Pauli Matrices as generators of SU(2)

[OSUPhysics]

Why is it that the Pauli spin matrices ( the operators of quantum spin in x,y,z) are the generators of a representation of SU(2)? I understand that we use the 2X2 representation as it is the simplest, but why is it that spin obeys this SU(2) symmetry and how is it that we come up with the Pauli matrices for the spin operators?

 

[Ben Niehoff]

SU(2) is isomorphic to Spin(3), which is the double cover of the rotation group in three dimensions. That is, its Lie algebra is identical to the rotation group, but the map between the groups themselves is 2-to-1.

The real group of interest here is Spin(3), which is constructed via Clifford algebra. In this case, it happens to be equivalent to SU(2).

Deriving the Pauli matrices from scratch involves some slightly more advanced ideas that maybe someone else has more time to relate.