Self-Energy and the Dyson Equation

Abstract

In the four preceding chapters, we have established the formalism of diagrammatic perturbation theory for the electron propagator, which allows one to derive successively higher-order contributions \(\varvec{G}^{(n)}(\omega )\). However, a finite perturbation expansion, e.g., through third order, $$\begin{aligned} \varvec{G}(\omega ) = \varvec{G}^0(\omega ) + \varvec{G}^{(2)}(\omega ) + \varvec{G}^{(3)}(\omega ) + O(4) \qquad (8.1) \end{aligned}$$ does not result in a useful approximation scheme to determine the physical quantities of interest, that is, ionization energies, electron affinities, and the corresponding spectral factors. The reason is that the components \(G_{pq}(\omega )\) are analytical functions, and a finite perturbation expansion does not recover the proper analytical structure ( 3.17), being a sum over simple poles, from which the desired information could be extracted. So the question is how to translate the diagrammatic perturbation expansion into a viable computational scheme. What is needed here is to sum the perturbation expansion, even if only partially, through infinite order. A possible path toward such infinite partial summations, recovering the proper analytical structure of the electron propagator, is provided by the Dyson equation, which we will address in this chapter.