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antum 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 1 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 2 schwinger functions are also referred to as euclidean correlation functions osterwalder schrader axioms edit 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 2 are numbered e0 e4 and have the following meaning e0 temperedness e1 euclidean covariance e2 positivity e3 symmetry e4 cluster property temperedness edit 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 edit euclidean covariance axiom e1 says that schwinger functions transform covariantly under rotations and translations namely s n x 1 x n s n r x 1 b r x n b displaystyle s_ n x_ 1 ldots x_ n s_ n rx_ 1 b ldots rx_ n b for an arbitrary rotation matrix r s o d displaystyle r in so d and an arbitrary translation vector b r d 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 edit symmetry axiom e3 says that schwinger functions are invariant under permutations of points s n x 1 x n s n x π 1 x π n displaystyle s_ n x_ 1 ldots x_ n s_ n x_ pi 1 ldots x_ pi n where π displaystyle pi is an arbitrary permutation of 1 n 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 edit cluster property e4 says that schwinger function s p q displaystyle s_ p q reduces to the product s p s q displaystyle s_ p s_ q if two groups of points are separated from each other by a large constant translation lim b s p q x 1 x p x p 1 b x p q b s p x 1 x p s q x p 1 x p q 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 x 0 0 displaystyle x 0 0 hyperplane while the vector b displaystyle b is parallel to it x 1 0 x p 0 0 x p 1 0 x p q 0 0 b 0 0 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 edit 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 τ 1 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 x displaystyle x let x θ displaystyle x theta be the reflected point about the τ 0 hyperplane then m n d d x 1 d d x m d d y 1 d d y n s m n x 1 x m y 1 y n f m x 1 θ x m θ f n y 1 y n 0 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 τ 0 displaystyle tau 0 hyperplane s 2 n x 1 x n x n θ x 1 θ 0 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 intuitive understanding edit 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 d ϕ f ϕ x f ϕ x θ e s ϕ d ϕ 0 ϕ τ 0 ϕ 0 d ϕ f ϕ e s ϕ ϕ τ 0 ϕ 0 d ϕ f ϕ θ e s ϕ displaystyle 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 displaystyle s_ which only depends on φ on the positive half space ϕ displaystyle phi _ and s displaystyle s_ which only depends upon φ on the negative half space ϕ displaystyle 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 osterwalder schrader theorem edit the osterwalder schrader theorem 4 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 edit this condition called e0 in 4 asserts that when the schwinger function of order n displaystyle n is paired with an arbitrary schwartz test function f displaystyle f which vanishes at coincident points we have the following bound s n f σ n f c n displaystyle s_ n f leq sigma _ n f _ c cdot n where c n displaystyle c in mathbb n is an integer constant f c n displaystyle f _ c cdot n is the schwartz space seminorm of order n c n displaystyle n c cdot n i e f n sup α n x r d 1 x n d α f x displaystyle f _ n sup _ alpha leq n x in mathbb r d 1 x n d alpha f x and σ n displaystyle sigma _ n a sequence of constants of factorial growth i e σ n a n b displaystyle sigma _ n leq a n b with some constants a b 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 edit 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 e0 ˇ displaystyle check text e0 4 this other theorem is rarely used since e0 ˇ displaystyle check text e0 is hard to check in practice 3 other axioms for schwinger functions edit axioms by glimm and jaffe edit 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 d μ displaystyle d mu on the space of distributions ϕ d r d displaystyle phi in d mathbb r d one then considers a generating functional s f e ϕ f d μ f d r d 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 z z 1 z n s i 1 n z i f i 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 z r n displaystyle z in mathbb r n for any collection of n displaystyle n compactly supported test functions f i d r d displaystyle f_ i in d mathbb r d intuitively this means that the measure d μ displaystyle d mu decays faster than any exponential os1 regularity this demands a growth bound for s f displaystyle s f in terms of f displaystyle f such as s f exp c d d x f x 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 s f displaystyle s f is invariant under euclidean