We study the exact equivalence between the self-dual model minimally coupled with a Dirac field and the Maxwell-Chern-Simons model with nonminimal magnetic coupling to fermions. We show that the fermion sectors of the models are equivalent only if a Thirring like interaction is included.Using functional methods we verify that, up to renormalizations, the equivalence persists at the quantum level.

We analyze the equivalence between the Thirring model and the fermionic sector of the theory of a Dirac field interacting via derivative coupling with two boson fields. For a certain choice of the parameters the two models have the same fermionic Green's functions.In a recent work1 we analyzed some properties of mass perturbation in the Thirring model as an example of a perturbative scheme in which the unperturbed system is not a free-field model but already incorporates some interaction. For practical reasons, instead of working directly with the Thirring model, it was convenient to use an equivalent theory, the derivative-coupling (DC) model. This theory describes an interaction of a Dirac field $ with two fields, one scalar, v, and the other pseudoscalar, 4.For specific values of the couplings, the fermionic Green's functions of the DC model turn out to be equal to those of the Thirring model as given, for example, by ~laiber.' This equivalence saved us a lot of technical complications making it possible, with relative ease to derive our results. However, in spite of this success, a basic question concerning the aforementioned equivalence still persists. Essentially, the problem is the following. The Thirring model has one degree of freedom, in the sense that the basic field can be written explicitly in terms of just one free scalar field. The DC model, on the other hand, has, in principle, three degrees of freedom, which can be taken to be 7, 4, and the potential, c, of the free vector current. So, the numbers of degrees of freedom of the two models do not match. It is our purpose to clarify this situation and establish the precise way in which the equivalence of the two models should be understood. Anticipating our results, we are going to prove that in the fermionic sector a certain combination of the fields 7, 4, and c is a spurion, or better it commutes with all the elements of the algebra generated by the fermionic components of the DC model. Besides that, to produce the same Green's functions as in the Thirring model we are forced to use another special combination of these fields. These two constraints effectively reduce the number of degrees of freedom from three to one.with the constants a, 5, a , and E given by 47T a==" [ F 1 ] , a=T [ b l ] , 4~where is the usual parameter of the sine-ord don^ model that is related to k by (6)