Analytic Properties of Nonstrictly Localizable Fields

Abstract

A class of test functions ``minimal with respect to causality'' is introduced. The corresponding fields are called local. Tempered and strictly localizable fields are local, but there is a large class of fields that are local but not strictly localizable. For local fields, the analytic properties of vacuum expectation values are studied. The local fields that are not strictly localizable are characterized by an arbitrary fast increase of Wightman functions near the light cone. With an adequate definition of local commutativity, other properties of tempered and strictly localizable fields seem to remain valid.

Distributional boundary values in the dual spaces of spaces of type 𝒮

“…According to property (C), there exists an n-tuple A =04,, ,A Λ ), Λ>0, /==l, ,n, such that A EC and φ G y α , Λ . Since C is a cone then y = (1/2)A G C We now proceed exactly as in the proof of [4,Lemma 2] to obtain the desired result; for we have obtained in our present setting exactly the properties that permit the proof of [4,Lemma 2] to hold. Further the method holds equally well for T a , a = (a u ,α n ), α, g 1, / = 1, ,rt.…”

“…GeΓfand and Shilov [10] have used these spaces for a study of the Cauchy problem; while Constantinescu [4] and Rieckers [11] have used them in their studies of quantum field theory.…”

“…Constantinescu [4] constructs local fields, which are a category of fields larger than the strictly localizable ones, and proves that the vacuum expectation values in a local field theory are boundary values of functions analytic in a tube domain corresponding to the forward light cone. These vacuum expectation values are in fact distributional boundary values of the analytic functions in the weak topology of the dual spaces of the spaces of type ϊf and are distributions in these dual spaces.…”

Quantum Field Theory in Terms of Fourier Hyperfunctions

“…The strictly localizability of fields A(f) connects intimately with the fact that f belongs to a function space which contains C°° functions with compact support. Such classes of test functions have been kept more or less in the concrete attempts to extend the Wightman axioms for quantum field theory made so far by several authors [1,2,8]. An abstract argument on the class of fields incorporated with the locality in extended sense has been given by Lomsadze and his coworkers [15].…”

“…Several authors have attempted to extend the Wightman axioms for quantum field theory so as to include into the theory a wider class of fields which, owing to singular (or nonrenormalizable) interactions, are no longer described by tempered distributions [1,2,8]. In the first paper of the present series [11], which will be quoted as NM I, we succeeded to formulate the quantum field theory in terms of Fourier hyperfunctions which had been studied extensively by Kawai [10].…”