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.
[1] 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).
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.
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.
OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group.
[2][3] Symmetry axiom (E3) says that Schwinger functions are invariant under permutations of points: where
Schwinger functions of fermionic fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation.
if two groups of points are separated from each other by a large constant translation: The limit is understood in the sense of distributions.
Assume fN has its support in the "time-ordered" subset of N points with 0 < τ1 < ... < τN.
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
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.
), 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.
The Osterwalder–Schrader theorem[4] 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.
This condition, called (E0') in,[4] asserts that when the Schwinger function of order
Linear growth condition is subtle as it has to be satisfied for all Schwinger functions simultaneously.
At first, Osterwalder and Schrader claimed a stronger theorem that the axioms (E0)-(E4) by themselves imply the Wightman axioms,[2] however their proof contained an error which could not be corrected without adding extra assumptions.
Two years later they published a new theorem, with the linear growth condition added as an assumption, and a correct proof.
[4] The new proof is based on a complicated inductive argument (proposed also by Vladimir Glaser),[5] by which the region of analyticity of Schwinger functions is gradually extended towards the Minkowski space, and Wightman distributions are recovered as a limit.
The linear growth condition (E0') is crucially used to show that the limit exists and is a tempered distribution.
Osterwalder's and Schrader's paper also contains another theorem replacing (E0') by yet another assumption called
[3] An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book.
, and show that these moments satisfy Osterwalder–Schrader axioms (E0)-(E4) and also the linear growth conditions (E0').
Then one can appeal to the Osterwalder–Schrader theorem to show that Wightman functions are tempered distributions.
Alternatively, and much easier, one can derive Wightman axioms directly from (OS0)-(OS4).
[6] Note however that the full quantum field theory will contain infinitely many other local operators apart from
are concerned, but weaker than then the full set of OS axioms since they don't say much about correlators of composite operators.
The crucial feature of these axioms is to consider the field restricted to a surface.
One of the axioms is Markov property, which formalizes the intuitive notion that the state of the field inside a closed surface depends only on the state of the field on the surface.