Each additional boson then corresponds to a decrease of ħ in the spin projection. Thus, the spin raising and lowering operators S + = S x + i S y and S − − = S x − − i S y , so that [ S + , S − − ] = 2 ℏ ℏ S z , correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct commutation relations for the spin operators, such that they act on a finite-dimensional space, unlike the original Fock space.