In the page 219 of Mahan's Many Particle Physics(3ed), there exists a transform
$$ S=c^{\dagger}c\sum_q\frac{M_q}{\omega_q}(a_q^{\dagger}-a_q)$$
In order to prove that the transformation relating to $e^{S}$ is $\textit{unitary}$, we should prove that
$$(e^S)^{\dagger}(e^S)=I$$
or equivalently,
$$S^{\dagger}=-S$$

However, in my opinion,
$$ S^{\dagger}=\big(c^{\dagger}c\big)^{\dagger}\sum_q\frac{M_q}{\omega_q}(a_q^{\dagger}-a_q)^{\dagger}=\big(cc^{\dagger}\big)\sum_q\frac{M_q}{\omega_q}(a_q-a_q^{\dagger})=\big(-c^{\dagger}c\big)\sum_q\frac{M_q}{\omega_q}\big(-(a_q^{\dagger}-a_q)\big)=S$$
What's wrong?

This post imported from StackExchange Physics at 2014-10-11 09:51 (UTC), posted by SE-user Roger209