On the Covariantization of the Chiral Constraints

We show that a complete covariantization of the chiral constraint in the FloreaniniJackiw necessitates an infinite number of auxiliary Wess-Zumino fields otherwise the covariantization is only partial and unable to remove the nonlocality in the chiral boson operator. We comment on recent works that claim to obtain covariantization through the use of Batalin-Fradkin-Tyutin method, that uses just one Wess-Zumino field. This work is supported by CNPq, Braśılia, Brasil Permanent address: Instituto de F́ısica, Universidade Federal do Rio de Janeiro, Brasil The quantization of chiral boson in two-dimensions is a very interesting theoretical problem, which has appeared originaly in the investigation of heterotic string [1], and became quite important in the study of fractional quantum Hall efect [2]. This problem has a simple solution in the Hamiltonian language while it has been beset with enormous difficulties in the Lagrangian side. One of these problems is the covariantization of the second-class chiral constraint, i.e., the transformation from second to first-class, which is the object of investigation in this paper. The usual Lagrangian route to chiral bosonization starts with a scalar field and projects out one of the chiral components by means of the chiral constraint ∂±φ ≈ 0. According to Dirac’s theory of constrained systems [3], this is a secondclass constraint, but in order to avoid the Lagrange multiplier to become dynamical, one needs it to be first-class. There has been two main routes to the covariantization of the chiral constraint. Siegel [4] proposed to set to zero one component of the energymomentum tensor resulting an action with the chiral constraint squared, which has a reparametrization invariance called as Siegel symmetry. In the quantum level, however, Siegel symmetry becomes second-class again on account of the central extension of the conformal algebra of the energy-momentum tensor [5]. A simple solution to the anomaly problem was given by Hull with the introduction of a new set of auxiliary fields called as no-movers [6]. Besides the anomaly problem, to produce first-class constraints by squaring second-class constraints has been criticized [7]. This sort of primary constraint does not produce the complete set of constraints when Dirac’s algorithm is employed, and is (infinitely) reducible. The second route to covariantization starts with the Floreanini-Jackiw model[8] which is a singular theory from Dirac’s point of view, the resulting constraint being the second-class chiral constraint. The constraint’s nature is changed à la FaddeevShatashvili[9] with the introduction of Wess-Zumino auxiliary fields. The covariantization of the chiral constraint in this route has been proposed in two conceptually differents papers: in Ref.[10], an infinite family of scalar fields, coupled by a combination of right and left chiral constraints carefully adjusted to be first-class from the start, was shown to have a single chiral boson in the spectrum by use of very elegant group theoretical methods. In Ref.[11], the FJ chiral boson was iteratively changed to modify the nature of chiral constraint to render it first-class. Following this route the modified Floreanini-Jackiw model with the original chiral field plus the set of Wess-Zumino fields will always have one left over second-class constraint that must as well be converted into first-class if a complete covariantization is desired. It is worth of mention that, as explained by Boyanovsky[12], the inclusion of constraint conversion terms do not change the physical spectrum of the theory but only the