  We extend a previously successful discussion of the constrained
Schr\"{o}dinger system through the Dirac--Bergmann algorithm to the case of the
Dirac field. In order to follow the analogy, first we discuss the classical
Dirac field as a spinorial variable, by introducing properly defined momenta
and a suitably modified, factor ordered Poisson bracket. According to the
Dirac--Bergmann algorithm two second class Hamiltonian constraints emerge,
leading to a factor ordered Dirac bracket on the full phase space. This becomes
the Poisson bracket on the reduced phase space in the canonical chart adapted
to the shell. The Dirac equation is recovered both as consistency condition on
the full phase space and as canonical equation on the reduced phase space.
Alternatively, considering the Dirac field as odd Grassmann variable, we
present the details of the Dirac--Bergmann algorithm (with either left and righ
derivatives acting on Grassmann valued superfunctions and involving a different
type of generalized Poisson and Dirac brackets). We propose a recipe for the
canonical second quantization of all three versions of the generalized Dirac
brackets, yielding the correct fundamental anticommutator.
