Spin of elementary particles

Question

Going by the explanation given by Stephen Hawking (as given in Brief History of Time) , the spin of a particle is no. of rotation you give to that particle so that it looks the same. Like you give half rotation to particle of spin 2 (just like a double headed arrow) and 1 complete rotation of particle of spin 1. But why does an electron with spin 1/2 Have two spins +1/2 and -1/2. If +1/2 spin means you need to give two complete rotation to the particle then what does -1/2 mean? I don't know if I am following the right approach. Can someone explain?

Answer 1

In the representation theory of angular momentum ($su(2)$ for spin and $so(3)$ for rotations: the algebras are equivalent) one labels states by their eigenvalue of the operator of the z-component, say, of angular momentum and their eigenvalue of the "square" of the angular momentum (would be the length of the angular momentum vector in classical mechanics). The latter operator, S^2, is the casimir of the group and commutes with the other operators.

So states are written $| s, m\rangle$ where the labels are the total spin, s, and its projection in the z direction. In particular, the electron is spin s= 1/2 and as such its projection in the z direction is m=+1/2 or m=-1/2. So two states, of spin 1/2, that differ only in the orientation of their spin with respect to the z axis.