Fermion Operators

At this point, we can hypothesize that the operators that create fermion states do not commute. In fact, if we assume that the operators creating fermion states anti-commute (as do the Pauli matrices), then we can show that fermion states are antisymmetric under interchange. Assume $b_r^\dagger$ and $b_r$ are the creation and annihilation operators for fermions and that they anti-commute.

$$\{b_r^\dagger,b_{r'}^\dagger\}=0$$

The states are then antisymmetric under interchange of pairs of fermions.

$$\bgroup\color{black} b_r^\dagger b_{r'}^\dagger \vert\rangle = -b_{r'}^\dagger b_{r}^\dagger \vert\rangle \egroup$$

Its not hard to show that the occupation number for fermion states is either zero or one.