Papers by Shigeaki Nagamachi
Journal of Mathematical Physics, Dec 1, 1992
A theory of Hilbert superspace over an infinite dimensional Grassmann algebra Λ is given. Axioms ... more A theory of Hilbert superspace over an infinite dimensional Grassmann algebra Λ is given. Axioms of Hilbert superspace are given and it is proven that a Hilbert superspace is isomorphic to H⊗Λ for some Hilbert space H. A natural topology on it called ε topology is defined and continuous Λ-linear operators, especially unitary operators are studied. A Hilbert subsuperspace is defined and it is proven that its orthogonal complement is a Hilbert subsuperspace.
Letters in Mathematical Physics, 1988
Superdistributions on superspace over an algebra A of generalized supernumbers are defined as con... more Superdistributions on superspace over an algebra A of generalized supernumbers are defined as continuous A-linear mappings to A from the test function space of G8-functions with compact support. A superdistribution is expressed in a form of the standard expansion whose coefficients are usual A-valued distributions.
Progress of Theoretical and Experimental Physics, 1973
The theory of neutral scalar field in interaction with a point source is studied exhaustive• ly i... more The theory of neutral scalar field in interaction with a point source is studied exhaustive• ly in the light of the algebraic formulation of quantum field theory. Emphases are laid on observing again the case of an extended source from the present viewpoint, and exhibiting the equivalence of two definitions of the renormalized Hilbert space by GNS construction and by infinite tensor product.
Journal of Mathematical Physics, May 1, 2008
We propose to look at (a simplified version of) Heisenberg's fundamental field equation (see [2])... more We propose to look at (a simplified version of) Heisenberg's fundamental field equation (see [2]) as a relativistic quantum field theory with a fundamental length, as introduced in [1] and give a solution in terms of Wick power series of free fields which converge in the sense of ultrahyperfunctions but not in the sense of distributions. The solution of this model has been prepared in [5] by calculating all n-point functions using path integral quantization. The functional representation derived in this part is essential for the verification of our condition of extended causality. The verification of the remaining defining conditions of a relativistic quantum field theory is much simpler through the use of Wick power series. Accordingly in this second part we use Wick power series techniques to define our basic fields and derive their properties.
Journal of mathematics, Tokushima University, 1975
Journal of Mathematical Physics, Feb 1, 2010
Some time ago we proposed a relativistic quantum field theory with a fundamental length in terms ... more Some time ago we proposed a relativistic quantum field theory with a fundamental length in terms of tempered ultrahyperfunctions ͓Brüning, E. and Nagamachi, S., "Relativistic quantum field theory with a fundamental length," J. Math. Phys. 45, 2199 ͑2004͒; e-print arXiv:math-ph/0804.1663; "Solutions of a linearized model of Heisenberg's fundamental equation II," J. Math. Phys. 49, 052304 ͑2008͔͒. The definition of the fundamental length by Brüning and Nagamachi ͓"Relativistic quantum field theory with a fundamental length," J. Math. Phys. 45, 2199 ͑2004͔͒ seems to depend on the Lorentz frame which is used to define it. In this article we show that this is actually not the case. In addition we discuss in some detail the geometric and analytic realization of the fundamental length and point out some important difference to standard relativistic quantum field theory in the sense of Gårding and Wightman.
