Elimination of Singulariries in Causal Green Functions for Generalized Klein-Gordon and Dirac Equations on Light Cone
Abstract
Klein-Gordon and Dirac equations are generalized to eliminate divergences in the integrals for Green functions of these equations. The generalized equations are presented as products of the operators for the Klein-Gordon equation with different masses and similarly for the operators of the Dirac equation. The homogeneous solutions of derived equations are sums of fields, corresponding to particles with the same values of the spin, the electric charge, the parities, but with different masses. Such particles are grouped into the kinds (families, dynasties) with members which are the particle generations. The Green functions of derived equations can be presented as sums of the products of Green functions for the Klein-Gordon equation (the Dirac equation) and the definite coefficients. The sums of these coefficients equal zero. The sums of the products of these coefficients and the particle masses to some powers equal zero too, i.e. for these coefficients some relations exist. In consequence of these relations the singularities in Green functions can be eliminated. It is shown that causal Green functions of derived equations сan be finite in all the space-time. This is possible if minimal quantities of the generations Nb and Nf for the bosons and the fermions equal 3 and 6, respectively. An absence of singularities in the Green functions on light cone is related to an attenuation of particle interactions on short distances. It is shown explicitly for the generalization of the Yukawa potential.
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References
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