  I investigate bosonization in four dimensions, using the smooth bosonization
scheme. I argue that generalized chiral ``phases'' of the fermion field
corresponding to chiral phase rotations and ``chiral Poincare transformations''
are the appropriate degrees of freedom for bosonization. Smooth bosonization is
then applied to an Abelian fermion coupled to an external vector. The result is
an exact rewriting of the theory, including the fermion, the bosonic fields,
and ghosts. Exact bosonization is therefore not achieved since the fermion and
the ghosts are not completely eliminated. The action for the bosons is given by
the Jacobian of a change of variables in the path integral, and I calculate
parts of this. The action describes a nonlinear field theory, and thus static,
topologically stable solitons may exist in the bosonic sector of the theory,
which become the fermions of the original theory after quantization.
