  A realistic interpretation of Schroedinger and Dirac equations for density
matrices is proposed, in which the difference between the position arguments of
the density matrix is considered as an objective extra space dimension.
"Particle" solutions are found, which are perfectly localized both in position
space and in momentum space (the position and momentum operators commute in the
density matrix representation); definitions for all observable quantities are
given and the values associated to the "particle" solutions are the correct
ones, both for the non-relativistic and the relativistic case. Finally, a
non-linear interaction (the electromagnetic one) is introduced in an attempt to
single out the "particle" solutions of the Dirac equation from all other
solutions; the dynamical evolution of the electromagnetic field is described by
the classical (unquantized) Maxwell equations.
