Dirac Equation in Terms of Hydrodynamic Variables

Yuri A.Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences,
101, bld.1 Vernadskii Ave. , Moscow, 119526, Russia.
e-mail: rylov@ipmnet.ru


The distributed system S, 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, pseudo-vector of the spin s, an action, and a pseudo-scalar k. In the quasi-uniform approximation, when all transversal derivatives (orthogonal to the flux vector j) are small, the dynamic system S turns to a statistical ensemble E of classical concentrated systems s. Under some conditions the classical system s 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 S and E 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 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 k responsible for quantum effects. It means that, setting k=0, the dynamic system S described by the Dirac equation turns to a statistical ensemble E of classical dynamic systems s.

There is text of the paper in English (pdf  ps), in Russian (pdf  ps).