Greens Function and Self-energy Operator

Green s functions are the key ingredients in many-body theory from which relevant physical information can be extracted. Given a non-relativistic N electron system under an external potential Uion ( ), the one-particle Green s function (for simplicity we henceforth omit the prefix one-particle ) is defined as 

As commented, the Green s function contains the information about one-particle excitations (we will see in Sect. 5.4 how to obtain ground-state properties). We start from the Lehmann representation of the Green s function  

G (uj) is the Fourier transform with respect to t = t —t, r] is a positive infinitesimal, p, is the Fermi energy of the system, and 

We may see that A (x, x uj) is just the superposition of delta functions with weights given by the amphtudes / (x) centred at each of the one-particle excitation energies f . That is, as anticipated above, the Green s function reflects the one-particle excitation spectra. Moreover, such weights - see (5.3), depend on the density of available eigenstates after the addition/removal of one electron. Further details about the role of A (uj) in the interpretation of photoemission experiments can be found in [14]. 

The spectral function actually selected diagonal matrix elements Ann ( ) in a suitable one-electron basis representation - may exhibit well-defined structures reflecting the existence of highly probable one-electron excitations. Due to the Coulomb interaction, we cannot assign each excitation to an independent particle (electron or hole) added to the system with the excitation energy. Nonetheless, some of these structures can be explained approximately in terms of a particle-like behaviour, so having a quasiparticle (QP) peak. Where a second peak is required we may have what is called a satellite. 

---