Exact Solitary Wave Solutions of the Coupled Nonlinear Scalar Field Equations

Because of the arbitrary property of the singular manifold, the usual singularity analysis can be extended to a different form. Using a nonstandard truncation approach, five special types of exact solution of the coupled nonlinear scalar field equations are obtained.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Covariant fields: Poincaré group representations and metric structure in the space of quantum states

The representations of the Poincaré group realized over the space of covariant fields transforming according to any irreducible representationD ^(m,n) of the Lorentz group are constructed explicitly with reference to a helicity basis. The representation is indecomposable in the massless case. The form of this representation together with the invariance of two-point Wightman functions of the field (which follows from a weak set of axioms) determines the metric structure in the space of quantum states of the field. This structure is explicitly determined for generalD ^(m,n). Certain particular cases (especially the symmetric traceless tensor field) are discussed in detail. Finally we consider the representation pertaining to massive fields, and examine the passage to the limit of vanishing mass. We present a limiting procedure which leads from the unitary representation of the massive field to the indecomposable non-unitary representation of the massless field.     

Nonstandard Fock spaces

A theory of nonstandard inner product spaces is developed using methods of nonstandard analysis. Various results concerning nonstandard operators and their spectra are proved. The theory is applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Moreover, a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Axiomatic Conformal Theory in Dimensions >2 and AdS/CT Correspondence

We formulate axioms of conformal theory (CT) in dimensions >2 modifying Segal’s axioms for two-dimensional CFT. (In the definition of higher-dimensional CFT, one includes also a condition of existence of energy-momentum tensor.) We use these axioms to derive the AdS/CT correspondence for local theories on AdS. We introduce a notion of weakly local quantum field theory and construct a bijective correspondence between conformal theories on the sphere S^d and weakly local quantum field theories on \({H^{d+1}}\) that are invariant with respect to isometries. (Here \({H^{d+1}}\) denotes hyperbolic space = Euclidean AdS space.) We give an expression of AdS correlation functions in terms of CT correlation functions. The conformal theory has conserved energy-momentum tensor iff the AdS theory has graviton in its spectrum.     

Program of quantization of nonlinear theories

In this paper, we concentrate our attention upon the troubles in the construction of nontrivial models in nonlinear quantum field theory which may be connected with the already known axioms of quantum field theory. We propose to come back to classical nonlinear (quasilinear) field theory for obtaining some information how to change these axioms. Simultaneously, we propose a program of quantization of these theories using Riemann waves, multiple simple waves, and simple elements.On leave of absence from the Institute of Philosophy and Sociology, Polish Academy of Science, 00-330 Warsaw, Nowy wiat 72, Poland.     

Nonstandard interactions in plasmon decay to a neutrino pair in a strongly magnetized medium

The neutrino luminosity of a stellar medium because of plasmon decay to a neutrino pair via nonstandard tensor interaction in a degenerate electron gas subjected to the effect of a magnetic field such strong that the electrons of the gas are in the lowest Landau level is calculated. Relative limits on nonstandard coupling constants are obtained from a comparison of the results of this calculation with the neutrino luminosity generated in the respective standard process proceeding under the same conditions.     

Upper entropy axioms and lower entropy axioms

The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak upper entropy axioms and weak lower entropy axioms. Their conditions are weaker than those of Shannon–Khinchin axioms and Tsallis axioms, while these conditions are stronger than those of the axiomatics based on the first three Shannon–Khinchin axioms. The subadditivity and strong subadditivity of entropy are obtained in the new axiomatics. Tsallis statistics is a special case of satisfying our axioms. Moreover, different forms of information measures, such as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can be unified under the same axiomatics.     

P(ϕ) 2 Quantum Field Theories and Segal’s Axioms

The purpose of this paper is to show that P(ϕ) ₂ Euclidean quantum field theories satisfy axioms of the type advocated by Graeme Segal.     

Nonexistence of quantum fields associated with two-dimensional spacelike planes

It is shown that in a relativistic quantum field theory satisfying Wightman's axioms, there are no nontrivial field-like operators, or even bilinear forms, associated to a two (or less)-dimensional spacelike plane in Minkowski space. This generalizes Wightman's result that fields can not be defined as operators at a point and stands in contrast to Borchers' result that field operators can be associated with one-dimensional timelike planes.     

