I. Electron Decay of the π-Meson. II. A Non-Linear Field Theory

Abstract

Part I. A π-meson decays into µ-meson and neutrino at least 1000 times faster than into an electron and a neutrino. After summarizing the difficulties in assuming that electrons or µ-mesons interact with nucleons through the intermediary of the π-meson, the decay of the π is discussed for the symmetric coupling scheme in which electrons and µ-mesons interact directly with nucleons. Selection rules rigorously forbid this decay for the most choices of the π-meson field and the form of nuclear β-decay. For the very special case of pseudoscalar meson and pseudovector β-decay (with arbitrary mixtures of scalar, vector and tensor) the decay rate for π → (µ,v) proceeds 10 4 times as fast as π → (µ,v) and 10 +3 as fast as π → (photon, e, v). This result is independent of perturbation theory. Agreement with the observed lifetime can be obtained if the divergent integral is cut off at the nucleon Compton wavelength. Part II. A unitary theory of particles is investigated, mostly on the classical level. The Dirac and the Klein-Gordon equations are augmented by simple non-linear terms. Interpreted as wave equations for classical fields they contain a much richer variety of solutions than the customary linear theories. Particles, instead of having independent existences as singularities, appear only as intense localized regions of strong field. Solutions of the field equations are subject to the boundary condition that the fields be regular everywhere and that all observable integrals be finite. For simple angular and temporal dependence the wave equation reduces to a set of ordinary differential equations. The boundary condition leads to a non-linear eigenvalue problem whose solutions are systematically described in the phase plane. Numerical solutions are found for some typical cases. The masses of the particles are positive; the number carrying unit charge is small. The scalar field variables can be interpreted in terms of operators according to the usual commutation rules, but the particles are unstable when perturbed by quantum fluctuations. The application of anti-commutation rules to the spinor fields has no classical limit. The lack of satisfactory recipe for quantizing classical spinor fields makes the interpretation of the particle-like solutions obscure.