Symmetries in the Hamiltonian formulation of string theory

TL;DR

This paper revisits the Hamiltonian formulation of string theory, demonstrating that diffeomorphism and Weyl symmetries can be naturally recovered without field-dependent reparametrizations, thus resolving a longstanding inconsistency.

Contribution

It shows that the standard Hamiltonian approach can preserve symmetries without losing covariance, clarifying the role of parametrization in string theory.

Findings

Diffeomorphism and Weyl symmetries emerge naturally via Castellani's procedure.

The failure of previous formulations is due to parametrization issues, not non-canonicity.

Standard Hamiltonian analysis can be consistent with Lagrangian symmetries.

Abstract

In the context of the Hamiltonian formulation of string theory, a widely acknowledged issue is the inability of the first-class constraints to accurately reproduce the Lagrangian symmetry transformations. We take a critical look at the Hamiltonian formulation of the Polyakov string and demonstrate, using Castellani's procedure, that diffeomorphism and Weyl symmetries naturally emerge without using any field-dependent reparametrization, thus preserving the theoretical consistency between Hamiltonian and Lagrangian descriptions. Additionally, we will review the standard Hamiltonian analysis of the symmetries in terms of lapse and shift functions and show that the reason behind this failure is not due to the non-canonicity of the variables but rather because this parametrization fails to preserve general covariance.