| Schwinger Function |
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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 be the reflected point about the τ=0 hyperplane. Then,
The Osterwalder-Schrader theorem states that Schwinger functions which satisfies these properties can be analytically continued into a Quantum Field Theory . Euclidean Path Integral s 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,
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 reflections, then the previous quantity has to be nonnegative. SEE ALSO |
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