Simplification of nonlinear equations for a field operator

TL;DR

The paper develops a framework to simplify nonlinear equations for interacting field operators by embedding them in a concrete Hilbert-like space ${\cal F}$ and by modifying parts of the Wightman reconstruction. It introduces a GNS-type construction for Wightman functionals and new spaces ${\cal L}^{\infty}$ and ${\cal DL}$ to host the field dynamics, enabling a reformulation of $\phi^3$ and QCD-like dynamics as algebraic or matrix equations on separable Hilbert spaces. Key contributions include a modified Wightman reconstruction with a cyclic representation, explicit group actions in the ${\cal L}^{\infty}$/${\cal DL}$ framework, a reduction of the $\phi^3$ motion equations to a quadratic matrix equation, and a proof of nontrivial solutions for a toy non-renormalizable model via fixed-point methods. This framework provides a nonperturbative, operator-centric toolkit for analyzing nonlinear field equations and exploring alternative renormalization perspectives with potential implications for high-energy theory.

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 $φ^3$ theory. Remarkably, we have managed to reduce the equation of motion for $φ^3$ 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.