The Green functions and the anomalous Green functions of the systems with the unsymmetrical interactions are calculated by using the stochastic quantization. By the correct treatment, the Dyson equations including the Green functions and the anomalous Green functions are obtained.
The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.
For systems governed by two kinds of interactions it is shown that these can be built successively into the Green’s functions describing the system’s response. Whereas for the ordinary Green’s function the Dyson equation to solve has the same form in each step, we derive the non-trivial second-step equation for the transfer or coupling function, which on the one hand is closely related to the self-energy and on the other hand of practical relevance in transport calculations. PACS numbers: 02.60.Cb, 02.90.+p, 72.10.Bg 1.
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.
Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.