DIVERGENCES OF INTEGRALS FOR GREEN FUNCTIONS OF KLEINGORDON AND DIRAC EQUATIONS AND NECESSARY EXISTENCE OF PARTICLE GENERATIONS
Yu.V. Kulish, E.V. Rybachuk
Full Text : (600 kB, Eng.)
Abstract
It is shown that the values of the infinite integrals for the Green functions of the Klein  Gordon and Dirac equations depend on the method used for its calculations, i.e., these integrals diverge. The Green functions proposed to eliminate these divergences include in the denominators the polynomials of the N degree instead of one factor m^{2}q^{2} or m for the Klein  Gordon equation or the Dirac equation, respectively. The corresponding generalizations of the Klein  Gordon and Dirac equations have 2N and N degree, respectively. The solutions of these generalizations for the Klein  Gordon and Dirac equations may be presented by the sum of N terms, each term corresponds to the contribution of one particle (one generation). All these particles have different masses but the same spin, parity, charge, isospin. Since the space  time is 4  dimensional one, the convergence of the integrals for proposed Green functions is possible only if the generation number N is not less than three for integer spin particles and not less than five for half  integer spin particles. It is shown that the proposed Green functions have no any singularities in the space  time. In particular, the interaction potentials must have the oscillator form at short distance. It is predicted that two (or greater) massive particles with quantum numbers of the photon and gluons must exist. Besides, five (or greater) fermions with quantum numbers of the electron, the neutrino, the quark, and quark must exist (e.g., e_{1}=e, e_{2}=μ, e_{3}=τ, e_{4}, e_{5},... and u_{1}=u, u_{2}=c, u_{3}=t, u_{4}, u_{5},...). The massless neutrino must be one. If higher neutrinos are enough heavy then the decay ν_{4,5} → eμν_{1,2} may be possible.

KEY WORDS: convergence, multiple integrals, Green functions, partial differential equations, oscillatory potentials, particle generations, massive photon, massive gluons, massive neutrino.
