Hyperfunction quantum field theory

Publications of the Research Institute for Mathematical Sciences, 1976

The Wightman axioms are extended to the quantum field theory in terms of Fourier hyperfunctions. The support concept of hyperfunctions is crucial for the formulation of locality and spectral condition. The complete equivalence is proved between modified Wightman axioms for relativistic theory and modified Osterwalder-Schrader axioms for Euclidean theory.

Letters in Mathematical Physics, 1976

A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.

Letters in Mathematical Physics, 2003

This note addresses the problem of localization in quantum field theory; more specifically we contribute to the ongoing discussion about the most appropriate concept of localization which one should use in relativistic quantum field theory: through localized test functions or through the fields directly without localized test functions. In standard quantum field theory, i.e., in relativistic quantum field theory in terms of tempered distributions according to Ga˚rding and Wightman, this is done through localized test functions. In hyperfunction quantum field theory (HFQFT), i.e., relativistic quantum field theory in terms of Fourier hyperfunctions this is done through the fields themselves. In support of the second approach we show here that it has a much wider range of applicability. It can even be applied to relativistic quantum field theories which do not admit compactly supported test functions at all. In our construction of explicit models we rely on basic results from the theory of quasi-analytic functions.

Reports on Mathematical Physics, 1979

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator , no longer a standard tempered operator-valued distribution, but a non-classical operatorvalued function. We prove using this novel approach that the quantum field theory with Hamiltonian exists and that the corresponding *algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the quantum field theory model is Lorentz covariant. 1.Introduction Remind that extending the classical real numbers ℝ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution as a nonstandard point function. Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials. Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis. For further information on classical nonstandard real analysis, we refer to [5]-[7].The technique of nonstandard analysis (NSA) in constructive quantum field theory (QFT) originally were considered in P. J. Kelemen and A. Robinson papers [8],[9]. The methods of nonstandard analysis are demonstrated for the construction of the nonstandard : : model. J. Glimm and A. Jaffe's results [10],[11] were analysed from the nonstandard point of view. For further information on methods of classical nonstandard analysis in QFT, we refer to [12],[13]. However methods of classical nonstandard analysis cannot resolve this problem in physical dimension d=4, in particular for the case of simplest scalar QFT model with interaction : : , see concise explanation in ref.. [15 Introduction, Remark 1.4] and ref.[17 sect.1, Remark 1.4]. Cardinally novel approach has been developed in author papers [14]-[19].This approach based on nonconservative extension of the model theoretical NSA. In this paper we consider a somewhat different hyperfinite cutoff theory, namely the theory in a periodic box. This gives a cutoff interaction which is translation invariant, and therefore it is useful for the study of the vacuum state. In a hyperfinite interval we prove that the total Hamiltonian is self #-adjoint and has a complete set of normalizable eigenstates.

Communications in Mathematical Physics, 1976

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

2008

We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be ‘F-quantum compatible’ with a field theory if it admits a ∗-algebra of local observables for that theory which satisfies a locality condition known as ‘F-locality’. Kay’s proposal is that, in formulating algebraic quantum field theory on (M,g), F-locality should be imposed as a necessary condition on the ∗-algebra of observables. The spacetimes studied are the 2and 4-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the 4dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the Fquantum compatibility of the 2-dimensional spacelike cylinder with both m...

Model
,  ≥  quantum field theory: A nonstandard approach based on nonstandard pointwise-defined quantum fields

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator , no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian exists and that the corresponding *algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the , ≥ 2 quantum field theory models are Lorentz covariant.

Publications of the Research Institute for Mathematical Sciences, 1981

The soft resolution (3:<o,i>)» d) of the sheaf Ok,i of slowly increasing holomorphic functions of (&,/) type is constructed so that the section modules £F(o, p)(£) are Frechet nuclear spaces. Using the above resolution, we construct the mixed type Fourier hyperfunctions which take their values in Frechet spaces. § 0. Introduction Recently the theory of Fourier hyperfunctions which take their values in a Hilbert space was developed by Y. Ito and S. Nagamachi [5], and it was applied to the formulation of the axiomatic quantum field theory by S. Nagamachi and N. Mugibayashi [9]. On the other hand P. D. F. Ion and T. Kawai [4] developed the theory of hyperfunctions which take their values not in a Hilbert space but in a Frechet space. There exist another kind of Fourier hyperfunctions which were announced in T. Kawai [6] to be published as modified Fourier hyperfunctions but actually was not published. It turned out that the new type Fourier hyperfunctions are useful in order to give the equivalent Euclidean formulation of the quantum field theory. We call this new Fourier hyperfunction the second type Fourier hyperfunction in distinction from the old one, the first type Fourier hyperfunction which was developed in T. Kawai [6], In S. Nagamachi and N. Mugibayashi [10], the mixed type Fourier hyperfunctions were used for above purpose. The mixed type Fourier hyperfunctions are those Fourier hyperfunctions which are of the first type in some variables and of the second type in

Reviews in Mathematical Physics, 1992

In the context of a linear model (the covariant Klein Gordon equation) we review the mathematical and conceptual framework of quantum field theory on globally hyperbolic spacetimes, and address the question of what it might mean to quantize a field on a non globally hyperbolic spacetime. Our discussion centres on the notion of F-locality which we introduce and which asserts there is a net of local algebras such that every neighbourhood of every point contains a globally hyperbolic subneighbourhood of that point for which the field algebra coincides with the algebra one would obtain were one to regard the subneighbourhood as a spacetime in its own right and quantize — with some choice of time-orientation — according to the standard rules for quantum field theory on globally hyperbolic spacetimes. We show that F-locality is a property of the standard field algebra construction for globally hyperbolic spacetimes, and argue that it (or something similar) should be imposed as a condition...