The action of the operator \( a_\alpha \) from the left on a state vector is to remove one particle in the state \( \alpha \). If the state vector does not contain the single-particle state \( \alpha \), the outcome of the operation is zero. The operator \( a_\alpha \) is normally called for a destruction or annihilation operator.

The next step is to establish the commutator algebra of \( a_\alpha^{\dagger} \) and \( a_\beta \).