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

TL;DR

This paper presents a rigorous nonperturbative construction of the $:\phi^4_4:$ quantum field theory in four-dimensional space-time, linking classical solution structures with quantum quantization.

Contribution

It introduces a novel approach connecting classical nonlinear wave solutions with quantum field construction, enabling nonperturbative development of the theory.

Findings

Successfully constructed Wightman functions and scattering matrix elements.

Established the nontriviality of the $:\phi^4_4:$ quantum field theory.

Linked classical solution structures with quantum quantization methods.

Abstract

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.