  We announce results about the nonperturbative mathematically rigorous
construction of the $:\!\phi^4_4\!:$ 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 $:\!\phi^4_4\!:$ 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.
