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Progress of Theoretical Physics

Note on Boson Expansion and Boson Coherent State for the su(2)-Spin System

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

Published Jan 1, 1996• 1 citation
Publisher

Abstract

With the aim of obtaining the classical counterpart of the su(2)-spin system and its quantal fluctuations in the frame of the boson coherent state, the Holstein-Primakoff and Marumori-Yamamura-Tokunaga boson expansions are investigated. The starting point is found in the Schwinger boson representation for the su(2)-spin system and its reformation. Dirac's canonical theory for constraint systems is also applied to the su(2)-spin system. In conclusion, the latter boson expansion is found to be superior to the former with respect to the use of the boson coherent state.

DOI
10.1143/ptp.95.79

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With the aim of obtaining the classical counterpart of the su(2)-spin system and its quantal fluctuations in the frame of the boson coherent state, the Holstein-Primakoff and Marumori-Yamamura-Tokunaga boson expansions are investigated. The starting point is found in the Schwinger boson representation for the su(2)-spin system and its reformation. Dirac's canonical theory for constraint systems is also applied to the su(2)-spin system. In conclusion, the latter boson expansion is found to be superior to the former with respect to the use of the boson coherent state.
In the framework of the Marumori-Yamamura-Tokunaga boson expansion, three forms for the su(2)-spin system are presented. The first is related with the well-known Holstein-Primakoff boson expansion and the second is the expansion reinvestigated in a previous paper by the present authors. The third is, in some sense, the original part of this paper. Corresponding with three forms, c-number counterparts are obtained by using three forms of wave packets defined in the su(2)-spin space. With the help of appropriate quantization rule, they go back to the original boson expansions. Furthermore, it is discussed if the c-number counterparts can be regarded as the classical counterparts. Finally, the relation to the su(1, 1)-algebraic model is discussed.
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 terms of the boson coherent state and a possible extension, three forms of wave packets are presented in the Schwinger boson representation. Each wave packet gives classical counterpart for a certain set of operators, for example, three components of the su(2)-spin. Furthermore, uncertainty relations which come from the boson coherent state and its extension are examined. The content discussed in this paper is, in a certain sense, an extension of that discussed in a paper presented recently by the present authors.