A nonperturbative solution of a nonlinear field equation of motion

Abstract

In a previous paper an infinite-component generalized free fieldϕ(x) was constructed which transforms locally under the Poincare group and has a propagator which is invariant under aTP transformation.ϕ(x) carries an infinite tower of unstable (excepting the lowest member) self-compounds which is defined by a spin trajectory. The propagator ofϕ(x) can be written as an infinite partial series (in spinj) which can be summed by a Sommerfeld-Watson transformation with a particular constraint on the spin trajectory. The present paper investigates in what sense the fundamental fieldϕ(x) fixed in such a way in formal detail can provide a nonperturbative solution of the nonlinear field equation of motion (in aTCP- and Lorentz-invariant manner) $$\begin{gathered} \square _x \left\langle {0\left| {\left\{ {\varphi (y),\varphi ^ + (x) + \varphi ( - y),\varphi ^ + ( - x)} \right\}} \right|0} \right\rangle = \hfill \\ g\left\langle {0\left| {\left\{ {\varphi (y),\varphi ^ + (x) \cdot \varphi ^ + (x) \cdot \varphi ^ + (x) + \varphi ( - y),\varphi ^ + ( - x) \cdot \varphi ^ + ( - x)} \right\}} \right|0} \right\rangle \hfill \\ \end{gathered} $$ withx0 =y0. It is found necessary in terms of a dynamical hypothesis to consider how the reaction of the two-particle states corresponding to the terms nonlinear inϕ†(x) leads to compounds as single-particle states. The consequently required covariant reduction of the nonlinear term, where it involves the product representations of discrete and continuum mass distributions, may be generalized from known theory with due recognition of the scale factor problem. The latter is solved in a way which appears to be of importance for the concept of a fundamental field. One then finds by a particular way of taking the momentum integrals involved off the mass shells, and considering the equation in the particular form above, that all the extra momentum integrals for all spin components are completely convergent. One ends up with purely dynamical equations involving the mass spectral functions nonlinearly, and giving the conditions forϕ(x) to be a solution of the nonlinear equation in this particular sense. These spectral-function equations are examined for convergence of the integrals overr and sums overj involved, and for the asymptotic behaviour inr andj. The examination shows that in principle a solution may be obtained which is consistent with the independent determination ofϕ(x) in the previous paper, provided that the scale factor function has a suitable dependence onr andj.