so nothing is changed. In the reverse case, we introduce a minus sign because we have to change the order of two operators: These can be combined, along with the anticommutation relations, to show or This equation, which is in the same form as the bosonic case above, is used in defining the contractions used in Wick's theorem. 2. The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice. For example: Multiple fermions For different fermions there are operators: : the fermion's creation operator. : the fermion's annihilation operator. Here . These satisfy the anti-commutation relations: where and denotes the Kronecker delta. These may be rewritten as: When calculating the normal order of products of fermion operators we must take into account the number of interchanges of neighbouring operators required to rearrange the expression. It is as if we pretend the creation and annihilation operators anticommute and then we reorder the expression to ensure the creation operators are on the left and the annihilation operators are on the right - all the time taking account of the anticommutation relations. Examples 1. For two different fermions () we have Here the expression is already normal ordered so nothing changes. Here we introduce a minus sign because we have interchanged the order of two operators. Note that the order in which we write the operators here, unlike in the bosonic case, does matter. 2. For three different fermions () we have Notice that since (by the anticommutation relations) the order in which we write the operators does matter in this case. Similarly we have Uses in quantum field theory The vacuum expectation value of a normal ordered product of creation and annihilation operators is zero. This is because, denoting the vacuum state by , the creation and annihilation operators satisfy (here and are creation and annihilation operators (either bosonic or fermionic)). Let denote a non-empty product of creation and annihilation operators. Although this may satisfy we have Normal ordered operators are particularly useful when defining a quantum mechanical Hamiltonian. If the Hamiltonian of a theory is in normal order then the ground state energy will be zero: . Free fields With two free fields φ and χ, where is again the vacuum state. Each of the two terms on the right hand side typically blows up in the limit