Technically, formula_39 is the Hermitian adjoint of the operator in the inner product space of ket vectors. The identification of formula_39 and as creation and annihilation operators comes from comparing conserved quantities for a state before and after one of these have acted upon it. formula_39 can for example be seen to add one particle, because it will add to the eigenvalue of the a-particle number operator, and the momentum of that particle ought to be since the eigenvalue of the vector-valued momentum operator increases by that much. For these derivations, one starts out with expressions for the operators in terms of the quantum fields. That the operators with formula_10 are creation operators and the one without annihilation operators is a convention, imposed by the sign of the commutation relations postulated for them.