# Two definitions for normal ordering of $c_{k+q}^\dagger c_k$

Consider the fermionic operator $$c_k, c^\dagger_k$$, and where $$k$$ is discrete and unbounded. (Note: This situation frequently arises in bosonization.) Let the vacuum $$|0\rangle$$ be the state with all $$k\leq 0$$ filled and all $$k>0$$ empty.

There are two definitions of normal ordering:

1. The normal ordering is obtained by substracting the vacuum expectation value.

2. The normal ordering is obatained by putting $$c_k$$ right to $$c_k^\dagger$$ for $$k > 0$$, and putting $$c_k^\dagger$$ right to $$c_k$$ for $$k \leq 0$$.

My question is that for $$c_{k+q}^\dagger c_k$$ with $$q>0$$, the two definitions above seem to be different. According to the first definition, $$:c_{k+q}^\dagger c_k:=c_{k+q}^\dagger c_k$$ for all $$k$$ since $$\langle0|c_{k+q}^\dagger c_k|0\rangle=0$$. However, if we use the second definition, $$:c_{k+q}^\dagger c_k:=c_k c_{k+q}^\dagger$$ when $$k\leq-q$$. Since $$c_k$$ and $$c_{k+q}$$ anticommute, this expression is minus the first expression!

Then what is the correct definition of the normal ordering?

• The normal ordering depends on what sort of vacuum you are concerned with. E.g. if there is BCS phase transition, $\langle0|c_{k+q}^\dagger c_k|0\rangle \neq 0$, and you have to resort to Bogoliubov-transformed operators to define vacuum and normal ordering. Jul 3 '19 at 15:09