function (section)

==Osterwalder–Schrader axioms<!--'Osterwalder–Schrader axioms' redirects here-->==

Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitian scalar field <math>\phi(x)</math>, <math> x\in \mathbb{R}^d</math>. Note that a typical [[quantum field theory]] will contain infinitely many local operators, including also [[Composite operator|composite operators]], and their correlators should also satisfy OS axioms similar to the ones described below. 

The Schwinger functions of <math>\phi</math> are denoted as

:<math>S_n(x_1,\ldots,x_n) \equiv \langle \phi(x_1) \phi(x_2)\ldots \phi(x_n)\rangle,\quad x_k \in \mathbb{R}^d.</math>

OS axioms from <ref name=":0" /> are numbered (E0)-(E4) and have the following meaning:  

* (E0) Temperedness 
* (E1) Euclidean covariance 
* (E2) Positivity 
* (E3) Symmetry 
* (E4) Cluster property 

=== Temperedness ===

Temperedness axiom (E0) says that Schwinger functions are [[Distribution (mathematics)|tempered distributions]] away from coincident points. This means that they can be integrated against [[Schwartz space|Schwartz]] test functions which vanish with all their derivatives at configurations where two or more points coincide. It can be shown from this axiom and other OS axioms (but not the linear growth condition) that Schwinger functions are in fact real-analytic away from coincident points.

=== Euclidean covariance ===
Euclidean covariance axiom (E1) says that Schwinger functions transform covariantly under rotations and translations, namely:

:<math>S_n(x_1,\ldots,x_n)=S_n(R x_1+b,\ldots,Rx_n+b)</math>

for an arbitrary rotation matrix <math>R\in SO(d)</math> and an arbitrary translation vector <math>b\in \mathbb{R}^d</math>. OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group.<ref name=":0" /><ref name="Kravchuk Qiao Rychkov 2021">{{Cite arXiv| last1=Kravchuk | first1=Petr | last2=Qiao | first2=Jiaxin | last3=Rychkov | first3=Slava | title=Distributions in CFT II. Minkowski Space | date=2021-04-05 | arxiv=2104.02090v1 }}</ref>

=== Symmetry ===

Symmetry axiom (E3) says that Schwinger functions are invariant under permutations of points:

:<math>S_n(x_1,\ldots,x_n)=S_n(x_{\pi(1)},\ldots,x_{\pi(n)})</math>,

where <math>\pi</math> is an arbitrary permutation of <math>\{1,\ldots,n\}</math>. Schwinger functions of [[Fermion|fermionic]] fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation.

=== Cluster property ===

Cluster property (E4) says that Schwinger function <math>S_{p+q}</math> reduces to the product <math>S_{p}S_q</math> if two groups of points are separated from each other by a large constant translation:

:<math>\lim_{b\to \infty} S_{p+q}(x_1,\ldots,x_p,x_{p+1}+b,\ldots, x_{p+q}+b)
=S_{p}(x_1,\ldots,x_p) S_q(x_{p+1},\ldots, x_{p+q})</math>.

The limit is understood in the sense of distributions. There is also a technical assumption that the two groups of points lie on two sides of the <math>x^0=0</math> hyperplane, while the vector <math>b</math> is parallel to it:

:<math>x^0_1,\ldots,x^0_p>0,\quad x^0_{p+1},\ldots,x^0_{p+q}<0,\quad b^0=0.</math>

=== Reflection positivity<!--'Reflection positivity' redirects here--> ===

Positivity axioms (E2) asserts the following property called (Osterwalder–Schrader) reflection positivity. Pick any arbitrary coordinate τ and pick a [[test function]] ''f''<sub>''N''</sub> with ''N'' points as its arguments. Assume ''f''<sub>''N''</sub> has its [[Support (mathematics)|support]] in the "time-ordered" subset of ''N'' points with 0 < τ<sub>1</sub> < ... < τ<sub>''N''</sub>. Choose one such ''f''<sub>''N''</sub> for each positive ''N'', with the f's being zero for all ''N'' larger than some integer ''M''. Given a point <math>x</math>, let <math>x^\theta</math> be the reflected point about the τ = 0 [[hyperplane]]. Then,

:<math>\sum_{m,n}\int d^dx_1 \cdots d^dx_m\, d^dy_1 \cdots d^dy_n S_{m+n}(x_1,\dots,x_m,y_1,\dots,y_n)f_m(x^\theta_1,\dots,x^\theta_m)^* f_n(y_1,\dots,y_n)\geq 0</math>

where * represents [[complex conjugation]]. 

Sometimes in theoretical physics literature reflection positivity is stated as the requirement that the Schwinger function of arbitrary even order should be non-negative if points are inserted symmetrically with respect to the  <math>\tau=0</math> hyperplane:

:<math>S_{2n}(x_1,\dots,x_n,x^\theta_n,\dots,x^\theta_1)\geq 0</math>.

This property indeed follows from the reflection positivity but it is weaker than full reflection positivity.

==== Intuitive understanding ====
One way of (formally) constructing Schwinger functions which satisfy the above properties is through the Euclidean [[Path integral formulation|path integral]]. In particular, Euclidean path integrals (formally) satisfy reflection positivity.  Let ''F'' be any polynomial functional of the field ''φ'' which only depends upon the value of ''φ''(''x'') for those points ''x'' whose ''τ'' coordinates are nonnegative. Then
: <math> \int \mathcal{D}\phi F[\phi(x)]F[\phi(x^\theta)]^* e^{-S[\phi]}=\int \mathcal{D}\phi_0 \int_{\phi_+(\tau=0)=\phi_0} \mathcal{D}\phi_+ F[\phi_+]e^{-S_+[\phi_+]}\int_{\phi_-(\tau=0)=\phi_0} \mathcal{D}\phi_- F[(\phi_-)^\theta]^* e^{-S_-[\phi_-]}. </math>

Since the action ''S'' is real and can be split into <math> S_+ </math>, which only depends on ''φ'' on the positive half-space (<math> \phi_+ </math>), and <math> S_- </math> which only depends upon ''φ'' on the negative half-space (<math> \phi_- </math>), and if ''S'' also happens to be invariant under the combined action of taking a reflection and complex conjugating all the fields, then the previous quantity has to be nonnegative.