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Progress of Theoretical Physics

The Extended Convolution Approximation as a Canonical Transformation in a Degenerate Boson System

Shuichiro Yamasaki, Masahiko HIROOKA

Published Jan 1, 1982• 2 citations
Publisher

Abstract

The convolution approximations to the three• particle and four-particle structure functions are incorporated in a non-linear canonical transformation applicable to a system of degenerate bosons.The transformation relates the quasi• particle operators bk to the density fluctuations P k and their canonical conjugates 'P k by way of a formal series in powers of N-ll ', where N is the number of particles.Higher order terms in the series require corrections to the convolution approximations.The next order terms are calculated to improve the approximations to the structure functions.The present formalism provides an unambiguous way of evaluating the residual interaction not taken account of in the trial state, the b•vacuum, which admits the convolution approximation or its extension. § 1. Introduction

DOI
10.1143/ptp.67.20

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The convolution approximations to the three• particle and four-particle structure functions are incorporated in a non-linear canonical transformation applicable to a system of degenerate bosons.The transformation relates the quasi• particle operators bk to the density fluctuations P k and their canonical conjugates 'P k by way of a formal series in powers of N-ll ', where N is the number of particles.Higher order terms in the series require corrections to the convolution approximations.The next order terms are calculated to improve the approximations to the structure functions.The present formalism provides an unambiguous way of evaluating the residual interaction not taken account of in the trial state, the b•vacuum, which admits the convolution approximation or its extension. § 1. Introduction
Creation and annihilation operators of quasi-particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non-zero momentum particles shows that the quasi-particles are phonons in the long wave-length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi-particles, approaches the Hartree-Fock state when the condensate is depleted completely at the critical temperature.
The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.
The infrared divergence of the interaction among Bogoliubov's quasi· particles in a degener­ ate boson system can be avoided by introducing improved quasi-particles. A formula is given which relates the single· particle operators ap directly to the improved quasi-particle operators bp ' which have been defined in terms of collective variables in a preceding paper. The operators ap are expressed in terms of bp as a series using N112 as a formal expansion parameter, where N is the number of particles in the system. Terms up to order N312 are given together with a tenn of order N2 which is necessary for the calculation of the ground· state energy to order N2 • The fonnula makes the reason clear why the operators b p enable us to eliminate the ultraviolet divergence haunting collective theories including the quantum hydrodynamics.
The three-particle structure function in the ground state of a boson system is studied in a formalism which establishes phonon type quasi-particles coupled to each other by the residual interaction preserving the linearity of the spectrum. The leading order results agree with those from the unquantized theories of Bogoliubov-Zubarev and Feenberg, et al. The next order corrections are calculated, and one of them is found to bear a strong resemblance to the result of Lee and Lee. Possible relations of the present formalism to the correlated basis function method in the uniform limit are discussed on a phenomenological level.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.