We present a derivation of abelian and non-abelian bosonization in a path integral setting by expressing the generating functional for current-current correlation functions as a product of a $G/G$-coset model, which is dynamically trivial, and a bosonic part which contains the dynamics. A BRST symmetry can be identified which leads to smooth bosonization in both the abelian and non-abelian cases.Comment: 19 pages, LaTex, La Plata 94-0

The existence of several nilpotent Noether charges in the decoupled formulation of twodimensional gauge theories does not imply that all of these are required to annihilate the physical states. We elucidate this matter in the context of simple quantum mechanical and field theoretical models, where the structure of the Hilbert space is known. We provide a systematic procedure for deciding which of the BRST conditions is to be imposed on the physical states in order to ensure the equivalence of the decoupled formulation to the original, coupled one.1997 Academic Press

We discuss the equivalence between the path integral representations of spin dynamics for anti-commuting (Grassmann) and commuting variables and establish a bosonization dictionary for both generators of spin and single fermion operators. The content of this construction in terms of the representations of the spin algebra is discussed in the path integral setting. Finally it is shown how a 'free field realization' (Dyson mapping) can be constructed in the path integral.The dynamics of a particle with spin coupled to an external magnetic field α i can be formulated in the path integral in terms of commuting variables. A transition matrix element is given by [1]Here s is the spin half representation, [ds] the invariant measure on the coset U(1)\SU(2), σ i the Pauli matrices, λ the spin of the particle, and α = α i σ i .On the other hand the dynamics can also be formulated in terms of a path integral over Grassmann variables η, a transition matrix element being given by [2]One expects these two formulations to be equivalent. Here we show that this is indeed the case by transforming a slightly more general form of the path integral representation (2) in terms of Grassmann variables into the path integral (1) in terms of commuting variables.This process is popularly known as bosonization and plays a particularly important role in many-body physics and field-theories in, mostly, 1+1 dimensions. Indeed, the model we are about to present is a toy model for non-abelian bosonization in 1+1 dimensions, leading to the Wess-Zumino-Witten model [3]. This model is also important in the manybody physics setting, and a discussion on its bosonization and the role it plays in the many-body problem can be found in [4].Naturally bosonization entails a set of rules that relate equivalent fermionic and bosonic quantities. This set of rules results immediately from our procedure. In particular it enables us to write down the bosonic equivalents of the generators of the spin algebra as well as the bosonic equivalents of single fermion operators in terms of vertex operators [5].

We show how to bosonize two-dimensional non-abelian models using finite chiral determinants calculated from a Gauss decomposition. The calculation is quite straightforward and hardly more involved than for the abelian case. In particular, the counterterm AĀ, which is normally motivated from gauge invariance and then added by hand, appears naturally in this approach.The path integral approach to bosonization of twodimensional field theories was developed some time ago [1] by exploiting the fact that gauge fields may be decoupled from the fermions by making local chiral transformations. It is crucial in this approach that the anomalous contributions from the fermionic measure under chiral rotations are properly taken into account.More recently an approach to bosonization with path integrals, refered to as smooth bosonization, was introduced [2]. Here a chiral gauge symmetry is first introduced, and the bosonization rules are then obtained by choosing an appropriate gauge. Again the anomalous contribution from the path integral measure is of great importance. For an infinitesimal chiral transformation this contribution may be calculated by the method of Fujikawa [3], but for the applications mentioned above it is necessary to obtain the anomalous Jacobian for a finite chiral transformation. One therefore requires the iteration of the infinitesimal anomaly to obtain the Jacobian for a finite chiral change of variables.For the abelian case this problem is rather simple, and the abelian chiral anomaly is easily integrated to yield the free scalar boson action [1]. The non-abelian case is, however, complicated and it is difficult to show directly that the non-abelian anomaly can be integrated to yield the WZW [4] action. In this paper we present a simple and direct derivation of this result for su(2), hardly more involved than the abelian case.Although it is well known that the finite chiral transformation yields the WZW action (or the gauged WZW action in the presence of a background field) [5], this result is of such importance in the path integral bosonization program that we consider it worthwhile to present an alternative derivation. The present derivation is not only more directly constructive than the conventional one, which proceeds indirectly by showing that the variation of the WZW model is in agreement with the anomaly [5], but it also serves to clarify some issues connected with regularization. In this context it is often stated that it is necessary to introduce a counter term of the type AĀ by hand as to ensure gauge invariance [5,6]. We show here that this term arises naturally when the covariant form of the anomaly is iterated.The Jacobian associated with the final chiral transformation is calculated by using the Gauss decomposition for the group. The calculation is then relatively simple and the iteration of the infinitesimal result is hardly more involved than the abelian case. Normally there is very little advantage in writing the WZW model using an explicit parameterization for the group, because the...