Editing Schwinger function (section)

=== 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.