arXiv subjects

Edward P. Osipov

Publications and source records attributed to Edward P. Osipov.

5 recordsLinked to original sources

arXiv1996

The $:ϕ^4_4:$ quantum field theory, II. Integrability of Wick kernels

We continue the construction of the $:ϕ^4_4:$ quantum field theory. In this paper we consider the Wick kernel of the interacting quantum field. Using the complex structure and the Fock-Bargmann-Berezin-Segal integral representation we prove that this kernel defines a unique operator--valued generalized function on the space $\Sc^α(\R^4)$ for any $α<6/5,$ i.e. the constructed quantum field is the generalized operator-valued function of localizable Jaffe class. The same assertion is valid for the outgoing quantum field. These assertions about the quantum field allow to construct the Wightman functions, the matrix elements of the quantum scattering operator and to consider their properties (positivity, spectrality, Poincare invariance, locality, asymptotic completeness, and unitarity of the quantum scattering).

hep-th↗

arXiv1996

Complex structure and the construction of the $:ϕ^4_4:$ quantum field theory in four-dimensional space-time

We announce results about the nonperturbative mathematically rigorous construction of the $:\!ϕ^4_4\!:$ quantum field theory in four-dimensional space-time. The complex structure of solutions of the classical nonlinear (real-valued) wave equation and quantization are closely connected among themselves and allow to construct non-perturbatively the quantum field theory with interaction $:\!ϕ^4_4\!:$ in four-dimensional space-time. We consider vacuum averages, in particular, we construct Wightman functions and matrix elements of the scattering operator as generalized functions for finite energies. The constructed theory is obviously nontrivial.

hep-th↗

arXiv1996

The $:ϕ^4_4:$ quantum field theory, I. Wave operator, holomorphity and Wick kernel

With the help of the complex structure and the wave operator of the nonlinear classical Klein-Gordon equation with the interaction $u^4_4$ we define the Wick kernel of the interacting quantum field in four-dimensional space-time and consider its properties. In particular, the diagonal of this Wick kernel is (real) solutions of the classical nonlinear Klein-Gordon equation with the interaction $u^4_4.$

funct-an↗

arXiv1996

Complex structure and solutions of classical nonlinear equation with the interaction $u^4_4$

We consider the (real) nonlinear wave equation $$\Box u + m^2 u + λu^3 = 0, \quad m > 0, \quad λ> 0, $$ on four-\-dimensional Minkowski space. We introduce the complex structure and show that the (nonlinear) operator of dynamics, the wave and scattering operators define complex analytic maps on the space of initial Cauchy data with finite energy. In other words, let $R(φ, π) = φ+ iμ^{-1}π$ be the map of initial data on the positive frequency part of the solution of the free Klein-\-Gordon equation with these initial data. The operators $RU(t)R^{-1},$ $RWR^{-1},$ and $RSR^{-1}$ are defined correctly and are complex analytic on the complex Hilbert space $H^1({\R}^3,\C).$

funct-an↗

arXiv1996

Quantum Interaction $ϕ^4_4$: the Construction of Quantum Field defined as a Bilinear Form

We construct the solution $ϕ(t,{\bf x})$ of the quantum wave equation $\Boxϕ+ m^2ϕ+ λ:\!\!ϕ^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!ϕ^3(t,{\bf x})\!: $ is defined as the normal ordered product with respect to the free $in$-field. The constructed solution is correctly defined as a bilinear form on $D_θ\times D_θ$, where $D_θ$ is a dense linear subspace in the Fock space of the free $in$-field. On $D_θ\times D_θ$ the diagonal Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.

hep-th↗