My first question is fairly basic, but I would like to clarify my understanding. The second question is to turn this into something worth answering.
Consider a relativistic electron, described by a spinor wave function $\psi(\vec x ,\sigma)$ and the Dirac equation. The conventional wisdom is that rotating everything by 360 degrees will map the spinor to its negative $\psi \mapsto -\psi$. However, it appears to me that this statement is "obviously false", because a rotation by 360 is, when viewed as an element of the group $SO(3)$, exactly equal to the identity map and cannot map anything to its negative.
Thus, to make sense of the behavior of spin under "rotation", I have to conclude the following
The rotation group $SO(3)$ does not act on the configuration (Hilbert) space of electrons. Only its double cover $SU(2)$ acts on the space of electrons.
Is this interpretation correct?
So, essentially, there is a symmetry group $SU(2)$ which acts on "physics", but its action on the spatial degrees of freedom is just that of $SO(3)$.
What other groups, even larger than $SU(2)$, are there that (could) act on "physics" and are an extension of $SO(3)$? Is it possible to classify all possibilities, in particular the ones that are not direct products?
Of course, gauge freedoms will give rise to direct products like $SO(3) \times U(1)$ (acting on space $\times$ electromagnetic potential), but I would consider these to be trivial extensions.