Construction of Schrödinger, Pauli and Dirac equations from Vlasov equation in case of Lorentz gauge

On the basis of the first principle – the law of probability conservation and the Helmholtz decomposition theorem the authors have succeeded to construct the Schrödinger, Pauli, Dirac equation, the Hamilton-Jacobi equation and the Maxwell equations. The approach described in this paper makes it possible to naturally connect the classical and quantum systems.