Symmetries in the Hamiltonian formulation of string theory

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.