Unified theory of effective interaction

We present a unified description of effective interaction theories in both algebraic and graphic representations. In our previous work, we have presented the Rayleigh-Schrödinger and Bloch perturbation theories in a unified fashion by introducing the main frame expansion of the effective interaction. In this work, we start also from the main frame expansion, and present various nonperturbative theories in a coherent manner, which include generalizations of the Brandow, Brillouin-Wigner, and Bloch-Horowitz theories on the formal side, and the extended Krenciglowa-Kuo and the extended Lee-Suzuki methods on the practical side. We thus establish a coherent and comprehensive description of both perturbative and nonperturbative theories on the basis of the main frame expansion.