Polyakov's string classical mechanics

TL;DR

This paper develops the Hamiltonian framework for Polyakov's classical string, clarifying its constraints and gauge symmetries, which will aid future studies in Wigner-covariant rest frame theory.

Contribution

It provides a detailed Hamiltonian analysis of Polyakov's string, including constraints, gauge fixing, and the classical Virasoro algebra, extending the understanding of string classical mechanics.

Findings

Hamiltonian scheme for Polyakov's string derived

Constraints and gauge symmetries analyzed and fixed

Classical Virasoro algebra identified as secondary constraints

Abstract

This paper is almost an exercise in which the Hamiltonian scheme is developed for Polyakov's classical string, by following the usual framework suggested by Dirac and Bergman for the reduction of gauge theories to their essential physical degrees of freedom. The results collected here will be useful in some forthcoming papers, where strings will be studied in the unusual context of Wigner-covariant rest frame theory. After a short introduction outlining the work, the Lagrangean scheme is presented in Section II, where the classical equivalence between Polyakov and Nambu-Goto string is rederived. In Section III the Hamiltonian framework is worked out, primary and secondary constraints are deduced. Then Lagrange multipliers are introduced; finally the Hamiltonian equations of motion are presented. In Section IV gauge symmetries are treated, by constructing their canonical generators; then some gauge degrees of freedom are eliminated by Dirac-Bergmann fixing procedure. In this paper only the gauge-freedom coming from primary constraints is fixed, while secondary constraints will need a deeper analysis. At the end of Section IV the classical version of Virasoro algebra is singled out as that of the secondary constraints, surviving to the first gauge fixing.