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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!

Jackson Burzynski
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  • 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. – Javier Nov 21 '16 at 18:20
  • Related: http://physics.stackexchange.com/q/96045/2451 and links therein. – Qmechanic Nov 21 '16 at 18:21
  • @Qmechanic this is exactly what I was looking for, thank you! – Jackson Burzynski Nov 21 '16 at 18:25
  • @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? – Alfred Centauri Nov 22 '16 at 00:38
  • @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? – Jackson Burzynski Nov 22 '16 at 03:39
  • @Qmechanic or is it that we are actually always discussing the projective representation of SO(3), but since the three dimensional representation of SU(2) corresponds to that of SO(3) we don't need to worry about it in the case of orbital angular momentum? – Jackson Burzynski Nov 22 '16 at 03:50
  • The latter question is a duplicate of http://physics.stackexchange.com/q/153369/2451 – Qmechanic Nov 22 '16 at 11:28

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