Local contractivity of the $\Phi_0^4$ mapping

TL;DR

This paper investigates the local contractivity of the $ ext{Phi}_0^4$ mapping in 0-dimensional quantum field theory, refining the structure of solution subsets, establishing contractivity near a specific sequence, and demonstrating rapid convergence to a fixed point.

Contribution

It introduces a new norm and iteration method for the $ ext{Phi}_0^4$ map, proving local contractivity and stability in the 0-dimensional case, with numerical validation.

Findings

Established local contractivity of the $ ext{Phi}_0^4$ map near a specific sequence.

Defined a new iteration method demonstrating rapid convergence.

Numerical results confirm stability and properties of the solution set.

Abstract

Previous results about the non trivial solution of the $\Phi^4$-equations of motion for the Green's functions in the Euclidean\ space (of $0\leq r\leq 4$ dimensions) in the Wightman Quantum Field theory framework, are reviewed in the 0 - dimensional case from the following three aspects: The structure of the subset $\Phi \subset {\cal B}$ characterized by the signs and "splitting" (factorization properties) is reffined and more explictly described by a subset $\Phi_0\subset \Phi$ The local contractivity of the corresponding $\Phi ^4_0$ mappping is established in a neighborhood of a precise nontrivial sequence $H_0\in \Phi_0$ using a new norm in the Banach space $ {\cal B}$. A new $\Phi ^4_0$ iteration is defined in the neighborhood of this sequence $H_0\in \Phi_0 $ and our numerical study displays clearly the stability of $\Phi_0$ (the splitting, the bounds and sign properties are perfectly illustrated), and a rapid convergence to the unique fixed point $H^* \in \Phi_0$.