Does Energy-momentum tensor from Polyakov action have anything to do with the energy and momentum of the string? I'm a bit confused with the idea of EMT from the Polyakov action. The EMT is derived by variation of the action with respect to the metric which provide the constraint to the theory, $T_{ab}=0$. Clearly, the energy and momentum of the string is no need to be zero. Is the EMT here related to the physical energy/momentum of the string? what are their connections?
 A: The energy-momentum tensor from which the (no-Weyl anomaly) constraint $\langle T^{\mu}_{\mu}(\sigma) \rangle=0$ is derived as $$T^{\mu \nu}= \frac{4\pi}{g(\sigma)^{1/2}}\frac{\delta}{\delta g_{\mu \nu}(\sigma)}S_{P},$$ here $S_{P}$ denotes the Polyakov action. Notice that this tensor is the energy-momentum tensor of the free boson CFT that lives on the worldsheet but certainly is not the energy momentum tensor of the worldsheet. Just to emphasize, the electromagnetic energy-momentum tensor is the EMT of a Maxwell-theory living on a given spacetime, but not the spacetime energy-momentum tensor.
The relevant EMT for a relativistic string moving in spacetime is proportional to the variation of the Nambu-Goto action with respect to the derivatives of the string embedding functions $X^{\mu}(\tau,\sigma)$. With it you may compute the hamiltonian density of a string if you wish (see section 7.3 in Zwiebach's textbook). But in principle the energy-momentum tensor of the string is different from the energy-momentum of the free boson CFT that lives on the worldsheet.
