Quantum factorization approximation for first order Coulomb energy

I’m working through "Advanced Quantum Mechanics" by Franz Schwabl, and he uses this G-correlation function to estimate the first order correction to the ground state energy in a Coulomb system (section 2.2.3). In the course of doing so, he runs into the following expectation value (where the implicit state is the filled Fermi sphere):
$$
left< a^dagger_{mathbf{p} + mathbf{q}, sigma’} (t)
a_{mathbf{k} + mathbf{q} , sigma} (t)
a_{mathbf{p} , sigma’} (t)
a^dagger_{mathbf{k} , sigma} (0)right>
$$
He does something called a "factorization approximation" which I am not familiar with to estimate this as:
$$
left< a^dagger_{mathbf{p} + mathbf{q}, sigma’} (t)
a_{mathbf{k} + mathbf{q} , sigma} (t)
a_{mathbf{p} , sigma’} (t)
a^dagger_{mathbf{k} , sigma} (0)right>
= left< a^dagger_{mathbf{p} + mathbf{q}, sigma’} (t)
a_{mathbf{k} + mathbf{q} , sigma} (t) right>
left< a_{mathbf{p} , sigma’} (t)
a^dagger_{mathbf{k} , sigma} (0)right>
$$
Additionally, he includes in a footnote that there is another possible factorization which becomes zero because there is an external condition that $mathbf{q} neq 0$:
$$
left< a^dagger_{mathbf{p} + mathbf{q} , sigma’} (t)
a_{mathbf{p} , sigma’} (t) right>
left< a^dagger_{mathbf{k} + mathbf{q} , sigma} (t)
a_{mathbf{k} , sigma} (0) right>
$$
How does this factorization process work (mechanically) in general, especially in the case when you have multiple factorizations that do not go to zero? Also, can someone explicitly explain the factorization which appears in the footnote? Somehow we start with an $a^dagger_{mathbf{k} , sigma} (0)$ guy and end up with an $a_{mathbf{k} , sigma} (0)$ guy without doing any kind of complex conjugation.

Lastly, can anyone recommend a pedagogical text which might explain why and under what conditions this kind of factorization approximation works?