Simplification of nonlinear equations for a field operator

Abstract

In this paper, we study different properties of the motion equations of interacting fields. In the second section, we prove that "Wightman's" fields (we use only a subset of Wightman's axioms) are unitarily equivalent to some operators on the vector space ${\cal F}$ (with one mathematical assumption). In the third section, we introduce $L^{\infty}$ and $DL$ Hilbert spaces, which are convenient for analyzing field equations, particularly the equations for $\phi^3$ theory. Remarkably, we have managed to reduce the equation of motion for $\phi^3$ to a quadratic matrix equation with matrices over a separable Hilbert space in the fourth section. Also, in the appendix, we have done the same for QCD. Furthermore, we prove the existence of solution to the motion equations of one toy model non-renormalizable theory in the fifth section.