The finite temperature Green functions of the fermion systems was calculated by using the stochastic quantization method. By properly approximation of the Langevin equations, Hartree-Fock, ring and ladder approximations of Green functions was obtained.
A new method of the generation of Green's function in external local and bilocal fields is proposed for the stochastic quantization of the 2D Hubbard model. The differential equations for Green's functions transform to the system of integral equations and it is solved numerically for a local Hubbard-Stratanovich field. The Trotter-Suzuki expansion and the matrix elements of the evolution operator are not used in this approach. For a 4 × 4 lattice with half-filled band the total energy E, the density n, the density–density 〈 ni↑ni↓〉 and spin–spin 〈SziSzj〉 correlation functions were calculated by stochastic quantization as the functions of one-site repusion U and inverse temperature β. The obtained results are in agreement with the previous Monte Carlo calculations.
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.