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Progress of Theoretical Physics

Modification of the Conventional Holstein-Primakoff Boson Representation for the su(1,1)-Algebra and Its Classical Counterpart

A. Kuriyama, João da Providência, Y. Tsue, M. Yamamura

Published May 1, 1998• 0 citations
Publisher

Abstract

In order to avoid unsatisfactory points of the conventional Holstein-Primakoff representation of the su(1,1)-algebra for the description of thermal effects in many-boson or oscillating systems, the conventional form is modified. The basic idea can be found in a possible canonical transformation from the conventional to the new form. Four forms are given, and with the aid of these modified forms, it is expected to describe, for example, fluctuations around a bose-condensed state.

DOI
10.1143/ptp.99.819

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In order to avoid unsatisfactory points of the conventional Holstein-Primakoff representation of the su(1,1)-algebra for the description of thermal effects in many-boson or oscillating systems, the conventional form is modified. The basic idea can be found in a possible canonical transformation from the conventional to the new form. Four forms are given, and with the aid of these modified forms, it is expected to describe, for example, fluctuations around a bose-condensed state.
With the aim of investigating dynamics of systems governed by su (1, 1)-algebra, a possible state which can give us a classical counterpart for the su (1, 1)-spin system properly is examined and constructed explicitly. By the use of canonicity conditions developed in the time-dependent Hartree-Fock theory, the c -number analogue of the Holstein-Primakoff boson representation is obtained for the su (1, 1)-algebra. It is noted that it is important to introduce a character of two-mode coherent state in order to present a suitable classical counterpart for the su (1, 1)-spin. Furthermore, it is shown that the spreading of uncertainty relations appears inevitably, which suggests that a certain fluctuation can be introduced.
With the aim of investigating dynamics of systems governed by su (1, 1)-algebra, a possible state which can give us a classical counterpart for the su (1, 1)-spin system properly is examined and constructed explicitly. By the use of canonicity conditions developed in the time-dependent Hartree-Fock theory, the c -number analogue of the Holstein-Primakoff boson representation is obtained for the su (1, 1)-algebra. It is noted that it is important to introduce a character of two-mode coherent state in order to present a suitable classical counterpart for the su (1, 1)-spin. Furthermore, it is shown that the spreading of uncertainty relations appears inevitably, which suggests that a certain fluctuation can be introduced.
In order to develop a previously proposed method to investigate thermal effects in quantum many-particle systems, basic ideas are formulated in more general systems obeying su(2)-algebra. It is shown that Hamiltonians of a single orbit shell model with pairing correlation, the Lipkin model, a symmetric rigid rotor model and an asymmetric rigid rotor model have almost the same structure, that is algebraic structure of su(1, 1), by the use of the Holstein-Primakoff boson representation for su(2)-algebra and so-called phase-space doubling introduced to study thermal phenomena. It is also discussed that the interaction term between intrinsic system and external environment approximately represents the interesting physical process in each system at finite temperature.
The basic idea presented in Parts (I) – (III) of this series for the deformed boson scheme is applied to the case of the su(2)- and the su(1, 1)-algebras for describing many-body systems consisting of four kinds of boson operators. A possible form of the coherent state given by the present authors is generalized and the su(2)q- and the su(1, 1)q-algebras are obtained in forms expressed in terms of four kinds of boson operators. As an illustrative example demonstrating the application of the deformed boson scheme, the framework for describing thermal effects observed in the two-level shell model with pairing correlation is proposed.