The Φ4 equations of motion. II. The zero-, one-, and two-dimensional solutions

Abstract

This is the second paper of the series concerning the solution of the system of Φ4 equations of motion for the Schwinger functions by a fixed-point method. These works constitute a part of a general program towards the construction of a Φ44 Wightman quantum field theory (QFT). In the previous paper [J. Math. Phys. 29, 2092 (1988)] (Paper I), a general outline of the program has been presented. Moreover, the ‘‘nice’’ properties of signs, splitting, and norms revealed ‘‘experimentally’’ by the Φ iteration have been analyzed. We have shown how this iterative procedure converges to the solution (if the coupling constant is fixed positive and smaller than a finite value) thanks to the conservation of these properties, which constitute in fact a complete system of self-consistent conditions. Taking into account this information, in the present paper, the answer to the zero-, one-, and two-dimensional problems is given. To be precise, the fixed-point method is constructed by formulating the properties of signs and splitting of the Φ iteration in terms of particular subsets Φ0Λ ⊆ℬ0 (zero dimensions) and ΦΛ⊆ℬ (one and two dimensions) of the appropriated Banach spaces ℬ0 and ℬ, defined exactly with the norms provided by the Φ iteration. The basic ingredients, both introduced already in I, are, on the one hand, the bounded positive sequences of the splitting constants and, on the other hand, the sweeping factors, which carry all the combinatorial information for the global terms. Their absolute and relative bounds yield the stability of the corresponding subsets and the conservation of the norms. F o r the zero-dimensional problem a simpler equivalent system (nonlinear map) of equations in the space of the splitting sequences is solved by ensuring the existence of the solution (resp. contractivity) inside the corresponding subset when the coupling constant Λ satisfies 0<Λ≲0.1 (resp. 0<Λ≲0.01). For the two- (or one-) dimensional problem it is shown that when 0<Λ≲0.006, the subset ΦΛ is stable under the nonlinear mapping (represented by the equations of motion), which in turn is contractive inside ΦΛ under weaker conditions on Λ. By this last result the convergence of the Φ iteration to the unique fixed point (presented in I) is reobtained in a direct way.