A question about normal ordering in bosonization

I am learning abelian bosonization, and I need to find the commutator of two density operatorS. In this process, I need to calculate $$\sum_k(C^{\dagger}_{k+q_1+q_2}C_{k}-C^{\dagger}_{k+q_1}C_{k-q_2})$$. By using the Wick's theorem, I have found out the difference of the contraction term, then I need to find the contribution of the normal ordered term$$\sum_k(:C^{\dagger}_{k+q_1+q_2}C_{k}:-:C^{\dagger}_{k+q_1}C_{k-q_2}:)$$. The answer is $$0$$, but I don't know how to get it.

$$C_k$$ is fermionic operator, the value of $$k$$ is $$\frac{2\pi n}{L}$$.

The normal ordering is with respect to the filled fermion sea, i.e.: $$|G\rangle=\Pi_{k<0}C^{\dagger}_k|vac\rangle$$

• Relabel the sum index $k\rightarrow k+q_2$ in the second term. Note that this relabeling is only allowed because the operators are normal ordered and so all terms are well defined. Without normal ordering you are adding and subtracting divergent terms and you will get an incorrect result. Feb 28 '21 at 13:59
• I ignore the convergence of normal odered operators, thank you very much! Feb 28 '21 at 14:30