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 real-analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms (named after Konrad Osterwalder and Robert Schrader). Schwinger functions are also referred to as Euclidean correlation functions.
Osterwalder–Schrader axioms
Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitian scalar field , . Note that a typical quantum field theory will contain infinitely many local operators, including also composite operators, and their correlators should also satisfy OS axioms similar to the ones described below.
The Schwinger functions of are denoted as
OS axioms from are numbered (E0)-(E4) and have the following meaning:
(E0) Temperedness
(E1) Euclidean covariance
(E2) Positivity
(E3) Symmetry
(E4) Cluster property
Temperedness
Temperedness axiom (E0) says that Schwinger functions are tempered distributions away from coincident points. This means that they can be integrated against Schwartz test functions which vanish with all their derivatives at configurations where two or more points coincide. It can be shown from this axiom and other OS axioms (but not the linear growth condition) that Schwinger functions are in fact real-analytic away from coincident points.
Euclidean covariance
Euclidean covariance axiom (E1) says that Schwinger functions transform covariantly under rotations and translations, namely:
for an arbitrary rotation matrix and an arbitrary translation vector . OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group.
Symmetry
Symmetry axiom (E3) says that Schwinger functions are invariant under permutations of points:
,
where is an arbitrary permutation of . Schwinger functions of fermionic fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation.
Cluster property
Cluster property (E4) says that Schwinger function reduces to the product if two groups of points are separated from each other by a large constant translation:
.
The limit is understood in the sense of distributions. There is also a technical assumption that the two groups of points lie on two sides of the hyperplane, while the vector is parallel to 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.