Schwinger function

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In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions and they are analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity.

Pick any arbitrary coordinate τ and pick a test function f_N with N points as its arguments. Assume f_N has its support in the "time-ordered" subset of N points with 0 < τ₁ < ... < τ_N. Choose one such f_N for each positive N, with the f's being zero for all N larger than some integer M. Given a point x, let $\scriptstyle \bar{x}$ be the reflected point about the τ = 0 hyperplane. Then,

$\sum_{m,n}\int d^dx_1 \cdots d^dx_m\, d^dy_1 \cdots d^dy_n S_{m+n}(x_1,\dots,x_m,y_1,\dots,y_n)f_m(\bar{x}_1,\dots,\bar{x}_m)^* f_n(y_1,\dots,y_n)\geq 0$

where * represents complex conjugation.

The Osterwalder-Schrader theorem states that Schwinger functions which satisfies these properties can be analytically continued into a quantum field theory.

Euclidean path integrals satisfy reflection positivity formally. Pick any polynomial functional F of the field φ which doesn't depend upon the value of φ(x) for those points x whose τ coordinates are nonpositive.

Then,

$\int \mathcal{D}\phi F[\phi(x)]F[\phi(\bar{x})]^* e^{-S[\phi]}=\int \mathcal{D}\phi_0 \int_{\phi_+(\tau=0)=\phi_0} \mathcal{D}\phi_+ F[\phi_+]e^{-S_+[\phi_+]}\int_{\phi_-(\tau=0)=\phi_0} \mathcal{D}\phi_- F[\bar{\phi}_-]^* e^{-S_-[\phi_-]}.$

Since the action S is real and can be split into S₊ which only depends on φ on the positive half-space and S₋ which only depends upon φ on the negative half-space 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.