  We search a canonical basis of Dirac's observables for the classical Abelian
Higgs model with fermions in the case of a trivial U(1) principal bundle. The
study of the Gauss law first class constraint shows that the model has two
disjoint sectors of solutions associated with two physically different phases.
In the electromagnetic phase, the electromagnetic field remains massless: after
the determination of the Dirac's observables we get that both the reduced
physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the
electromagnetic field becomes massive and in terms of Dirac's observables we
get a local, but nonanalytic in the electric charge (or equivalently in the sum
of the electromagnetic mass and of the residual Higgs field), physical
Hamiltonian; however the associated Lagrangian is nonlocal. Some comments on
the R-gauge-fixing, the possible elimination of the residual Higgs field and on
the Nielsen-Olesen vortex solution close the paper.
