Chiral Poincaré transformations and their anomalies ∗ Jan

We consider global transformations of a Dirac fermion field that are generated by the generators of Poincaré transformations, but with a γ5 appended. Using proper time regularization in Minkowski space, we find that these chiral translations and chiral Lorentz transformations in general have anomalies, and we calculate these in a theory of a massive fermion coupled to an external Abelian vector field. For this calculation, a certain freedom in the regularization scheme has been fixed to ensure both phase and Poincaré invariance. We show that theories which are regularized to give covariant anomalies does not respect Poincaré invariance. If we interpret SJ in the expression J = exp(iSJ) for the Jacobian of finite chiral transformations as an action for a theory where the transformation parameters are the dynamical fields, then these anomalies describe the decay into one, two, or three external vectors. The chiral degrees of freedom thus realizes bosonization in four dimensions. We argue that this may be relevant for an effective description of positronium in the case of QED, and the non-Abelian generalization may play a role in effective theories of QCD. PACS numbers: 11.30.Rd; 11.15.Tk; 11.30.Qc