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

Abstract

We announce results about the nonperturbative mathematically rigorous construction of the $:{\varphi }_4^4:$ 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 $:{\varphi }_4^4:$ 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.

10 pages, LaTeX