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 <math>\mathbb R^d</math> 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 <math>\phi(x)</math>, <math> x\in \mathbb{R}^d</math>. 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 <math>\phi</math> are denoted as

:<math>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.</math>

OS axioms from

Symmetry

Symmetry axiom (E3) says that Schwinger functions are invariant under permutations of points:

:<math>S_n(x_1,\ldots,x_n)=S_n(x_{\pi(1)},\ldots,x_{\pi(n)})</math>,

where <math>\pi</math> is an arbitrary permutation of <math>\{1,\ldots,n\}</math>. 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 <math>S_{p+q}</math> reduces to the product <math>S_{p}S_q</math> if two groups of points are separated from each other by a large constant translation:

:<math>\lim_{b\to \infty} S_{p+q}(x_1,\ldots,x_p,x_{p+1}+b,\ldots, x_{p+q}+b)

=S_{p}(x_1,\ldots,x_p) S_q(x_{p+1},\ldots, x_{p+q})</math>.

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 <math>x^0=0</math> hyperplane, while the vector <math>b</math> is parallel to it:

:<math>x^0_1,\ldots,x^0_p>0,\quad x^0_{p+1},\ldots,x^0_{p+q}<0,\quad b^0=0.</math>

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 <math>x</math>, let <math>x^\theta</math> be the reflected point about the τ = 0 hyperplane. Then,

:<math>\sum_{m,n}\int d^dx_1 \cdots d^dx_m\, d^dy_1 \cdots d^dy_n S_{m+n}(x_1,\dots,x_m,y_1,\dots,y_n)f_m(x^\theta_1,\dots,x^\theta_m)^* f_n(y_1,\dots,y_n)\geq 0</math>

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 <math>\tau=0</math> hyperplane:

:<math>S_{2n}(x_1,\dots,x_n,x^\theta_n,\dots,x^\theta_1)\geq 0</math>.

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

: <math> \int \mathcal{D}\phi F[\phi(x)]F[\phi(x^\theta)]^* e^{-S[\phi]}=\int \mathcal{D}\phi_0 \int_{\phi_+(\tau=0)=\phi_0} \mathcal{D}\phi_+ F[\phi_+]e^{-S_+[\phi_+]}\int_{\phi_-(\tau=0)=\phi_0} \mathcal{D}\phi_- F[(\phi_-)^\theta]^* e^{-S_-[\phi_-]}. </math>

Since the action S is real and can be split into <math> S_+ </math>, which only depends on φ on the positive half-space (<math> \phi_+ </math>), and <math> S_- </math> which only depends upon φ on the negative half-space (<math> \phi_- </math>), 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, 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 <math>\check{\text{(E0)</math>. In this approach one assumes that one is given a measure <math>d\mu </math> on the space of distributions <math> \phi \in D'(\mathbb{R}^d)</math>. One then considers a generating functional

:<math> S(f) =\int e^{\phi(f)} d\mu,\quad f\in D(\mathbb{R}^d)</math>

which is assumed to satisfy properties OS0-OS4:

(OS0) Analyticity. This asserts that

:<math>z=(z_1,\ldots,z_n)\mapsto S\left(\sum_{i=1}^n z_i f_i\right)</math>

is an entire-analytic function of <math>z\in \mathbb{R}^n</math> for any collection of <math>n</math> compactly supported test functions <math>f_i\in D(\mathbb{R}^d)</math>. Intuitively, this means that the measure <math>d\mu</math> decays faster than any exponential.

(OS1) Regularity. This demands a growth bound for <math>S(f)</math> in terms of <math>f</math>, such as<math>|S(f)|\leq \exp\left(C \int d^dx |f(x)|\right)</math>. See See also their description in the book of Barry Simon. Like in the above axioms by Glimm and Jaffe, one assumes that the field <math> \phi \in D'(\mathbb{R}^d)</math> is a random distribution with a measure <math>d\mu </math>. This measure is sufficiently regular so that the field <math> \phi</math> has regularity of a Sobolev space of negative derivative order. 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.