Publications of The Research Institute for Mathematical Sciences, 1981
The soft resolution G£?'o t p), d) of the sheaf Ok,i of rapidly decreasing holomorphic functions ... more The soft resolution G£?'o t p), d) of the sheaf Ok,i of rapidly decreasing holomorphic functions of (k, /) type is constructed. Using the above resolution, we prove ^(V^O^i) § I. Introduction In the first part of the present paper (S. Nagamachi [4]) ,which will be referred to as [I], we defined the mixed type Fourier hyperfunctions which take values in a Frechet space E, The purpose of this second part is to prove that the space H£(V, E Ojc,i) of ^-valued Fourier hyperf unctions with support contained in a compact set K is isomorphic to the space L (O ki t (K) , E) of continuous linear mappings of 0 &>Z (/C) into E. We proved this theorem in [I] only for E=C (Theorem 5.13 of [I]). In Section 2, we study the Fourier transformation for slowly increasing C°° functions and rapidly decreasing distributions. In Section 3, we prepare the theory of cohomology with bounds in an appropriate form. In Section 4, we construct a soft resolution of the sheaf Ok.i of rapidly decreasing holomorphic functions (Theorem 4. 9) , where S[ 0tP) is the sheaf subordinate to the presheaf {S[^P) (•$)} of (Q,p)-forms whose coefficients are rapidly decreasing distributions in J2
Reports on Mathematical Physics, Oct 1, 1979
Letters in Mathematical Physics, Jul 1, 1987
An infimte-dimensional topological algebra is defined as an inductive limit of fimte-dimenslonal ... more An infimte-dimensional topological algebra is defined as an inductive limit of fimte-dimenslonal a-commutative Banach algebras. This algebra has some desirable properties for the algebra of supernumbers, on which we can develop a satisfactory theory of superanalysis.
Reviews in Mathematical Physics, Aug 20, 2017
It turns out that a parametrization of degenerate density matrices requires a parametrization of ... more It turns out that a parametrization of degenerate density matrices requires a parametrization of F = U (n)/(U (k 1) × U (k 2) × • • • × U (k m)) n = k 1 +• • •+k m where U (k) denotes the set of all unitary k × k-matrices with complex entries. Unfortunately the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods directly, but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra back ground and conclude with two concrete examples.
Journal of Mathematical Physics, May 1, 1993
On a Fock superspace, a linear canonical supertransformation is implemented by an inner-product-p... more On a Fock superspace, a linear canonical supertransformation is implemented by an inner-product-preserving operator which is necessarily not continuous.
Proceedings of the Japan Academy. Series A, Mathematical sciences, 1983
On developpe la theorie des matrices dont des entrees sont des elements d'une algebre τ-commu... more On developpe la theorie des matrices dont des entrees sont des elements d'une algebre τ-commutative et on etudie les groupes et les algebres constitues par ces matrices
Progress of Theoretical and Experimental Physics, Sep 1, 1973
The measure-theoretic formulation of canonical commutation relations is applied to the (,P2) •+1 ... more The measure-theoretic formulation of canonical commutation relations is applied to the (,P2) •+1 quantum field theory for s~3. This formulation makes it possible to understand profoundly the structure of the mass shif(model in a simple way. In particular the van Hove phenomenon of the model is exhibited, and the transition to the inequivalent representation for boson with a different mass is performed.
Progress of Theoretical and Experimental Physics, Jun 1, 1975
Conditions are given for a locally Markov field as a limit of Markov fields to have a Hamiltonian... more Conditions are given for a locally Markov field as a limit of Markov fields to have a Hamiltonian H*. It is shown that H* is unitarily related to the total Hamiltonian minus the vacuum energy, when the latter is properly defined. The present formulation is conveniently illustrated by constructing the Euclidean version of the theory of neutral scalar field interacting with a fixed source.
Journal of Mathematical Physics, 2001
This paper addresses the following problem of relativistic quantum field theory: Given a relativi... more This paper addresses the following problem of relativistic quantum field theory: Given a relativistic quantum field, construct a net of local observable algebras over space–time with “natural” properties. A few years ago we started a project which suggests to look at this problem in the framework of relativistic quantum field theory in terms of Fourier hyperfunctions. Accordingly we present the relevant analyticity results, some modular aspects of our theory for the Bisognano–Wichman argument, and a concrete suggestion for the definition of the local observable algebras. Finally, we construct a class of models of hyperfunction quantum fields for which the algebras of bounded operators assigned to nonempty regions in space–time are not trivial. It is remarkable that our models are not tempered quantum fields and do not admit “test functions” of compact support.
Journal of Mathematical Physics, Sep 1, 1986
The analysis over σ-commutative algebras (generalized supercommutative algebras), that is, differ... more 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.
Communications in Mathematical Physics, Jun 1, 1976
The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function spa... more 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.
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Papers by Shigeaki Nagamachi