Bosonization in four dimensions: The smooth way∗

I investigate bosonization in four dimensions, using the smooth bosonization scheme. I argue that the generalized chiral “phases” of the fermion field corresponding to chiral phase rotations and chiral Poincaré transformations are the appropriate degrees of freedom for bosonization. ’Chiral Poincaré transformations’ are Lorentz transformations and translations with a γ5 included in the generators. The methods of smooth bosonization are then generalized from two to four dimensions, and applied to a theory of a massive Abelian fermion coupled to an external vector field. The result is an exact rewriting of the theory, including the fermion, the bosonic fields, and ghosts. The fermion is eliminated from the theory only if the energy-momentum tensor and a corresponding axial tensor are symmetrical. This, however, would leave a theory of bosons and ghosts, hence bosonization is not exact. The action SJ for the bosons is given by the Jacobian J = exp(iSJ) of a change of variables in the path integral, and I calculate parts of this. The action describes a nonlinear field theory which includes four-derivative terms, and it is therefore possible that static, topologically stable solitons exist in the bosonic sector of the theory, which become the fermions of the original theory after quantization. PACS numbers: 11.15.Tk; 11.30.Rd; 11.10.Lm