Bosonization of massless free fermion fields We are going to write an explicit expression for a massless free fermion field as a non-local function of a free massless pseudo-scalar

The kinematical constraints on field theories in two space-time dimensions lead to many unusual effects which have no analogue in higher dimensions. Perhaps the strangest of these is the equivalence between large classes of Bose and Fermi field theories. This correspondence, which has been dubbed bosonization, provides a transparent unified method for solving two dimensional Fermi theories [ l**, 2-41. In this paper we will apply bosonization to the non-abelian SU(N) Thirring models [5,6,7+]. Although we will not solve these models, we will find several amusing relations between them and other two-dimensional theories. Our most striking result is for the case N = 2: we show that the SU(2) Thirring model is equivalent to the theory of a free massless scalar field and a Sine-Gordon field. The bare coupling constant of the Sine-Gordon theory is fixed at f12 = 87r, the value which makes the model exactly renormalizable. According to the results of Coleman [2], this implies a correspondence between the SU(2) model and another fermion model a free massless Fermi field and a massive abelian Thirring model. This work is divided into three parts. In sect. 2 we set up a precise operator scheme which implements the bosonization of free massless Fermi fields. Our formulas are very similar to those proposed by Dell’Antonio et al. [I] and by Mandelstam [3]. In sect. 3 we apply our formalism to the SU(N) models in the interaction picture and give a heuristic derivation of the equivalent boson Lagrangians for these models. Sect. 4 is devoted to a proof of the equivalence for the SU(2) case