Local contractivity of the ${\Phi }_0^4$ mapping

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 $0\le r\le 4$ 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 $\Phi \subset \mathcal{B}$ 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 ${\Phi }_0^4$ mappping is established in a neighborhood of a precise nontrivial sequence $H_0\in {\Phi }_0$ using a new norm in the Banach space $\mathcal{B}$. A new ${\Phi }_0^4$ 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 $H^{\ast }\in {\Phi }_0$.

33 pages, 3figures