Not necessarily. The previous answers seemed to be missing the mark somehow,
so I wanted check Mr Vodka's prior knowledge of Lie groups, QM, and their use in
the classification of elementary particles.
Ballentine covers the rotation group SO(3) much earlier, so he assumes the
reader already knows about how those noncommuting ##J_x,J_y,J_z## operators
form the Lie algebra of so(3) (or su(2), but let's skip that for the moment).
He also assumes that by section 7 the reader knows about the relationship between
a Lie group like SO(3) and the Lie algebra so(3) which "generates" the group.
From what you said above, I get the impression you're not yet clear on the intimate
connection between angular momentum and the rotation group SO(3) ?
No, it's "complete" because there's only one "Casimir" invariant operator for
the SO(3). (A Casimir invariant is an operator which commutes with all the
group operators, other than the trivial identity operator.) Since this
derivation focuses only
with angular momentum and nothing else, he
doesn't need to consider other stuff at this point.
OK, that's close enough for present purposes.
The feeling I got reading through earlier parts of this thread is that you
were having trouble transferring the principles in the quantum theory of
angular momentum over to more general cases of classification of elementary
But... to ensure none of us waste our time..., maybe you should restate your
question(s) as clearly as you can now?