Nonlinear Spinor Field

Abstract

A nonlinear spinor field, suggested by the symmetric coupling between nucleons, muons, and leptons, has been investigated in the classical approximation. Solutions of the field equations having simple angular and temporal dependence were obtained, subject to the boundary conditions that the fields be regular and that all observable integrals be finite. These b.c. lead to a nonlinear eigenvalue problem, whose solutions may be systematically discussed in the phase plane. Numerical solutions were obtained with a differential analyzer. If charge and mass of the particle-like solutions are defined in terms of $\ensuremath{\int}{s}_{4}d\mathbf{x}$ and $\ensuremath{\int}{T}_{44}d\mathbf{x}$, then the number of masses corresponding to the same charge turns out to be small in all cases investigated. For certain lagrangians the nonlinearity leads to solutions having positive energy only. The mass ratio between the lightest stable particle and the heaviest unstable particle can be taken of the order of ${10}^{\ensuremath{-}3}$, if the nonlinear coupling constant is properly chosen. Although our specific model is too simple to meet certain obvious requirements, a theory of this general type has some interesting features.