Mean Field Approximation to two-subsystems: Examples

Using the Funtional Integrals Formulation is developes a self-consistent mean field expansion to evolution operators of a system composed by two subsystems. This is a general expansion and can be generalized for more of two subsystems, which can be as system composed by fermions and bosons with an interaction between their ($\sigma$-model , $\sigma-\omega$ model or maser model) as a fermionic system with two or more different interactions between fermions particles, (NJL model ).