Schwinger function

Schwinger function   (Redirected from Osterwalder–Schrader theorem) Jump to navigation Jump to search In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points.[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.

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 [2] 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.[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 _{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})} .

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 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 {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. 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 {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 {displaystyle S_{+}} , which only depends on φ on the positive half-space ( {displaystyle phi _{+}} ), and {displaystyle S_{-}} which only depends upon φ on the negative half-space ( {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 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 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|_{Ccdot n},} where {displaystyle Cin mathbb {N} } is an integer constant, {displaystyle |f|_{Ccdot n}} is the Schwartz-space seminorm of order {displaystyle N=Ccdot n} , i.e. {displaystyle |f|_{N}=sup _{|alpha |leq N,xin 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 [2] claimed a stronger theorem that the axioms (E0)-(E4) by themselves imply the Wightman axioms, however their proof contained an error which could not be corrected without adding extra assumptions. Two years later [4] they published a new theorem, with the linear growth condition added as an assumption, and a correct proof. The proof in [4] is based on a complicated inductive argument (proposed also by Vladimir Glaser in [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. Ref. [4] also contains another theorem replacing (E0') by yet another assumption called {displaystyle {check {text{(E0)}}}} . This other theorem is rarely used, since {displaystyle {check {text{(E0)}}}} is hard to check in practice. See e.g. [3] for a review of these facts. Other axioms for Schwinger functions 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 dmu } 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)}dmu ,quad fin 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 Sleft(sum _{i=1}^{n}z_{i}f_{i}right)} is an entire-analytic function of {displaystyle zin 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 dmu } 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(Cint 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}),dmu )} . See [6] for the precise condition. Relation to Osterwalder–Schrader axioms 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 {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).[6] 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.[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 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.

See also Wick rotation Axiomatic quantum field theory Wightman axioms References ^ 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. ^ Jump up to: a b c d Osterwalder, K., and Schrader, R.: "Axioms for Euclidean Green’s functions," Comm. Math. Phys. 31 (1973), 83–112; 42 (1975), 281–305. ^ Jump up to: a b Kravchuk, Petr; Qiao, Jiaxin; Rychkov, Slava (2021-04-05). "Distributions in CFT II. Minkowski Space". Retrieved 2021-05-10. ^ Jump up to: a b c d e Osterwalder, Konrad; Schrader, Robert (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics. Springer Science and Business Media LLC. 42 (3): 281–305. doi:10.1007/bf01608978. ISSN 0010-3616. ^ Glaser, V. (1974). "On the equivalence of the Euclidean and Wightman formulation of field theory". Communications in Mathematical Physics. Springer Science and Business Media LLC. 37 (4): 257–272. doi:10.1007/bf01645941. ISSN 0010-3616. ^ Jump up to: a b c d 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. Retrieved 2021-05-10. ^ Simon, Barry (1974). The P(phi)_2 Euclidean (quantum) field theory. Princeton, New Jersey: Princeton University Press. ISBN 0-691-08144-1. OCLC 905864308. hide vte Quantum field theories Theories Axiomatic QFTConformal field theoryLattice field theoryLocal QFTNoncommutative QFTGauge theoryQuantum field theory in curved spacetimeString theoryThermal quantum field theoryTopological QFTTwo-dimensional conformal field theory Models Regular Born–InfeldEuler–HeisenbergGinzburg–LandauNon-linear sigmaProcaQuantum electrodynamicsQuantum chromodynamicsQuartic interactionScalar electrodynamicsScalar chromodynamicsSolerYang–MillsYang–Mills–HiggsYukawa Low dimensional 2D Yang–MillsBullough–DoddGross–NeveuSchwingersine-GordonThirringThirring–WessToda Supersymmetric Wess–ZuminoN = 1 super Yang–MillsSeiberg–WittenSuper QCD Conformal 2D free massless scalarLiouvilleMinimalPolyakovWess–Zumino–Witten Superconformal 6D (2,0)ABJMN = 4 super Yang–Mills Topological BFChern–Simons Particle theory ChiralFermiMSSMNambu–Jona-LasinioNMSSMStandard ModelStueckelberg Related Casimir effectCosmic stringHistoryLoop quantum gravityLoop quantum cosmologyOn shell and off shellQuantum chaosQuantum dynamicsQuantum foamQuantum fluctuations linksQuantum gravity linksQuantum hadrodynamicsQuantum hydrodynamicsQuantum informationQuantum information science linksQuantum logicQuantum thermodynamics See also: Template:Quantum mechanics topics Categories: Axiomatic quantum field theory

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