Klein-Gordon-Fock equation for a particle under classical magnetic field effect in light front coordinate

This article is intended to present, from a didactic perspective, the details and mathematical procedures necessary for the construction of the quantum differential equation, under a relativistic regime, called the Klein-Gordon-Fock Equation (KGFE), admitting the front coordinate system of light. To do so, initially, we highlight some algebraic properties of Einstein’s Special Theory of Relativity as well as briefly present the coordinates of the light front in terms of cartesian coordinates and the temporal evolution parameter, that is, of Minkowski space, thus obtaining the KGFE both in usual coordinates of Minkowski space and in coordinates of the light front. Then, for these last coordinates, we develop in detail the algebraic structure of KGFE for an electrically charged quantum particle in a special relativity regime and under influence of classical magnetic field configurations. Furthermore, we obtain the KGFE with an algebraic structure similar to a Schrödinger quantum differential equation. Finally, as an immediate application of the use of these coordinates, we developed the necessary algebraic procedures and steps to obtain the annihilation operator of the integral type of motion, which is configured as the necessary mathematical structure for the construction of coherent quantum states.

Keywords
Klein-Gordon-Fock Equation; Light Front Coordinates; Coordinate Systems; Creation and Annihilation Operators