Sur certaines difficultés des théories spinorielles non-linéaires

Abstract

Two problems arising in non-linear spinor field theories are discussed. One of these problems concerns general non-linear field equations and is connected with the behaviour of the anticommutator at the origin of the light cone. We show that the resolution of the nonlinear equation by means of the new Tamm-Dancoff method is not self-consistent, if we assume anticommutativity of the fields at the same point. A consistent resolution of the equation requires, as a consequence of microscopic causality, an anticommutator with at least δ-type singularities on the light cone. The other problem concerns Heisenberg's theory of elementary particles; we show that, in this theory, the nucleon-nucleon scattering amplitude vanishes. In order to get some insight on the validity of the Tamm-Dancoff approximation we compute, in an appendix, first and second order contributions to the Möller scattering amplitude in quantum electrodynamics by means of the Tamm-Dancoff method and compare with the results of perturbation theory. The calculations indicate that the first-order contribution is in accordance with perturbation theory ( e 2 terms) while this is no longer the case for the second-order contribution.