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For fermions, we define the operator by:

(A-18)

where the newly created state |ui〉 appears first in the list of states in the ket on the right-hand side. If we start from a ket where the individual state |ui〉 is already occupied (ni = 1), the action of leads to zero, as in this case (A-10) gives:

(A-19)

Formulas (A-16) and (A-17) are also valid for fermions, with all the occupation numbers equal to 0 or 1 (or else both members are zero).

Comment:

Definition (A-18) must not depend on the specific order of the individual states uj, .., uk, .., ul, .. in the ket on which the operator acts. It can be easily verified that any permutation of the states simply multiply by its parity both members of the equality. It therefore remains valid independently of the order chosen for the individual states in the initial ket.