Dirac equation in terms of hydrodynamic variables

The distributed system ${\mathcal{S}}_D$ described by the Dirac equation is investigated simply as a dynamic system, i.e. without usage of quantum principles. The Dirac equation is described in terms of hydrodynamic variables: 4-flux $j^i$, pseudo-vector of the spin $S^i$, an action $\mathrm{\hslash }\varphi$ and a pseudo-scalar $\kappa$. In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector $j^i$) are small, the system ${\mathcal{S}}_D$ turns to a statistical ensemble of classical concentrated systems ${\mathcal{S}}_{dc}$. Under some conditions the classical system ${\mathcal{S}}_{dc}$ describes a classical pointlike particle moving in a given electromagnetic field. In general, the world line of the particle is a helix, even if the electromagnetic field is absent. Both dynamic systems ${\mathcal{S}}_D$ and ${\mathcal{S}}_{dc}$ appear to be non-relativistic in the sense that the dynamic equations written in terms of hydrodynamic variables are not relativistically covariant with respect to them, although all dynamic variables are tensors or pseudo-tensors. They becomes relativistically covariant only after addition of a constant unit timelike vector $f^i$ which should be considered as a dynamic variable describing a space-time property. This “constant” variable arises instead of $\gamma$-matrices which are removed by means of zero divizors in the course of the transformation to hydrodynamic variables. It is possible to separate out dynamic variables $\kappa$, $\kappa ^i$responsibleforquantumeffects\mathrm{.}Itmeansthat,setting$\kappa ,\kappa ^i\equiv 0$,thedynamicsystem$$\mathcal{S}$_D$ described by the Dirac equation turns to a statistical ensemble ${\mathcal{E}}_{Dqu}$ of classical dynamic systems ${\mathcal{S}}_{dc}$.