Editing Wigner's classification (section)

==Theory of projective representations==
Physically, one is interested in irreducible [[projective representation|''projective'' unitary representations]] of the Poincaré group. After all, two vectors in the quantum Hilbert space that differ by multiplication by a constant represent the same physical state. Thus, two unitary operators that differ by a multiple of the identity have the same action on physical states. Therefore the unitary operators that represent Poincaré symmetry are only defined up to a constant—and therefore the group composition law need only hold up to a constant.

According to [[Projective representation#Infinite-dimensional projective unitary representations: Bargmann's theorem|Bargmann's theorem]], every projective unitary representation of the Poincaré group comes from an ordinary unitary representation of its universal cover, which is a double cover. (Bargmann's theorem applies because the double cover of the [[Poincaré group]] admits no non-trivial one-dimensional [[Group extension#Central extension|central extension]]s.)

Passing to the double cover is important because it allows for half-odd-integer spin cases. In the positive mass case, for example, the little group is SU(2) rather than SO(3); the representations of SU(2) then include both integer and half-odd-integer spin cases.

Since the general criterion in Bargmann's theorem was not known when Wigner did his classification, he needed to show by hand (§5 of the paper) that the phases can be chosen in the operators to reflect the composition law in the group, up to a sign, which is then accounted for by passing to the double cover of the Poincaré group.