On local field products in special Wightman theories

Abstract

We shall try to define local field products under assumptions imposed only on the fourpoint-function. This idea is based on the work of Schlieder and Seiler [1].In our framework we shall prove that the two-point-function carries the strongest singularity whenever two arguments in a Wightman function coincide. This will be generalized to the case when more arguments coincide. We shall define "regulated" rc-point-functions and study their properties in detail. This will lead us to the definition of arbitrarily high powers of the field-operators as operatorvalued distributions over ^(1R 4

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

“…This continuity property will then be translated into familiar language and it means essentially: When k of the Schwinger points come together in any n-point function then they produce a singularity which is independent of the remaining n -k points, as long as they stay apart. This is a property which seems to be closely related to the existence of a Wilson-Zimmermann expansion [5,6].…”

“…S 2 , ...)• The necessity of (6), i.e. the proof that the distribution estimate in Theorem 3.7 leads to the estimate (6) for the function 6 n can be shown in the same way as in [17], Theorem 4.1.…”

All massless, scalar fields with trivialS-matrix are wick-polynomials

“…Therefore F n (λξ) is a polynomial in λ and as shown in [5] we can take N = D. Now the intersection of τ^_ x with {ξ\ \\ξ\\ <R n } is open so all but finitely many a^ξ) vanish identically. Therefore F n (ξ) is a polynomial and we get the following representation:…”

“…This is the basic assumption for a series of papers -initiated by Schlieder and Seiler [4] -on Wilson-Zimmermann-Expansions. We refer to [5] for the proof of the following property of the rc-point-function W n of φ: Lemma 3. For every n^2 the functions…”

Lokale Algebren und Operatorprodukte am Punkt

“…Ein anderer, axiomatischer Ansatz zur Definition von Normalprodukten auf der Grundlage der Operatorproduktentwicklung stammt von Baumann [Bau75] auf der Grundlage der Ergebnisse von Schlieder und Seiler [SSe73]. Hier wird das Normalprodukt nicht durch Subtraktion von Termen im raumartigen Limes definiert, sondern durch Multiplikation des raumartigen Produkts mit einem geeigneten c-Zahl-Faktor, so daß der resultierende Ausdruck auf einer Nullumgebung holomorph ist.…”