# Schwinger function

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 (named after Julian Schwinger) and they are analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity.

## Details

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

$\sum _{m,n}\int d^{d}x_{1}\cdots d^{d}x_{m}\,d^{d}y_{1}\cdots d^{d}y_{n}S_{m+n}(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})f_{m}({\bar {x}}_{1},\dots ,{\bar {x}}_{m})^{*}f_{n}(y_{1},\dots ,y_{n})\geq 0$

where * represents complex conjugation.

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