Projective representations of the spin group

  • #1

Main Question or Discussion Point

To define spinors in QM, we consider the projective representations of SO(n) that lift to linear representations of the double cover Spin(n). Why don't we consider projective representations of Spin?
 

Answers and Replies

  • #2
I answered my own question (I think).

Modulo redefinition of the phases of operators, projective representations are in correspondence with central extensions (as both are built out of nontrivial algebraic 2-cocycles). For n>2, Spin(n) is a universal cover, so the phase of any of its projective representations is a coboundary, which is to say that operators can be redefined to make the representation linear.

tl;dr: There are no nontrivial projective representations of Spin.

What happens in n=1,2? I haven't thought about it.
 
  • #3
Ah, these must be anyons.
 
  • #4
dextercioby
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By the analysis of Wigner and especially Bargmann, Spin(3) of QM is isomorphic to SU(2) which is known to be semi-simple, hence its Lie algebra has no non-trivial central extensions. This implies that the projective representations of SU(2) are trivially related to its linear representations.
 
  • #5
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It is true that in semi-simple lie algebra one can always remove the central charges by a redefinition of generators, but sometimes it is possible to remove the central charges even if the algebra is not semi-simple as is the case with Poincare group by using some special argument.
For n>2, we have our spin groups as double cover of the SO(n) and they are simply connected, hence they have no nontrivial central extension. However this is not the case for n≤2, like for n=2 the group is U(1) and it is infinitely connected.
 
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