Shigeaki Nagamachi

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC∞ functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.

Hyperfunction quantum field theory: Basic structural results

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 of QFT 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 of QFT 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 ‘‘standar...

Relativistic quantum field theory with a fundamental length

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).

Usage of infinite-dimensional nuclear algebras in superanalysis

An infinite-dimensional topological algebra is defined as an inductive limit of finite-dimensional σ-commutative Banach algebras. This algebra has some desirable properties for the algebra of supernumbers, on which we can develop a satisfactory theory of superanalysis.

Lie groups and Lie algebras with generalized supersymmetric parameters

Matrices with σ‐symmetric parameters (the most general extension of supersymmetric parameters) are investigated. The superdeterminants of such matrices are defined. Lie groups consisting of these matrices and their Lie algebras are studied.

Hyperfunction quantum field theory II. Euclidean Green's functions

The axioms for Euclidean Greens functions are extended to hyperfunction fields without being supplemented by any condition like the linear growth condition of Osterwalder and Schrader.

Hyperfunctions and renormalization

A condition is found so that the Wick‐ordered power series of scalar fields in four‐dimensional space‐time is defined as a Fourier‐hyperfunction field, and the derivative coupling model is investigated in the framework of hyperfunction quantum field theory.

Characteristic functions and invariants of supermatrices

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.

Analysis on generalized superspaces

The analysis over σ‐commutative algebras (generalized supercommutative algebras), that is, differentiation and integration for functions defined on superspace over a σ‐commutative algebra, is studied.

The Haag-Ruelle formulation of scattering in hyperfunction quantum field theory

Abstract The Kallen-Lehmann representation for two-point Wightman Fourier hyperfunctions and the cluster property for truncated vacuum expectation values are established in the framework of hyperfunction quantum field theory. With some additional assumptions these properties allow one to verify the Haag-Ruelle asymptotic conditions.