Finding free electron gas Green function in Fourier space

[kakaho345]

Homework Statement

See below

Relevant Equations

See below

As in title:

Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971:

Up to here everything is very straight forward, in particular, since we are working on free electron gas, $E=\hbar \omega$

However, I have no idea how to arrive here:

I understand that $e^{ik\cdot(x-x')}$ is from terms like $\psi=e^{ikx}c$, however, the term $e^{-i\omega_k(t-t')}$ the sign doesn't look right to me for the two time region should have different signs in the exponential. Also, I don't know how to deal with the exponential sandwiched between the field operator. The step function in time is from the two pieces of time regions, but I am not sure on the step function in k. I may be from the filled Fermi sea.

I understand this is a very simple question. However, I have been sitting whole day dealing with this. Any help will be appreciated.

 

[vanhees71]

You simply have to think about, how the ground state looks like! Note that at $T=0$ the system is in a pure state of lowest possible energy under the given constraints. First think what is the constraint here!