Regular self-congruent solutions of charged spinor field and scalar field with logarithmic potential

Abstract

It was shown that interaction of scalar field(which has Log–potential) with electric and spinor fields gives rise to regular field configuration with finite energy. This means that an observer at infinity sees Coulomb “point” charge, regularized at the center of the object. Using of Log–potential in our model is justified by the fact that this kind of potential is found to be the simplest generalization of quantum mechanics and it was shown that corresponding classical equation for nonlinear spinor field has regular solutions. Description of the model was made using equations of Maxwell, Dirac and nonlinear equations for nonlinear scalar field with Log–potential. Solution of this equations was obtained in Wolfram Mathematica CAS from nonliear eigenvalue problem. Besides, there was examined  behavior of corresponding functions at infinity and near zero and it was illustrated that spinor  and scalar fields decrease  exponentially, while electrical field decrease according to Coulomb law.