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The Green’s functions defined in Chap. 10 have analytical properties that are similar but not identical to the Green’s functions defined in Chap. 2 corresponding to second-order (in time) differential equations. They can all be expressed in terms of a generalized DOS and the Fermi or Bose thermal equilibrium distributions. From the Green’s functions (or the generalized DOS) one can easily obtain all thermodynamic quantities and linear response functions like the conductivity. The poles of an appropriate analytic continuation of in the complex -plane can be interpreted as the energy (the real part of the pole) and the inverse lifetime (the imaginary part of the pole) of quasiparticles. The latter are entities that allow us to map an interacting system to a noninteracting one.

The problem of finding the eigenvalues and eigenfunctions of a Hamiltonian ℌ = ℌ + ℌ can be solved in three steps: 1) Calculate the Green’s function () corresponding to ℌ. 2) Express () as a perturbation series in terms of () and ℌ, where () is the Green’s function associated with ℌ. 3) Extract from () information about the eigenvalues and eigenfunctions of ℌ.

Some applications of the many-body Green’s functions are briefly presented. They include the study of a high-density electronic system moving in a positive background; this model describes approximately electrons in metals. A low-density Fermi system with short-range interactions is also examined. The employment of the Green’s function formalism to justify the widely used independent-particle approximation is emphasized. Superconductivity and the Hubbard model are also examined.