Chapter 4 Su(2) 4.1 Representations of Su(2)

Thus our generators are not quite canonically normalized, but are all normalized equally, and β is positive definite. This is related to the fact, which we have already seen, that the group is compact. In writing down our generators, we have chosen, arbitrarily, one direction to make diagonal. Any rotation can be, by similarity transformation with at rotation, rotated into the z direction, so as we have a choice as to which of the infinite number of equivalent representations to choose, we may choose L3 to be diagonal. If we had several generators which commuted with each other, we could have chosen all of them to be diagonal. In general, we will take a maximal set of commuting generators, called the Cartan subalgebra, and represent them as diagonal. For SU(2), no rotations about any other axis commute with L3, so the Cartan subalgebra is one dimensional.