transformations f x f r x b displaystyle f x mapsto f rx b os3 reflection positivity take a finite sequence of test functions f i d r d displaystyle f_ i in d mathbb r d which are all supported in the upper half space i e at x 0 0 displaystyle x 0 0 denote by θ f i x f i θ x displaystyle theta f_ i x f_ i theta x where θ displaystyle theta is a reflection operation defined above this axioms says that the matrix m i j s f i θ f j 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 d r d d μ displaystyle d mathbb r d d mu see 6 for the precise condition relation to osterwalder schrader axioms edit although the above axioms were named by glimm and jaffe os0 os4 in honor of osterwalder and schrader they are not equivalent to the osterwalder schrader axioms given os0 os4 one can define schwinger functions of ϕ displaystyle phi as moments of the measure d μ displaystyle d mu 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 6 note however that the full quantum field theory will contain infinitely many other local operators apart from ϕ displaystyle phi such as ϕ 2 displaystyle phi 2 ϕ 4 displaystyle phi 4 and other composite operators built from ϕ displaystyle phi and its derivatives it s not easy to extract these schwinger functions from the measure d μ displaystyle d mu and show that they satisfy os axioms as it should be the case nelson s axioms edit 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 ϕ d r d displaystyle phi in d mathbb r d is a random distribution with a measure d μ displaystyle d mu this measure is sufficiently regular so that the field ϕ 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 see also edit wick rotation axiomatic quantum field theory wightman axioms references edit streater r f wightman a s 2000 pct spin and statistics and all that princeton n j princeton university press isbn 978 0 691 07062 9 oclc 953694720 1 2 3 4 osterwalder k and schrader r axioms for euclidean green s functions comm math phys 31 1973 83 112 42 1975 281 305 1 2 kravchuk petr qiao jiaxin rychkov slava 2021 04 05 distributions in cft ii minkowski space arxiv 2104 02090v1 hep th 1 2 3 4 osterwalder konrad schrader robert 1975 axioms for euclidean green s functions ii communications in mathematical physics 42 3 springer science and business media llc 281 305 bibcode 1975cmaph 42 281o doi 10 1007 bf01608978 issn 0010 3616 s2cid 119389461 glaser v 1974 on the equivalence of the euclidean and wightman formulation of field theory communications in mathematical physics 37 4 springer science and business media llc 257 272 bibcode 1974cmaph 37 257g doi 10 1007 bf01645941 issn 0010 3616 s2cid 121257568 1 2 3 4 glimm james jaffe arthur 1987 quantum physics a functional integral point of view new york ny springer new york isbn 978 1 4612 4728 9 oclc 852790676 nelson edward 1973 01 01 construction of quantum fields from markoff fields journal of functional analysis 12 1 97 112 doi 10 1016 0022 1236 73 90091 8 issn 0022 1236 simon barry 1974 the p phi _2 euclidean quantum field theory princeton new jersey princeton university press isbn 0 691 08144 1 oclc 905864308 v t e quantum field theories theories algebraic qft axiomatic qft conformal field theory lattice field theory noncommutative qft gauge theory qft in curved spacetime string theory supergravity thermal qft topological qft two dimensional conformal field theory models regular born infeld euler heisenberg ginzburg landau non linear sigma proca quantum electrodynamics quantum chromodynamics quartic interaction scalar electrodynamics scalar chromodynamics soler yang mills yang mills higgs yukawa low dimensional 2d yang mills bullough dodd gross neveu schwinger sine gordon thirring thirring wess toda conformal 2d free massless scalar liouville logarithmic minimal polyakov wess zumino witten supersymmetric 4d n 1 n 1 super yang mills seiberg witten super qcd wess zumino superconformal 6d 2 0 abjm n 4 super yang mills supergravity pure 4d n 1 4d n 1 4d n 8 higher dimensional type i type iia type iib 11d topological bf chern simons particle theory chiral fermi mssm nambu jona lasinio nmssm standard model stueckelberg related casimir effect cosmic string history loop quantum gravity loop quantum cosmology on shell and off shell quantum chaos quantum dynamics quantum foam quantum fluctuations links quantum gravity links quantum hadrodynamics quantum hydrodynamics quantum information quantum information science links quantum logic quantum thermodynamics see also template quantum mechanics topics retrieved from https en wikipedia org w index php title schwinger_function oldid 1355851000 category axiomatic quantum field theory hidd...