Singular-perturbation—strong-coupling field theory and the moments problem

Abstract

Motivated by recent work of Bender, Cooper, Guralnik, Mjolsness, Rose, and Sharp, a new technique is presented for solving field equations in terms of singular-perturbation---strong-coupling expansions. Two traditional mathematical tools are combined into one effective procedure. Firstly, high-temperature lattice expansions are obtained for the corresponding power of the field solution. The approximate continuum-limit power are subsequently obtained through the application of Pad\'e techniques. Secondly, in order to reconstruct the corresponding approximate global field solution, one must use function-moments reconstruction techniques. The latter involves reconsidering the traditional moments problem of interest to pure and applied mathematicians. The above marriage between lattice methods and reconstruction procedures for functions yields good results for the ${\ensuremath{\varphi}}^{4}$ field-theory kink, and the sine-Gordon kink solutions. It is argued that the power are the most efficient dynamical variables for the generation of strong-coupling expansions. Indeed, a momentum-space formulation is being advocated in which the long-range behavior of the space-dependent fields are determined by the small-momentum, infrared, domain.