Hi Everybody! I am working on QFT and learning representation theory from Coleman's lecture notes. Just the necessary stuff to go to the Dirac equation. To my question: From the generators of SO(3) I get through exponentiation an element of SO(3), this holds naturally for any Lie group. Now, the generators themselves are n x n dimensional. Then, if I take as my generators the pauli sigma matrices which satisfy the commutation relations of the Lie group SO(3) then the resulting representations of the elements of SO(3) should be rotations which act on two component spinors. BUT how do I get an element of SO(3) which acts on vectors? Or better say which acts on a multiplett such that it transforms as a vector. A 3 x 3 representation of the Lie algebra should give me an element of SO(3) which acts on 3 compenent spinors I guess, but not vectors. Let me repeat, when constructing the represention of SO(3) with a 3 x 3 representation of the Lie algebra, I should get a rep that transforms 3 spinors, but how do I get a rep that transforms 3 vectors? Does it depend on the choice of the representation of the Lie algebra? I saw the words vector representation and spinor representation, are these in any way related to my problem? Thanks for any help in advance!!