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Since strings are extended objects, is all angular momentum in string theory essentially "orbital" angular momentum? Or is there still a kind of intrinsic angular momentum assigned to a string?

Either way, is there anything that prevents the "intrinsic spin" of a particle represented by a string from being arbitrarily large?

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The orbital momentum of a string may be arbitrarily large. Whether it should be called "orbital" or "intrinsic" depends on the perspective. The right answer is the formula, such as $$J_{ij} = \int d\sigma [x_i (\sigma) p_j (\sigma) - p_i(\sigma) x_j(\sigma) + \gamma_{ij}^{ab} \theta_a(\sigma) \theta_b(\sigma)]$$ I added some superstring term, too. The $xp$ terms may be viewed as a density of the orbital angular momentum on the string; the fermionic term is its most direct fermionic generalization. However, both of these terms, and especially the latter, become "intrinsic angular momentum" when you expand the fields $x,p,\theta$ to Fourier modes and interpret the string as a particle with internal oscillations. In particular, the intrinsic spin-1/2 always comes from the quantization and/or excitations of the fermionic degrees of freedom.

The formula says much more than just some confusing dogmatic word "intrinsic" vs "orbital", at least to a person who wants to understand the terms accurately enough - at the level of maths.

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