EQUIVALENCE OF THE LAGRANGIAN AND HAMILTONIAN BRST CHARGES FOR THE BOSONIC STRING IN THE HARMONIC AND CONFORMAL GAUGES

Abstract

We consider the BRST formalism for the bosonic string in arbitrary gauges, both from the Hamiltonian and from the Lagrangian point of view. In the Hamiltonian formulation we construct the BRST charge Q(H) following the Batalin-Fradkin-Fradkina-Vilkovisky (BFFV) formalism in phase space. In the Lagrangian formalism, we use the Noether procedure to construct the BRST charge Q(L) in configuration space. We then discuss how to go from configuration to phase space and demonstrate that the dependence of Q(L) on the gauge fixing disappears and that both charges become equal. We work through two gauges in detail: the conformal gauge and the de Donder (harmonic) gauge. In the conformal gauge one must use equations of motion, and a simple canonical transformation is found which exhibits the equivalence. In the de Donder gauge, nontrivial canonical transformations are needed. Our results overlap with work by Beaulieu, Siegel and Zwiebach on the de Donder gauge, but since we only require BRST invariance and not anti-BRST invariance, we need simpler field redefinitions; moreover, we stay off-shell.