# Why do we classify states under covering groups instead of the group itself?

Why do we always classify states under covering group representations instead of the group itself? For example see the following picture I lifted from 'Symmetry in physics' by Gross

So in the first example, why not classify states under SO(3) instead of SU(2)?

EDIT: From a physicists perspective, I know enough that we consider projective representations of groups in quantum mechanics because those are the most general thing that keeps $|\langle \phi|\psi\rangle|^2$ invariant. But Gross above seems to indicate that representations of the covering group are a better way to go. Furthermore, covering groups are always simply connected, to where-as groups with projective representations are in general not simply connected - is this property of (universal) covering groups related to why we use them in these sorts of cases?

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A group is an abstract mathematical object. Measurements/Experiments are in the real field. So, if you want to draw meaningful conclusions that you can verify experimentally, you have to work with operator representations (in $\mathbb{R}$) that are appropriate. Ultimately, if you cannot prove something (or its consequences) that can be experimentally verified, it is worthless. While this does not directly answer your question, I think it is a useful reminder on why we work with representations instead of abstract objects. Hence the comment. –  Antillar Maximus Dec 28 '12 at 1:22
The reason we use covering groups is because finding projective representations of a Lie group $G$, is equivalent to finding linear representations of its universal cover $G'$, and we usually prefer to work with single valued objects. For example half-integer spin representations are linear wrt. $SU(2)$, but projective wrt. $SO(3)$. However $G$ and $G'$ have isomorphic Lie algebras, so one can alternatively consider representations of the Lie algebra, which is what physicist usually prefer to do. –  Heidar Dec 28 '12 at 11:25
@Heidar that could (and, I think, should) be an answer, or at least the start of one. –  David Z Dec 29 '12 at 5:06