I am currently studying the paper by Witten on supersymmetry and Morse theory. In the introduction it is stated that when supersymmetry is not broken, i.e. $Q|0\rangle=0$, the Hilbert space contains bosons and fermions of equal mass. I thought initially that this was the case simply because $[Q,H]=0$ (take f.e. a boson $|b\rangle$ with energy E, then $HQ|b\rangle=QH|b\rangle=EQ|b\rangle$ so its fermion counterpart has energy E as well) but I don't see why you can not use this to argue that there are also equal mass bosons/fermions when the symmmetry is broken (which is of course not the case). My question thus comes down to the fact that I don't understand why the $Q|0\rangle=0$ requirement is necessary and how symmetry breaking leads to different masses. In the paper there is also this sentence
Now in any quantum field theory if a symmetry operator (an operator which commutes with the Hamiltonian) annihilates the vacuum state, then the one particle states furnish a representation of the symmetry. In the case of a supersymmetric theory, if a solution of (8) does exist, then the Hilbert space of the theory contains bosons and fermions of equal mass.
which I find a bit vague but might give an answer. There is actually already thread about this sentence here but I don't think the answers there help me with my questions.