it: Reflection positivity Positivity axioms (E2) asserts the following property called (Osterwalder–Schrader) reflection positivity. 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 , let be the reflected point about the τ = 0 hyperplane. Then, 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 hyperplane: . 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. 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 Since the action S is real and can be split into , which only depends on φ on the positive half-space (), and 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. Osterwalder–Schrader theorem The Osterwalder–Schrader theorem states that Euclidean Schwinger functions which satisfy the above axioms (E0)-(E4) and an additional property (E0') called linear growth condition can be analytically continued to Lorentzian Wightman distributions which satisfy Wightman axioms and thus define a quantum field theory. Linear growth condition This condition, called (E0') in, asserts that when the Schwinger function of order is paired with an arbitrary Schwartz test function which vanishes at coincident points, we have the following bound: where is an integer constant, is the Schwartz-space seminorm of order , i.e. and a sequence of constants of factorial growth, i.e. with some constants . Linear growth condition is subtle as it has to be satisfied for all Schwinger functions simultaneously. It also has not been derived from the Wightman axioms, so that the system of OS axioms