Longitudinal electric field: from Maxwell equation to non-locality in time and space

Abstract: In this paper we use the classical electrodynamics to show that the Lorenz gauge can be incompatible with some particular solutions of the d Alembert equations for electromagnetic potentials. In its turn, the d Alembert equations for the elec- tromagnetic potentials is the result of application of the Lorenz gauge to general equations for the potentials. The last ones is the straightforward consequence of Maxwell equations. Since the d Alembert equations and the electromagnetic poten- tials are necessary for quantum electrodynamics formulation, one should oblige to satisfy these equations also in classical case. The solution of d Alembert equations, which modifies longitudinal electric field is found. The requirement of this modifi- cation follows from the necessity to satisfy the physical condition of impossibility of instantaneous transferring of interaction in space.