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This question already has an answer here:

I have been studying the group theoretic formalism of quantum mechanics and I have yet to find a satisfying explanation for the need for projective representations in the theory of angular momentum. I understand the connection between SO(3) and angular momentum, and I understand the the projective representation of SO(3) is given by the usual representation of the double cover, SU(2). The thing that I am missing is $\textit{why}$ the projective representation is important in the first place.

Furthermore, I understand the consequences of studying SU(2) rather than SO(3), namely that for half-integer spin particles a rotation of $2\pi$ under SU(2) transforms the wavefunction to its negative, but for integer spin particles the rotation is equal to the identity, but as I said, I see this a a consequence rather than motivation for studying SU(2) over SO(3). Is there a more satisfying answer to my question other than "we do it this way because it works"? Thanks!

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marked as duplicate by Qmechanic quantum-mechanics Nov 22 '16 at 11:29

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  • $\begingroup$ You've almost got it, the argument just goes backwards: we study projective representations because we don't care if the state picks up a phase when rotating. $\endgroup$ – Javier Nov 21 '16 at 18:20
  • $\begingroup$ Related: physics.stackexchange.com/q/96045/2451 and links therein. $\endgroup$ – Qmechanic Nov 21 '16 at 18:21
  • $\begingroup$ @Qmechanic this is exactly what I was looking for, thank you! $\endgroup$ – Jackson Burzynski Nov 21 '16 at 18:25
  • $\begingroup$ @Qmechanic, since the OP has effectively had his/her question answered, given the comment just above, what is the proper action, if any, to be taken for this question? $\endgroup$ – Alfred Centauri Nov 22 '16 at 0:38
  • $\begingroup$ @Qmechanic As a follow up, I now understand why the projective representation is used for spin, but I don't see why it isn't also used for orbital angular momentum. Why do we not care about extraneous phases in the case of spin but retain them in the orbital case? $\endgroup$ – Jackson Burzynski Nov 22 '16 at 3:39