A Method of a Quasi-Linear Canonical Transformation for the Many-Body Problem. I

Abstract

A general method of a quasi-linear canonical transformation is introduced as an approach . to the many-body problems. In this method the nobody problem can be solved if the(n-1)-body problem is assumed to be solved. In the application of this method to the system of N bosons, there appears no logarithmic divergent term in the physical quantities which correspond to the self-energy operators III and .s02 appearing in the. method of Green's function. One of the most familiar methods of approaching the many-body problems is the Hartree-Fock approximation. This approximation has been used to obtClin many useful results for the Case of the many-electron atom. However, in the case of a many-particle system enclosed in a large cubic box, the effect of the non­ diagonal parts of the Hamiltonian with respect to the occupation number of particles is important and the divergent results are often encountered in the perturbation calculation. In particular for the case of the hard-sphere interaction we cannot apply directly the Hartree-Fock approximation to this system since the matrix elements of the hard-spherepotehtial cannot be defined well. Landau!) proposed a concept of the quasi-particles or elementary excitations which is very useful to deal with the many-particle system at low temperatures. However, it is the semi-phenomenological theory, and, for example, in the Fermi liquid 2 ) the way of obtaining the energy coefficients between the quasi-particles has not been elucidated. Recently, by several authors, the Landau theory has been studied by using the niethod of Green's func·tion. 3 ) In the case of a Bose system, no theory corresponding to the theory of the Fermi liquid has been established so far. Orie of the difficulties lies in the treatment of the number of the particles which are condensed into the one-particle state of zero momentum; It was discovered by Kapitza 4 ) that the liquid helium reveals the superfluidity . below the transition temperature T~2°K and the cCin~iti~n for the superfluidity . was shown by Landau.!) According to his theory the excitation energy of the supcrfluid system must have the phonon spectrum or the gap for small 'momenta in order that the liquid may revea'l the superfluidity. Bogoliubov 5 ) showed firstly by the quantum mechanical calculation. that the excitation energy of the interacting