### 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
#### Abstract

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).