Part I: Green's Functions in Mathematical Physics - Time-Independent Green's Functions / Time-dependent Green's Functions / Part II: One-Body Green's Functions - Physical Significance of One-Body Green's Functions / Green's Functions and Perturbation Theory / Green's Functions for Tight-Binding Hamiltonians / Single Impurity Scattering / Extension Theory for Lattice Green's Functions / Part III: Many-Body Green's Functions - Field Operators and Three Pictures / Definition and Properties of Many-Body Green's Functions / The Diagram Technique for Zero-Temperature Green's Functions / Definition and Properties of Matsubara Green's Functions / Diagram Technique for the Matsubara Green's Functions / Three Approximation Schemes of the Diagram Technique / Linear Response Theory / The Equation of Motion Technique for the Green's Functions / Magnetic Systems Described by Heisenberg Model / The Green's Functions for Boson Systems with Condensation / Superconductors with Weak Interaction Between Electrons / Nonequilibrium Green's Functions / Electronic Transport through a Mesoscopic Structure / Appendixes / Index.
The diagram technique, which was developed to solve a hierarchy of equations of motion for the reduced density matrices, is applied in order to solve a hierarchy of the equations of motion for the two-time Green's functions. The diagrammatical representations of the two-time Green's functions are obtained either in terms of the equilibrium one- and many-particle reduced density matrices or in terms only of the equilibrium one-particle reduced density matrix. The latter representation for the one-particle two-time Green's function and the formula by which the reduced density matrix is calculated from the Green's function give us a set of coupled equations which determines both the reduced density matrix and the two-time Green's function. The calculation with the aid of the present technique is illustrated for an example of Hubbard's calculation for the “Hubbard Hamiltonian”.
A new technique is presented for evaluating the electron Green function Gk(t) in an insulating electron–phonon system. The technique is particularly suitable for those electron–phonon interactions in which the long wavelength phonons are of prime importance such as the polaron and piezoelectric interactions.An exact formal expression is obtained for the time and temperature dependent Green function in the form[Formula: see text]where S is a functional of the differential operator d/dk. The Green function is then expressed as [Formula: see text], and from the exact result a series expansion is derived for the action function Ak(t).This series expansion has a very special property: to whatever order Ak(t) is evaluated the resulting expression for Gk(t) includes, in some way, every perturbation theory diagram. By this we mean that as in conventional techniques (the Hartree–Fock approximation, for example) an infinite subset of diagrams are summed exactly and in contrast to these techniques the remaining infinity of diagrams are not discarded but are evaluated approximately.The technique is applied to the polaron interaction.