How is $j=1/2$ representation, $U(R(theta,hat{bf n}))=e^{i{sigma}cdot{hat {bf n}}theta/2}$, is a projective representation of ${rm SO}(3)$?

Question

A projective unitary representation of ${rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{iphi(R_1,R_2)}U(R_1R_2)tag{1}$$ where $R_1,R_2in {rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(theta,hat{bf n}))=e^{i{sigma}cdot{hat {bf n}}theta/2}$, is a projective representation of ${rm SO}(3)$ i.e., satisfies the condition $(1)$. To do this, one has to show that for $$R_1R_2=R_3Rightarrow U(R_1)U(R_2)=e^{iphi}U(R_3).tag{3}$$ Any suggestions how to show this or at least check this?

ZeroTheHero · Answer

Take $hat{bf{n}}_1=hat{bf{n}}_2$, and $theta_1+theta_2=2pi$.  As an $SO(3)$ element, you should have $U(R_1)U(R_2)=U(2pi)=1$ but here you get $-1$.
Thus, you get $U(R_1)U(R_2)=e^{iphi}1$, where $e^{iphi}=-1$ (or $phi=pi$)

Qmechanic · Answer

OP describes projective representations in terms of a 2-cocycle. An alternative description is in terms of a quotient $$PSU(2)~:=~ SU(2)/mathbb{Z}_2~cong~SO(3),$$
where $SU(2)$ denotes the 2-dimension $j=1/2$ non-projective defining/fundamental/spinor representation and
$$mathbb{Z}_{2}~cong~{pm {bf 1}_{2 times 2}}.$$
In other words, in this latter description the 2-dimensional representation of $SO(3)$ is double-valued, i.e. $pm U$ represents the same $SO(3)$ rotation.

$vec{alpha}=thetahat{bf n}$ is a rotation-vector in the axis-angle representation. Note that the $4pi$-periodicity of $SU(2)$ becomes the familiar  $2pi$-periodicity of $SO(3)$. See also e.g. this & this related Phys.SE posts.

How is $j=1/2$ representation, $U(R(theta,hat{bf n}))=e^{i{sigma}cdot{hat {bf n}}theta/2}$, is a projective representation of ${rm SO}(3)$?

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