What is the physical meaning of SU(2) being the double covering of SO(3)? Say we have a magnetic field oriented in some direction, is the effect of this field somehow associated with rotations in 3 dimensional space?
The magnetic field which will have an effect on the spin of some physical system will be represented by some linear comb. of the Pauli matrices. How does this relate to SO(3)?
A good example to understand this is to take an electron, using simply SO(3) will give a direction in the 3D space but won't give the projection of the spin, so you have an ambiguity of the sign here. One fermion represented in SO(3) will have 2 elements in SU(2), or you could will correspond to a spinor, thus this is topographically equivalent to say that that SU(2) is a double cover of SO(3). Another example is that if you took an electron (spin 1/2), we need to take two full rotations in the 3D space to go back to the initial state (double cover).
In the Stern-Gerlach experiment, when you apply the magnetic field you will separate the fermions into two groups, an up group and a down group and this is an effect of the ambiguity over the sign of the projection of the spin.
SO(3) is perfect to describe bosons, because then the spin won't have an effect in the description of the state. But in case of fermions, SO(3) isn't accurate and you will always have a projection of the state you're trying to determine.
