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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?

Ramiro Hum-Sah
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Palmy
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  • Related: https://physics.stackexchange.com/q/449169/2451 – Qmechanic Jun 19 '20 at 16:32
  • The stress tensor $T_{ab}$ describes the response of the string under a change of the worldsheet metric. On the other hand, the energy and momentum of a string is the response of the string under a change of the target space metric, i.e. they have to do with target space isometries, NOT with worldsheet diffeomorphisms/isometries. There are two different metrics here so you need to be careful. – Prahar Jun 19 '20 at 16:39

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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.

Ramiro Hum-Sah
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    "this tensor is the energy-momentum tensor of the free boson CFT that lives on the worldsheet" - this definition is found in Ralph Blumenhagen, Erik Plauschinn, "Introduction to Conformal Field Theory With Applications to String Theory", eqn. (2.84). – Joseph Shtok Jan 17 '21 at 12:16