The state of a particle will generally change if you rotate it. The details of how the state changes under an infinitesimal rotation are contained in the angular momentum operator J. This operator can be divided into two parts: an "orbital" angular momentum and a "spin" angular momentum.

The orbital angular momentum tells you how the state changes under a rotation due to wavefunction of the particle. I think this is pretty intuitive: if the wavefunction has some angular variation, then rotating the particle will change its state, so there ought to be some contribution to J due to the wavefunction. Spin angular momentum on the other hand is a lot stranger, at least to me. If you rotate a particle with spin, its state will change even if its wavefunction is completely isotropic.

Maybe I'm being naive, but to me this fact implies that particles cannot be point-like but must have some extended structure. If particles were really point-like, their only degrees of freedom would be their position, so how could something like spin arise?

Could string theory explain spin then? Or is spin introduced into string theory in as ad hoc a way as it's introduced into non-string physics?

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    $\begingroup$ Comment to the question (v1): OP is essentially repeating in disguise the faq How can a quantum point particle have intrinsic angular momentum a.k.a. spin? See physics.stackexchange.com/q/1/2451 and links therein. If OP doesn't appreciate this fact for point particles, it is likely not going to be helpful to think about string theory. $\endgroup$ – Qmechanic Mar 28 '13 at 15:46
  • $\begingroup$ Hi Qmechanics. The "answers" in the FAQ don't actually answer the question. They simply vigorously restate the claim. The question is, how do particles have internal angular momentum (ie spin)? The "answer" given is that particles have internal angular momentum. $\endgroup$ – occasional_comment Mar 28 '13 at 16:34
  • $\begingroup$ The one point that was addressed is that angular momentum in QM has nothing to do with rotating relative to a point. In QM, angular momentum is all about how states transform under rotation. But point particles CAN'T change under rotation except by their wavefunction, so while the classical picture of angular momentum is flawed, the fact that point particles have internal angular momentum is still mysterious. $\endgroup$ – occasional_comment Mar 28 '13 at 16:35
  • $\begingroup$ Spin is not postulated in all theories, in fact the Dirac equation requires particles to have some spin property. How string theory elaborates on this you can find a summary of here. Since I'm not an expert I can't tell you anything more than what is in the link. But as you might expect intuitively, string theory apparently describes spin as a rotation of the string. $\endgroup$ – Wouter Mar 28 '13 at 16:38
  • $\begingroup$ Hi Wouter. As I understand it, while Dirac initially thought that QM + special relativity lead necessarily to spin, the modern understanding is that it's perfectly acceptable to construct a relativistic theory of spin-zero particles and that the Dirac equation only comes about the you consciously try to make a theory for spin-1/2 fields. $\endgroup$ – occasional_comment Mar 28 '13 at 21:54

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