Since there are indications (from string theory and concrete models) that one must consider relativistic quantum field theories with a fundamental length the question of a suitable framework for such theories arises. It is immediately evident that quantum field theory in terms of tempered distributions and even in terms of Fourier hyperfunctions cannot meet the (physical) requirements. We argue that quantum field theory in terms of ultra-hyperfunctions is a suitable framework. For this we propose a set of axioms for the fields and for the sequence of vacuum expectation values of the fields, prove their equivalence, and we give a class of models (analytic, but not entire functions of free fields).
The characteristic functions and the invariants of supermatrices are studied. It is shown that the Euclidean algorithm is useful in obtaining a system of invariants.Proof: The proof is easy. Let A(X) = A[X]"", = {f IgVEA[X ],gEA o [X ],gElf!JJ} be the ring of quotients of A [X]. The addition and the multiplication of A(X) are defined in a usual way, and J.Ig l =/zlg2inA(X)ifandonlyifJ.g2 =gl/zinA[X].A polynomialjEA [X] is invertible iff #0, because/ -fis nilpotent. The even part Ao (X) = Ao [X 1.9 = {f Igl!.gEA o [X ],gElf!JJ} 2726
The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not contain C 00 functions with compact support. In spite of this defect the support concept of //-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory. § 1. Introduction
Closure of field operators, asymptotic Abelianness, and vacuum structure in hyperfunction quantum field theory The choice of the class E' of generalized functions on space-time in which to formulate general relativistic quantum field theory (QFT) is discussed. A first step is to isolate a set of conditions on E' that allows a formulation ofQFT in otherwise the same way as the original proposal by Wightman [Ark. Fys. 28, 129 (1965)], where E' is the class of tempered distributions. It is stressed that the formulation ofQFT in which E' equals the class of Fourier hyperfunctions on space-time meets the following requirements: (A) Fourier hyperfunctions generalize tempered distributions thus allowing more singular fields as suggested by concrete models; (B) Fourier hyperfunction quantum fields are localizable both in space-time and in energy-momentum space thus allowing the physically indispensable standard interpretation of Poincare covariance, local commutativity, and localization of energy-momentum spectrum; and (C) in Fourier hyperfunction quantum field theory almost all the basic structural results of "standard" QFT (existence of a PCT operator, spin-statistics theorems, existence of a scattering operator, etc.) hold. Finally, a short introduction to that part of Fourier hyperfunction theory needed in this context is given.
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