Editing Schwinger function (section)

==== 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.