The Bogoliubov-Tyablikov green's function method in the quantum theory of magnetism

We briefly review the half-a-century period (1959-2009) of the creation, development, and application of the method of two-time temperature retarded and advanced Green's functions, introduced by Bogoliubov and Tyablikov in 1959. This method has not exhausted its resources by far, as is evidenced by its effective modification proposed by Plakida in 1970 and Tserkovnikov in 1971 and then, in a much more advanced form, by Tserkovnikov in 1981. Currently, a self-consistent description of both one-particle (transverse) and collective (longitudinal) Green's functions and correlation functions is achieved based on the Green's function method in the Bogoliubov-Tyablikov-Plakida-Tserkovnikov formulation.