The Φ4 equations of motion. I. A new constructive method: The Φ iteration

Abstract

A new method for the construction of a nontrivial Φ44 model consistent with the general principles of a Wightman quantum field theory is proposed. The renormalized equations of motion for the connected Green’s functions in the Euclidean momentum space are considered. Using a fixed-point method in an appropriate Banach space, the existence of a unique nontrivial solution of these equations is proved when the coupling constant Λ is fixed positive and smaller than a finite value. As a first step that avoids the difficulties of the renormalization operation, the corresponding problem in two dimensions is solved first. Also, in order to deal with all technical complexities that stem from the purely combinatorial nature of the equations, the corresponding zero-dimensional problem was studied previously. In all cases, it has been proved that the nontrivial solution exists inside particular subsets of the corresponding Banach space, characterized by the alternating signs and the factorization (or ‘‘splitting’’) properties of the Green’s functions at zero external momenta. These properties first appeared ‘‘experimentally’’ by the iteration of the two-dimensional system of equations called ‘‘Φ iteration’’ in the paper, and have been crucial for the conservation of the norms and for the contractivity of the nonlinear mappings.