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:

${\displaystyle S_{n}(x_{1},\ldots ,x_{n})=S_{n}(Rx_{1}+b,\ldots ,Rx_{n}+b)}$

for an arbitrary rotation matrix ${\displaystyle R\in SO(d)}$ and an arbitrary translation vector ${\displaystyle b\in \mathbb {R} ^{d}}$. OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group.^[2][3]

Symmetry

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

${\displaystyle S_{n}(x_{1},\ldots ,x_{n})=S_{n}(x_{\pi (1)},\ldots ,x_{\pi (n)})}$,

where ${\displaystyle \pi }$ is an arbitrary permutation of ${\displaystyle \{1,\ldots ,n\}}$. Schwinger functions of fermionic fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation.

Cluster property

Cluster property (E4) says that Schwinger function ${\displaystyle S_{p+q}}$ reduces to the product ${\displaystyle S_{p}S_{q}}$ if two groups of points are separated from each other by a large constant translation:

${\displaystyle \lim _{b\to \infty }S_{p+q}(x_{1},\ldots ,x_{p},x_{p+1}+b,\ldots ,x_{p+q}+b)=S_{p}(x_{1},\ldots ,x_{p})S_{q}(x_{p+1},\ldots ,x_{p+q})}$.

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 ${\displaystyle x^{0}=0}$ hyperplane, while the vector ${\displaystyle b}$ is parallel to it:

${\displaystyle x_{1}^{0},\ldots ,x_{p}^{0}>0,\quad x_{p+1}^{0},\ldots ,x_{p+q}^{0}<0,\quad b^{0}=0.}$

Reflection positivity

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 < τ₁ < ... < τ_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 ${\displaystyle x}$, let ${\displaystyle x^{\theta }}$ be the reflected point about the τ = 0 hyperplane. Then,

${\displaystyle \sum _{m,n}\int d^{d}x_{1}\cdots d^{d}x_{m}\,d^{d}y_{1}\cdots d^{d}y_{n}S_{m+n}(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})f_{m}(x_{1}^{\theta },\dots ,x_{m}^{\theta })^{*}f_{n}(y_{1},\dots ,y_{n})\geq 0}$

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 ${\displaystyle \tau =0}$ hyperplane:

${\displaystyle S_{2n}(x_{1},\dots ,x_{n},x_{n}^{\theta },\dots ,x_{1}^{\theta })\geq 0}$.

This property indeed follows from the reflection positivity but it is weaker than full reflection positivity.

Linear growth condition

This condition, called (E0') in,^[4] asserts that when the Schwinger function of order ${\displaystyle n}$ is paired with an arbitrary Schwartz test function ${\displaystyle f}$ which vanishes at coincident points, we have the following bound:

${\displaystyle |S_{n}(f)|\leq \sigma _{n}|f|_{C\cdot n},}$

where ${\displaystyle C\in \mathbb {N} }$ is an integer constant, ${\displaystyle |f|_{C\cdot n}}$ is the Schwartz-space seminorm of order ${\displaystyle N=C\cdot n}$, i.e.

${\displaystyle |f|_{N}=\sup _{|\alpha |\leq N,x\in \mathbb {R} ^{d}}|(1+|x|)^{N}D^{\alpha }f(x)|,}$

and ${\displaystyle \sigma _{n}}$ a sequence of constants of factorial growth, i.e. ${\displaystyle \sigma _{n}\leq A(n!)^{B}}$ with some constants ${\displaystyle A,B}$.

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.

History

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 ${\displaystyle {\check {\text{(E0)}}}}$.^[4] This other theorem is rarely used, since ${\displaystyle {\check {\text{(E0)}}}}$ is hard to check in practice.^[3]

Axioms by Glimm and Jaffe

An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book.^[6] In this approach one assumes that one is given a measure ${\displaystyle d\mu }$ on the space of distributions ${\displaystyle \phi \in D'(\mathbb {R} ^{d})}$. One then considers a generating functional

${\displaystyle S(f)=\int e^{\phi (f)}d\mu ,\quad f\in D(\mathbb {R} ^{d})}$

which is assumed to satisfy properties OS0-OS4:

• (OS0) Analyticity. This asserts that

${\displaystyle z=(z_{1},\ldots ,z_{n})\mapsto S\left(\sum _{i=1}^{n}z_{i}f_{i}\right)}$

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

• (OS1) Regularity. This demands a growth bound for ${\displaystyle S(f)}$ in terms of ${\displaystyle f}$, such as${\displaystyle |S(f)|\leq \exp \left(C\int d^{d}x|f(x)|\right)}$. See ^[6] for the precise condition.
• (OS2) Euclidean invariance. This says that the functional ${\displaystyle S(f)}$ is invariant under Euclidean transformations ${\displaystyle f(x)\mapsto f(Rx+b)}$.
• (OS3) Reflection positivity. Take a finite sequence of test functions ${\displaystyle f_{i}\in D(\mathbb {R} ^{d})}$ which are all supported in the upper half-space i.e. at ${\displaystyle x^{0}>0}$. Denote by ${\displaystyle \theta f_{i}(x)=f_{i}(\theta x)}$ where ${\displaystyle \theta }$ is a reflection operation defined above. This axioms says that the matrix ${\displaystyle M_{ij}=S(f_{i}+\theta f_{j})}$ has to be positive semidefinite.
• (OS4) Ergodicity. The time translation semigroup acts ergodically on the measure space ${\displaystyle (D'(\mathbb {R} ^{d}),d\mu )}$. See ^[6] for the precise condition.

Nelson's axioms

These axioms were proposed by Edward Nelson.^[7] See also their description in the book of Barry Simon.^[8] Like in the above axioms by Glimm and Jaffe, one assumes that the field ${\displaystyle \phi \in D'(\mathbb {R} ^{d})}$ is a random distribution with a measure ${\displaystyle d\mu }$. This measure is sufficiently regular so that the field ${\displaystyle \phi }$ 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.