Diagrammatical Method for the Two-Time Green's Function for Many-Body Problems at Finite Temperatures

Abstract

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”.