Local contractivity of the $\Phi_4^4$ mapping

TL;DR

This paper proves the existence and uniqueness of solutions to a non-linear system modeling the dynamics of $ abla$ Green's functions in Euclidean quantum field theory, using local contractivity in a specialized Banach space.

Contribution

It introduces a new mapping approach demonstrating local contractivity for the $ abla$ $ ext{Φ}_4^4$ system, establishing solution existence and uniqueness.

Findings

Existence of a unique solution to the $ ext{Φ}_4^4$ system.

Solution characterized by specific analyticity and asymptotic properties.

The new mapping is locally contractive near a tree-type Green's function sequence.

Abstract

We show the existence and uniqueness of a solution to a $\Phi_4^4$ non linear renormalized system of equations of motion in Euclidean space. This system represents a non trivial model which describes the dynamics of the $\Phi_4^4$ Green's functions in the Axiomatic Quantum Field Theory (AQFT) framework. The main argument is the local contractivity of the so called \emph{"new mapping"} in the neighborhood of a particular "tree type" sequence of Green's functions. This neighborhood (and the $\Phi_4^4$ non trivial solution) belongs to a particular subset of the appropriate Banach space characterized by signs, splitting (analogous to that of the $\Phi_0^4$ solution), axiomatic analyticity properties and "good" asymptotic behavior with respect to the four-dimensional euclidean external momenta.