Role for a fundamental equation in fundamental theory of matter

Abstract

Starting from a fundamental spinor field equation of the type i ∇ψ(x) = l 2T( ψ(x)O iϱψ(x))O i·ϱψ(x) , equations relating τ-functions of successive orders may be deduced. Using the lowest order equation of the latter set, and a minimal representation of the fourth order τ-functions involved in terms of the relevant baryon scattering amplitudes, a set of algebraic equations is obtained in terms of the masses of the basic baryon fields and their coupling constants to the exchanged mesons concerned. On the basis of the expectation that the algebraic equations would yield a threefold multiplicity for the positive basic baryon masses, and the belief that the fundamental equation should have as many basic fields as the multiplicity of its basic mass solutions, the present paper uses three basic baryon fields. Then ϱ is an eight component vector of 3 x 3 matrices giving a formally symmetric self-interaction in the fundamental equation. It is considered possible, nevertheless, that the basic masses may be different due to the fact that the coupling constants as determined by the mentioned algebraic equations depend on the mass ratios in general. The implications of the formal symmetry of the fundamental equation and the possible asymmetry of the solution for the multiplicity of the vacuum states are discussed. Corresponding to the γ 5 invariance of the fundamental equation at least a corresponding multiplicity of leptonic mass solutions is predicted. The scalar parts of the baryon scattering amplitudes uses a double dispersion integral representation of a Regge type amplitude in the integration of the τ-function equation. The importance of the oscillatory behaviour of the Regge type amplitude for the elimination of the usual type of divergences in the Feynman integrations involved is discussed.