Dirac equation solution in the light front via linear algebra and its particularities

Abstract

In undergraduate and postgraduate courses, it is customary to present the Dirac equation defined in a space of four dimensions: three spatial and one temporal. This article discusses aspects of the Dirac equation (QED) on the light front. This proposal of coordinate transformations comes from Dirac who originally introduced three distinct forms of relativistic dynamics possible depending on the choice we make of the different hypersurfaces constant in time. The first he called instantaneous, the most common form, the hypersurface of which is specified by the boundary conditions set at  . The second, known as the point form, has as its characterizing surface, a hyperboloid, described by the initial conditions in , being  one constant (chosen as the time of this system). The third relativistic form, known as the light front form, has its hypersurface tangent to the light cone; being defined by the initial conditions at , and  is the time in the light front system. The method of this work is deductive. Therefore, one obtains the solution of the Dirac equation for the Free Electron and for the positron in the coordinates in the light front with the particularity of the energy associated with the system being given by , and for moments  we have the electron and  we have the positron. The result of this is that the positive energy states in the light front  and negative   are independently described in the equation, and with additional, the problem at the limit that does not converge.