From two-body Green's function to one-body in perturbation theory

In summary, the speaker is attempting to derive the Hedin's equations for GW on their own, but has come across an equation that they cannot derive or find a source for. They are asking for help in understanding the derivation or for a link to a document with a full derivation.
  • #1
forever_physicist
7
1
Hi guys! I am trying to derive the Hedin's equations used for GW on my own, and I found this equation, but I cannot really derive it, nor find a source where they explain how this can be derived (they link to other papers that in the end don't show where this is coming from). The equation I am talking about is:
$$G(1,3,2,3^+) = G(1,2)G(3,3^+)-\frac{\delta G(1,2)}{\delta \phi(3)}$$
Can somebody tell me how this can be derived or just link me to a document with a full derivation?
Thanks!
 
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1. What is the two-body Green's function in perturbation theory?

The two-body Green's function is a mathematical tool used in perturbation theory to describe the behavior of a system with two interacting particles. It is a function that relates the positions and momentums of the two particles to the total energy of the system.

2. How is the two-body Green's function related to the one-body Green's function?

In perturbation theory, the two-body Green's function can be used to derive the one-body Green's function, which describes the behavior of a single particle in the presence of an external potential. This is done through a process called reduction, where the two-body Green's function is simplified to only include interactions between the particle and the potential.

3. What is the significance of the one-body Green's function in perturbation theory?

The one-body Green's function is a fundamental quantity in perturbation theory, as it allows for the calculation of various physical observables, such as the particle's energy and wave function. It also provides a way to analyze the effects of external potentials on the behavior of a single particle.

4. How is perturbation theory used to calculate the one-body Green's function?

Perturbation theory involves expanding the one-body Green's function in terms of the two-body Green's function and the external potential. This expansion can be carried out to different orders, allowing for increasingly accurate calculations of the one-body Green's function.

5. What are the limitations of using perturbation theory to calculate the one-body Green's function?

Perturbation theory is only applicable when the interactions between particles are small compared to the external potential. Additionally, higher order calculations may become increasingly complex and time-consuming. In some cases, alternative methods such as numerical simulations may be more suitable for calculating the one-body Green's function.

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