Hello, I'm reading Ryder's Quantum Field Theory book, and I'm reading the preliminary part where he discusses a little bit of Groups before he introduces the Dirac equation. So, as an example, he is talking about the identification between the SU(2) group and the O(3) group. Before he gets there though, he talks about the transformation properties of the spinors. He makes a statement in equation 2.40 that the outter product of the spinor with it's hermitian conjugate transforms as: [tex]\xi\xi^\dagger \rightarrow U\xi\xi^\dagger U^\dagger [/tex] Later, he introduces another matrix that transforms like this one in equation 2.47. It's the outter product between [itex](\xi_1, \xi_2)[/itex] (imagine this is a column vector, I forget how to make a column vector on Latex) and [itex](-\xi_2, \xi_1)[/itex] (this is a row vector). I get that the outter product of these two objects is a matrix which transforms according to the above law, but, I don't see why Ryder has to make this matrix. He later makes the identification with O(3) by identifying this matrix with one he creates from an O(3) vector dotted with the pauli matrices. I just don't get why he uses this matrix instead of the one in equation 2.40.