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

Abstract

We apply the method of quasi-linear canonical transformation proposed in a previous paper to the system of N interacting bosons, and we calculate the excitation energy for the dilute hard-sphere Bose gas. In our formulation the total number of the particles are exactly conserved in contrast with the Bogoliubov theory. There appears no logarithmic divergent term in the physical quantities which correspond to the self-energy operators .Ell and .s02 appearing in the method of Green's function. In the prevIOUS paper referred to as I, the general theory of the: quasi-linear transformation has been investigated. In this paper we shall give its application to the system of N bosons at the zero temperature, and we evaluate the excitation energy and the chemical potential in the hard-sphere Bose gas. The system considered is enclosed in the cubic box of volume S2. The number of the par­ ticles, N, and the volume S2 are so large that we consider the 'problem. in the limit N ---,) 00, S2---') 00 as keeping N / S2 to be constant. In our formulation the total number of the particles are exactly conserved. It is assumed that the ground state of the system is given by the completely degenerate state of the quasi-particles denoted by (ao+)NIO) where ao+ is the creation operator for the quasi-particles with zero momentum, and the operators a p and a p + are introduced by the quasi-linear transformation. This assumption is very plausible as seen in the next section. Some theorems for the matrix' elements of the field operators are derived from the properties of the q~asi-linear transformation and from the assumption that' almost all the quasi-particles are condensed into the zero momentum state at the zero temperature. Usingthese theorems it can be shown that the excitation energy exhibits no gap, and the volocity of the quasi-particles with small momenta is equal to the hydrodynamical sound velocity. Moreover, it can be seen that there appears no divergent term in the physical quantities which corresponds to the self-energy operators appearing in the method of Green's function. For the case of the dilute hard-sphere Bose gas, we obtain the excitation energy for a small momentum and the chemical potential which