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 \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.

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 \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. S_n(x_1,\ldots,x_n)=S_n(R x_1+b,\ldots,Rx_n+b) for an arbitrary rotation matrix R\in SO(d) and an arbitrary translation vector b\in \mathbb{R}^d . OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group. S_n(x_1,\ldots,x_n)=S_n(x_{\pi(1)},\ldots,x_{\pi(n)}) , where \pi is an arbitrary permutation of \{1,\ldots,n\} . Schwinger functions of fermionic fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation. =S_{p}(x_1,\ldots,x_p) S_q(x_{p+1},\ldots, x_{p+q}) . 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 x^0=0 hyperplane, while the vector b is parallel to it: x^0_1,\ldots,x^0_p>0,\quad x^0_{p+1},\ldots,x^0_{p+q} Positivity axioms (E2) asserts the following property called (Osterwalder–Schrader) reflection positivity. Pick any arbitrary coordinate τ and pick a test function f N with N points as its arguments. Assume f N has its support in the "time-ordered" subset of N points with 0 1 N . Choose one such f N for each positive N, with the f's being zero for all N larger than some integer M. Given a point x , let x^\theta be the reflected point about the τ = 0 hyperplane. Then, \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 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 \tau=0 hyperplane: S_{2n}(x_1,\dots,x_n,x^\theta_n,\dots,x^\theta_1)\geq 0 . This property indeed follows from the reflection positivity but it is weaker than full reflection positivity. \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_-]}. Since the action S is real and can be split into S_+ , which only depends on φ on the positive half-space ( \phi_+ ), and S_- which only depends upon φ on the negative half-space ( \phi_- ), 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. where C\in \mathbb{N} is an integer constant, |f|_{C\cdot n} is the Schwartz-space seminorm of order N=C\cdot n , i.e. |f|_{N} = \sup_{|\alpha|\leq N, x\in \mathbb{R}^d} |(1+|x|)^N D^\alpha f(x)|, and \sigma_n a sequence of constants of factorial growth, i.e. \sigma_n \leq A (n!)^B with some constants A,B . 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 (E0)-(E4) plus the linear growth condition (E0') appears to be stronger than the Wightman axioms. ===Axioms by Glimm and Jaffe=== An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book. In this approach one assumes that one is given a measure d\mu on the space of distributions \phi \in D'(\mathbb{R}^d) . One then considers a generating functional S(f) =\int e^{\phi(f)} d\mu,\quad f\in D(\mathbb{R}^d) which is assumed to satisfy properties OS0-OS4: (OS0) Analyticity. This asserts that z=(z_1,\ldots,z_n)\mapsto S\left(\sum_{i=1}^n z_i f_i\right) is an entire-analytic function of z\in \mathbb{R}^n for any collection of n compactly supported test functions f_i\in D(\mathbb{R}^d) . Intuitively, this means that the measure d\mu decays faster than any exponential. (OS1) Regularity. This demands a growth bound for S(f) in terms of f , such as |S(f)|\leq \exp\left(C \int d^dx |f(x)|\right) . See for the precise condition. (OS2) Euclidean invariance. This says that the functional S(f) is invariant under Euclidean transformations f(x)\mapsto f(R x+b) . (OS3) Reflection positivity. Take a finite sequence of test functions f_i\in D(\mathbb{R}^d) which are all supported in the upper half-space i.e. at x^0>0 . Denote by \theta f_i(x)=f_i(\theta x) where \theta is a reflection operation defined above. This axioms says that the matrix M_{ij}=S(f_i+\theta f_j) has to be positive semidefinite. (OS4) Ergodicity. The time translation semigroup acts ergodically on the measure space (D'(\mathbb{R}^d),d\mu) . See for the precise condition. ==See also== Wick rotation Axiomatic quantum field theory Wightman axioms ==References==

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