New approach to the solution of nonlinear problems

Abstract

In this talk we introduce a new technique, called the delta expansion, which can be used to solve nonlinear problems in both classical and quantum physics. The idea of the delta expansion is to expand in the power of a nonlinear term. For example, to treat a y4 term, we introduce a small parameter delta and consider a (y2(1+δ)) term. When we expand in powers of delta, the resulting perturbation series appears to have a finite radius of convergence and numerical results are superb. We illustrate the delta expansion by applying it to various difficult nonlinear equations taken from classical physics, the Lane-Emden, Thomas-Fermi, Blasius, Duffing, Burgers, and Kordeweg-de Vries equations. In the study of quantum field theory, the delta expansion is a powerful calculational tool that provides nonperturbative information. The basic idea is to expand in the power of the interaction term. For example, to solve a λφ4 theory in d-dimensional space-time, we introduce a small parameter δ and consider a λ(φ2)1+δ field theory. We show how to expand such a theory as a series in powers of δ using graphical methods. The resulting perturbation series appears to have a finite radius of convergence and numerical results for low-dimensional models are good. We compute the Green's functions for a scalar quantum field theory to first order in delta, renormalize the theory, and see that when the space-time dimension is four or more, the theory is free. This conclusion remains valid to second order in delta, and we believe that it remains valid to all orders in delta. The delta expansion is consistent with global supersymmetry invariance. We examine a supersymmetric quantum field theory for which we do not know of any other means for doing analytical calculations and compute the ground-state energy density and the fermion-boson mass ratio to second order in delta. Last, we show how to use the delta expansion to solve theories having a local gauge invariance. We compute the anomaly in two-dimensional electrodynamics and discuss the calculation of g-2 in four-dimensional electrodynamics.