A non-perturbation method for non-linear field theories

A non-perturbation method is developed for approximating the field energy and the N-body propagators in non-linear field theories. In the present work the method is applied to the scalar meson field with the addition of λϕ4 to the Lagrangian. The particular problems considered (in first approximation) are: the field energy in the presence of an external source; the one- and two-meson propagators; and meson-pair theory. In all cases the answer is renormalized and the approximation is shown to give a sensible account of the specific non-linear features of the problem which come into play when either the coupling constant λ, the external source, or the particle momentum becomes large. Moreover, in the ‘linear’ region (i. e. when the parameters are small), perturbation theory is valid and the method is shown to agree-with it. However, if the renormalization is performed by the customary limiting process, the external source energy and the two-body propagator have spurious singularities which may be loosely described as ‘ghost states’. It is not clear whether this failing is inherent in the field theory itself or in the method. Finally, a brief discussion is given of the results of the second approximation, and of the possible usefulness of the method in the more realistic pseudo-scalar meson-nucleon problem.