07.06 Green's functions for independent electrons - The Dyson equation скачать в хорошем качестве

07.06 Green's functions for independent electrons - The Dyson equation

In physics we often encounter a system that can be described by an Hamiltonian that can be written as a sum of a part that is easy to solve (H0) an a part that is hard to solve (H1). Such that the total Hamiltonian (H=H0+H1) is hard to solve. A standard example would be the Hamiltonian for interacting electrons where H0 could be the Hartree-Fock approximated mean-field Hamiltonian of an interacting system and H1 the two particle Coulomb interaction. We can also use examples for electrons in potential. For a single impurity in an otherwise periodic potential, H0 could be fully periodic, such that one can diagonalise the Hamiltonian by a Fourier transform to crystal momentum and H1 the Hamiltonian representing the impurity. In this case we can use the Dyson equations to obtain the full Green's functions from the Green's functions of H0 in a series expansion. This series expansion can be expressed with the use of Feynman diagrams. Such a series expansion is related to perturbation theory and one needs to be careful with the range of convergence. The advantage over the Dyson equations with respect to standard perturbation theory is that one often can sum to infinite order and with the use of analytical continuation obtain results that are valid even outside the range of convergence.