Configuration-interaction formulation of the Dyson equation

The Dyson equation represents a well-defined connection between the one-particle Green’s function and the self-energy. By means of the Dyson equation, the evaluation of the one-particle Green’s function can be reduced to the evaluation of the self-energy. As it is well known, the one-particle Green’s function is composed of an advanced and a retarded part, which can be represented in the (N+1)- and (N-1)-particle configuration spaces, respectively. The Dyson equation combines these two spaces into a matrix in the union space. This is difficult to understand in configuration-interaction language, where the Hamiltonian matrices for systems containing different numbers of particles are naturally separated. Starting with the usual matrix representations of the Hamiltonian in the (N+1)- and (N-1)-particle spaces, it is shown that it is possible to define a class of unitary transformations that mixes the two parts and gives rise to an effective Hamiltonian matrix in the union space. This effective Hamiltonian leads to all possible representations of the Dyson equation. The transformations used arise from suitable combinations of the matrices of residues of the advanced and retarded Green’s functions. Several specific members of the above class of unitary transformations are discussed in some detail also in connection with possible approximation schemes for the Green’s function. Special attention is paid to those unitary transformations that depend on the ground state of the N-particle system as the only unknown quantity. This investigation has given rise to some general conclusions and to an alternative approach to the one-particle Green’s function.