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If $| \psi \rangle$ is a string mode, how do you compute $\langle \psi | \hat{T}^{\mu\nu}(\vec{x}) | \psi \rangle$ where $\vec{x}$ is a point in target space? This information will tell us the energy distribution of a string. In string theory, the size of a string grows logarithmically as the worldsheet regulator scale. In the limit of zero regulator size, all strings are infinite in size, and this ought to show up in the stress-energy distribution.

What about multi-string configurations? The vacuum has virtual string pairs. Only the spatial average $\int d^9x \hat{T}^{00}(\vec{x}) |0\rangle = 0$. $\hat{T}^{00}(0) | 0 \rangle \neq 0$ even though $\langle 0 | \hat{T}^{00}(0) | 0 \rangle = 0$. Here, $|0\rangle$ is the string vacuum. The stress-energy operator can create a pair of strings. It doesn't preserve the total number of strings.

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String theory isn't field theory--- the string is all there is. You compute S-matrix elements, not off-shell things, and local T is a field theory concept. – Ron Maimon May 14 '12 at 7:02

There's no stress-energy tensor in string theory. If there were, the Weinberg-Witten argument can be applied to the graviton string modes, and that would lead to a contradiction.

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This is a really good answer +1, but it should be mentioned that the stress-energy issues are directly analogous to those of General Relativity, and that Weinberg Witten argument is based on Case and Gasiorowicz 1962 argument. – Ron Maimon May 14 '12 at 7:00

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