Title: Self-consistency, relativism and many-particle system

Abstract: The interrelation between the concepts of self-consistency, relativism and many-particle systems is considered within the framework of a unified consideration of classical and quantum physics based on the first principle of the probability conservation law. The probability conservation law underlies the Vlasov equation chain. From the first Vlasov equation, the Schr\"odinger equation, the Hamilton-Jacobi equation, the equation of motion of a charged particle in an electromagnetic field, the Maxwell equations, the Pauli equation and the Dirac equation are constructed. The paper shows with mathematical rigor that quantum systems with a time independent function of quasi-density probability in phase space are not capable to emit electromagnetic radiation. It is shown that at the micro-level a quantum object may be considered rather as an {\guillemotleft}extended{\guillemotright} object than a point one. And the hydrodynamic description of continuum mechanics is applicable for such an object. A number of exact solutions of quantum and classical model systems is considered, demonstrating a new insight at the quantum mechanics representation.