# Schwinger function Contents 1 Osterwalder–Schrader axioms 1.1 Temperedness 1.2 Euclidean covariance 1.3 Symmetry 1.4 Cluster property 1.5 Reflection positivity 1.5.1 Intuitive understanding 2 Osterwalder–Schrader theorem 2.1 Linear growth condition 2.2 History 3 Other axioms for Schwinger functions 3.1 Axioms by Glimm and Jaffe 3.1.1 Relation to Osterwalder–Schrader axioms 3.2 Nelson's axioms 4 See also 5 References Osterwalder–Schrader axioms Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitean scalar field {displaystyle phi (x)} , {displaystyle xin 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 {displaystyle phi } are denoted as {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.

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 Rin SO(d)} and an arbitrary translation vector {displaystyle bin mathbb {R} ^{d}} . OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group. 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 _{bto 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})} .

Given (OS0)-(OS4), one can define Schwinger functions of {displaystyle phi } as moments of the measure {displaystyle dmu } , 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). Note however that the full quantum field theory will contain infinitely many other local operators apart from {displaystyle phi } , such as {displaystyle phi ^{2}} , {displaystyle phi ^{4}} and other composite operators built from {displaystyle phi } and its derivatives. It's not easy to extract these Schwinger functions from the measure {displaystyle dmu } and show that they satisfy OS axioms, as it should be the case.

To summarize, the axioms called (OS0)-(OS4) by Glimm and Jaffe are stronger than the OS axioms as far as the correlators of the field {displaystyle phi } are concerned, but weaker than then the full set of OS axioms since they don't say much about correlators of composite operators.

Nelson's axioms These axioms were proposed by Edward Nelson. See also their description in the book of Barry Simon. 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 dmu } . 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.