A conceptual question about Green's function's treatment of interaction Here we have electron gas and some other stuff. We expand the Hamiltonian to the 1st order of one single harmonic oscillator's displacement $\vec{u}$. Its equilibrium position is at the origin. Then we get an effective coupling Hamiltonian $\vec{j}(\vec{r})\times\vec{f}(\vec{r})\cdot \vec{u}$, wherein $\vec{j}(\vec{r})$ is the electron density, $\vec{f}(\vec{r})$ is some effective potential.

I assumed a particular mode (frequency $\Omega$ for x,y,z) of $\vec{u}$. And I tried to work out the above diagram. It's doable. 


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*In this electron Green's function calculation, do we take into account of the interaction's influence on the oscillator? Is it damped or not?

*I'm confused about where the momentum transfer $q$ come from. Potential $\vec{f}(\vec{r})$ or oscillator's motion $\vec{u}$? I only Fourier transform $\vec{j}(\vec{r})$ and $\vec{f}(\vec{r})$ in the Hamiltonian, therefore this $q$ merely appears in terms that come from $\vec{j}(\vec{r})\,,\vec{f}(\vec{r})$. So I suppose this momentum transfer $q$ comes from the potential. Is this correct?

 A: I am assuming that those are bare Green's function in the diagram, and the dotted line is the harmonic oscillator $<a^\dagger a>$ ? 
1) The diagram you drew does not "know" about the damping of the harmonic oscillator, (although it is closely related by the optical theorem and such to the diagrams that would calculate the damping, I suppose that is what Isidore is saying). If it is necessary to include the damping then you would have to think of a way to self consistently include it. 
2) I would recommend you write everything down in real space coordinates, get the second order term and then Fourier transform it. That should make it clear.
A: *

*The electron motion does feed back to the oscillator, but that is another diagram, known as the bubble diagram, in which you calculate the self-energy correction of the oscillator. That self-energy presumably contains imaginary part, which is then interpreted as the damping of the oscillator. You can either calculate the self-energy corrections self-consistently, or you can simply neglect it in the weak coupling limit away from the non-fermi-liquid criticality. 

*The momentum $q$ should be the momentum of the phonons (quanta of the oscillator motion).The potential $f(r)$ also carries a momentum, but this momentum is on the vertex (where the electron emits/absorbs the phonon). More precisely, due to the presence of the potential $f(r)$, momentum is not conserved on the vertex, the non-conserved amount of momentum is provided by the potential scattering.
