Dirac equation in terms of hydrodynamic variables

Abstract

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 $\hbar \phi $ 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 $, $\kappa ^i$ responsible for quantum effects. It means that, setting $\kappa ,\kappa ^i\equiv 0$, the dynamic system $\mathcal{S}_D$ described by the Dirac equation turns to a statistical ensemble $\mathcal{E}_{Dqu}$ of classical dynamic systems $\mathcal{S}_{dc}$.