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Progress of Theoretical Physics

On the Collective Variables in Many-Boson System

Shuichiro Yamasaki, T. Kebukawa, Sigenobu Sunakawa

Published May 1, 1975• 4 citations
Publisher

Abstract

A new well-defined velocity operator is proposed for the collective description of a many-boson system. It is shown that the new velocity operator is irrotational. The Hamiltonian of the system is successfully described in terms of the density-fluctuation operator and the velocity operator. The interaction term in the order of N-1 has a different form from that in previous papers. The range of validity of the present formalism is discussed in the final section in answer to the questions raised so far on the collective formalism of the many-boson system.

DOI
10.1143/ptp.53.1243

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arXiv:1603.07420•International Journal of Modern Physics E•Jul 26, 2016•2 citations
A new well-defined velocity operator is proposed for the collective description of a many-boson system. It is shown that the new velocity operator is irrotational. The Hamiltonian of the system is successfully described in terms of the density-fluctuation operator and the velocity operator. The interaction term in the order of N-1 has a different form from that in previous papers. The range of validity of the present formalism is discussed in the final section in answer to the questions raised so far on the collective formalism of the many-boson system.
Green's functions of phonons for many-boson systems at high density are obtained from the phonon Hamiltonian given in a previous paper on the basis of the coherent-state representation. A dispersion equation derived from these Green's functions gives rise to the excitation spectrum leading to the Feynman energy in the low-momentum limit. It is shown that the dynamic structure function obtained here satisfies the sum rules in the low-momentum limit. As an illustrative example, the one-dimensional Lieb-Liniger model is considered. The relationships to the previous theories, in particular, to the velocity field approach of Yamasaki, Kebukawa and Sunakawa and to our density and phase operator approach are elucidated by the use of a unitary transformation. Discussion will be made on the D-wave portion of the roton-roton interaction in relation to the resonance state of roton pair studied in one of our previous papers. Finally, it is pointed out that the zeroth Fourier transform of the phase operator is related to the superfluid velocity in the two-fluid theory of liquid helium II.
The behavior of an individual particle interacting with phonons in a many-boson system has been investigated in the scheme of the collective description by using the number density operator and the phase operator as the collective variables. For a model-interaction potential which is strong but of long-range, it has been shown that the third order anharmonic terms to appear in this collective description give rise to the backflow effect of the individual particle which has been discarded in the random phase approximation. The Feynman-Cohen wave function and the dipole-type interaction between individual particles transmitted by phonons have been obtained. In relation to Puff's theory of sum rules the spectral function associated with the density-density correlation function for individual particles has also been investigated.
In the preceeding paper, introducing [Formula: see text]-dependent density operators and defining exact momenta (collective variables), we could get an exact canonically momenta approach to a one-dimensional (1D) neutron–proton (NP) system. In this paper, we attempt at a velocity operator approach to a 3D NP system. Following Sunakawa, after introducing momentum density operators, we define velocity operators, denoting classical fluid velocities. We derive a collective Hamiltonian in terms of the collective variables.