Euclidean formulation of quantum field theory without positivity

General properties of local quantum field theories (QFT) without positivity are discussed in connection with their euclidean formulation. Modified euclidean axioms for local QFT's without positivity are presented, which allow us to recover by analytic continuation Wightman functions satisfying the modified Wightman axioms for indefinite metric QFT's.     

A note on the Hilbert space formalism of quantum mechanics

In [5] M.J. Maczyński showed that the Hilbert space formalism of quantum mechanics (see [3]) can be derived from a set of seven axioms involving only the probability function p(A, α, E) and the complex field postulate. The purpose of this note is to give the next axiom that is equivalent to the analogous real field postulate.     

Ultradistributions and quantum fields: Fourier-Laplace transforms and boundary values of analytic functions

The distribution theory is the mathematical framework of the axiomatic quantum field theory of A.S. Wightman. The axioms are satisfied in the case of the free field and in some non-trivial models studied in the constructive quantum field theory introduced by J. Glimm and A. Jaffe (two and three dimensions). No non-trivial example in four dimensions is known. The vacuum expectation values in the Wightman theory can have at most polynomial increase in momentum space. A. Jaffe has extended the axioms in order to allow non-polynomial increase in momentum space. In this paper we discuss the ultradistribution framework which is the most general framework for Jaffe fields (strictly localizable fields). The ultradistributions have been introduced by A. Beurling, G. Björck and independently by C. Roumieu. Ultradistribution theory is a natural generalization of the distribution theory. We study the Fourier-Laplace transform of ultradistributions, extenting results of A. Jaffe [8,9] in several directions. A Bochner-Schwartz theorem for ultradistributions is also shown to be valid. We expect ultradistribution theory to play a role in constructive quantum field theory.     

Determination of the surface anisotropy contribution to the magnetic anisotropy field of a nanocrystalline barium ferrite powder at various temperatures

The temperature dependence of the average anisotropy field of a nanocrystalline barium hexaferrite powder was studied by treating remanent-magnetization curves with inclusion of thermal fluctuations. The contribution of surface anisotropy was isolated; its nonstandard temperature dependence is intimately related to the specific features of formation of near-surface regions in ultrathin particles.     

On the axiomatic approach to quasi-classical field theory

We study the connection between quasi-classical field theory and axiomatic statements of the quantum field theory, Schwinger source theory, and the Lehmann-Symanzik-Zimmermann (LSZ) formalism. The classical Schwinger source is connected with the classical field; the LSZ R-function is connected with the quantum field operator. The axioms of the quantum field theory are written in the context of the quasi-classical expansion. In the considered approach, the stationary action principle and canonical commutation relations for field operators are obtained as corollaries and are not postulated as initial statements of the theory.     

Gödel axiom mappings in special relativity and quantum-electromagnetic theory

Exponential mappings into an imaginary space or number field for the axioms of a theory, which are in the form of propositional constants and variables, make possible: (a) an understanding of the meaning and differences between the Lorentz transformation constants, such that their product is still equal to one, but the axioms at each end of the transformations are logically inverse and separately consistent; (b) an interpretation of the psi function phase factor which is part of the axiomE=hf; (c) the unification of the quantum-mechanical psi function and the electromagnetic wave function. Thus, those statements whose mechanisms are unknown (the axioms of the theory) are to be assigned the axiom propositional number symbol and are to be associated with the complex probability eⁱ, which is a uniform factor of the energy equations expressing the physical state. Such probabilistic axiom functions can be associated with both the special theory of relativity and the quantum-electromagnetic theory.     

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.     

The existence of the: exp ϕ:4 quantum field theory in a finite volume

The quantum field theory with an exponential interaction in the four-dimensional space-time in a periodic box is considered. It is shown that the vacuum expectation values exist as tempered distributions and satisfy some Wightman axioms.     

A rigorous formulation of LSZ field theory

An exact formulation of LSZ field theory is given. It is based on the Wightman axioms, asymptotic completeness, and a technical assumption stating the existence of retarded products of field operators. Reduction formulae are derived directly from the strong asymptotic condition. The GLZ-theorem, which states the conditions under which a given set of retarded functions defines a field theory, is formulated and proved in a rigorous way.