We consider the Dirac equation with two different four-potentials of the plane electromagnetic waves. We derive the partial differential equation for the wave function, which is generalized form of the Volkov equation. We find the solutions of the Dirac equation for two orthogonal plane waves. We determine the modified Compton formula for the scattering of two photons on an electron.

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the th...

We considered an extension of the standard functional for the Einstein-Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler-Lagrange equations provide a new type of EinsteinDirac coupling. We establish a special method for con...

In the model of topological particles we have four types topologically stable dual Dirac monopoles with soft core and finite mass. We discuss steps how to get a equation for these particles. show free interacting case that arrive at in limit, where solitons approach singular monopoles.

A symmetry reduction of the Dirac equation is shown to yield the system of ordinary differential equations whose integrability by quadratures is closely connected to the stationary mKdV hierarchy. Consider the Dirac equation of an electron

Dirac equation in (1+2) dimensions for quasi-particles in graphene and quantum field theory of their Coulomb interaction. Abstract There is evidence for existance of massless Dirac quasi-particles in graphene, which satisfy Dirac equation in (1+2) dimensions near the so called Dirac points which lie at the corners at the graphene's bril-luoin zone. Certain subtle points which are peculiar to od...

By analogy with the octonionic electromagnetic and gravitational interactions, the octonionic strong and weak interactions and their quantum interplays are discussed. In weak interaction, the study deduces some conclusions of field sources and intermediate particles, which are consistent with the Dirac equation, Yang-Mills equation, some new intermediate particles and three kinds of colors etc....