Evolution of coupled fermions under the influence of an external axial-vector field

The evolution of coupled fermions interacting with external axial-vector fields is described with the help of the classical field theory. We formulate the initial conditions problem for the system of two coupled fermions in (3+1)-dimensional space-time. This problem is solved using perturbation theory. We obtain in the explicit form the expressions for the leading and next to leading order terms in the expansion over the strength of the external fields. It is shown that in the relativistic limit the intensity of the fermion field coincides with the transition probability in the two neutrinos system interacting with moving and polarized matter. PACS 14.60.Pq; 03.50.-z