Principle of Compensation of Dangerous Diagrams for Boson Systems. I. Maximum Overlap

Abstract

The principle of compensation of dangerous diagrams (PCDD), used by Bogoliubov to determine the coefficients in the canonical transformation to quasiparticles in boson systems, is obtained by maximizing the overlap between the true ground‐state vector and the quasiparticle vacuum state. The zero‐momentum state is treated exactly, which implies that the sum of all diagrams leading from the vacuum to a one‐quasiparticle state must be zero, in addition to the diagrams leading from the vacuum to the two quasiparticle state. Other criteria, such as the diagonalization of the quasiparticle reaction operator up to terms cubic in the operators, and the absence of one and two quasiparticle contributions to the true ground‐state wavefunction, are also shown to lead to the PCDD. A generalization of the Hartree procedure of minimizing the ground‐state energy is obtained by replacing the bare quasiparticle interaction with the quasiparticle reaction operator, and is shown to be equivalent to the PCDD. Finally, a perturbation expansion of the PCDD is obtained, and the reducibility of diagrams is discussed.