# Which matrices represent unitary projective representations of ${\rm SO(3)}$?

I was reading this post which triggered the following question.

The group $${\rm SO(3)}$$ is real orthogonal. However, it is possible to consider representations of $${\rm SO(3)}$$ on a complex vector space. In particular, any element $$R_a\in {\rm SO(3)}$$ can be represented by a unitary matrix $$U(R_a)$$ by choosing the generators $$J_i$$ to be hermitian and satisfying the Lie algebra $$[J_a,J_b]=i\epsilon_{abc}J_b.\tag{1}$$ For example, choosing $$J_a=\frac{1}{2}\sigma_a$$, we obtain a Unitary representation $$R_a\mapsto U(R_a)=\exp\Big[\frac{-i\theta_a \sigma_a}{2}\Big]\tag{2}.$$ This is a two dimensional Unitary representation with $$j=1/2$$. We can also arrive at higher dimensional unitary representations of with $$j=3/2,5/2,...$$ etc.

$$1.$$ What should I call the representation $$(2)$$ (and all those with $$j=3/2,5/2,...$$) ? Should I call it a unitary representation of $${\rm SO(3)}$$ or call it the fundamental unitary representation of $${\rm SU(2)}$$?

$$2.$$ If $$(2)$$ is a complex unitary representation of $${\rm SO(3)}$$, is it projective? I do not see that because $$R_z(\pi)R_z(\pi)=R_z(2\pi)\tag{3}$$ is mapped to $$U(R_z(\pi))U(R_z(\pi))=U(R_z(2\pi))$$ which looks non-projective or ordinary. However, according to Wikipedia, half-integral spin representations are projective representations of $${\rm SO(3)}$$.

$$3.$$ If $$(2)$$ and its higher-dimensional representations with $$j=3/2,5/2,...$$ etc are all ordinary non-projective representations of $${\rm SO(3)}$$, it seems that it is not necessary to talk about the $${\rm SU(2)}$$ group in quantum mechanics. Apparently, we shall not lose any physics by confining ourselves to the ordinary non-projective complex unitary representations of $${\rm SO(3)}$$.

1. In the defining 3-dimensional representation of the 3D rotation group $$SO(3)$$ a $$2\pi$$-rotation is the identity element $$R(2\pi)~=~\mathbb{1}_{3\times 3}.\tag{1}$$
3. For $$2\times 2$$-matrices, we calculate using various properties of the Pauli matrices that $$U(R(2\pi))~=~-\mathbb{1}_{2\times 2},\tag{2}$$ which shows that $$U$$ cannot be a genuine group representation of $$SO(3)$$.
• $\uparrow$ Yes. – Qmechanic Sep 1 at 8:43