Topological Solitons in Discrete Space-Time as the Model of Fermions

Abstract

In the present work we discuss arguments in favour of the view that massive fermions represent dislocations (i.e. topological solitons) in discrete space-time, with Burgers vectors parallel to the axis of time. If we assume that the symmetrical parts of tensors of distortions (i.e. derivatives of atomic displacements on coordinates) and the mechanical stresses are equal to zero, then the equations of the field theory of dislocations assume the form of the Maxwell equations. If we consider these tensors as symmetrical, we obtain the equations of the theory of gravitation, and it appears that the sum of the tensor of distortions and of the pseudo-Euclidean metrical tensor is the analogue of the metrical tensor. It is shown that we can also get Dirac equation with four-fermion interaction in the framework of the field theory of dislocations. This model explains quantization of electrical charge: this is proportional to the topological charge of dislocation, and this charge accepts quantized values because of the discrete structure of the 4-dimensional lattice.