  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.
