Schwinger function

In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered n-tuples in

R

d

{\displaystyle \mathbb {R} ^{d}}

that are pairwise distinct. 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

ϕ

(

x

)

{\displaystyle \phi (x)}

,

x

∈

R

d

{\displaystyle x\in \mathbb {R} ^{d}}

. 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

ϕ

{\displaystyle \phi }

are denoted as

S

n

(

x

1

,

…

,

x

n

)

≡

⟨

ϕ

(

x

1

)

ϕ

(

x

2

)

…

ϕ

(

x

n

)

⟩

,

x

k

∈

R

d

.

{\displaystyle S_{n}(x_{1},\ldots ,x_{n})\equiv \langle \phi (x_{1})\phi (x_{2})\ldots \phi (x_{n})\rangle ,\quad x_{k}\in \mathbb {R} ^{d}.}

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.