N=1 susy with massive supermultiplets; therefore we have two pairs of raising/lowering operators. Define a1 dagger as raising j3 by half and a2 dagger as lowering j3 by half. Take vacuum state with j, j3 where j3 has 2J+1 possible values. If we act with just a1 dagger we get superposition of states, both with j3+1/2 but one has j+1/2 and one has j-1/2 . If we act just with a2 dagger we get a similar situation but with j3-1/2 . But if we act with both a1 and a2 dagger we only get a state with j, j3. I understand that the j3 is additive, but surely we should have states with j plus or minus one as well? It seems to be postulated that the spin can only go up or down by half but not by one?