ar X iv : h ep - t h / 98 06 06 7 v 3 3 S ep 1 99 8 Current Algebra in the Path Integral framework

In this letter we describe an approach to the current algebra based in the Path Integral formalism. We use this method for abelian and non-abelian quantum field theories in 1+1 and 2+1 dimensions and the correct expressions are obtained. Our results show the independence of the regularization of the current algebras. PACS number: 03.70.+k, 11.40.Ex Typeset using REVTEX E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] 1 Abelian bosonization and current algebras play an important role in the description of two-dimensional quantum field theories as non-perturbative methods and they are an important ingredient in order to show the equivalence between different (two-dimensional) models [1–3] (for a complete review of the most important references in the field see [4]). The non-abelian extension of the bosonization is, however, a more technical problem that was solved in [5]. Essentially, the solution given by Witten for N free fermionic fields was to show the equivalence with a Wess-Zumino-Witten [6] theory with the current algebra describing a SU(N) Kac-Moody algebra. In the abelian or non-abelian bosonization, the current commutators are normally computed using a point splitting regularization plus the Bjorken-Johnson-Low (BJL) limit, in order to have an equal-time commutator. Although it seems a technical point, the computation of the current-current commutator using different regularizations could shed some light on the independence of the regularization of the current algebra [8]. The purpose of this paper is to present an explicit calculation of the current algebra in two and three dimensions based in the path integral approach. This procedure allows translating the definition of the product of two operators at the same point, to a regularization of a functional determinant where many other regularizations are available. In order to compute the current algebra, let us start considering a massless fermion in 1+1 dimensions coupled to a gauge field Aμ L = ψ̄iD/ψ, (1) where Dμ = ∂μ + Aμ. The gauge field Aμ is an external auxiliary field that can be set equal to zero at the end of the calculation. Thus, from the euclidean partition function Z [A] = ∫