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Progress of Theoretical Physics

Boson Expansion for Many-Fermion System as a Canonical Theory with Constraints. II

M. Yamamura, A. Kuriyama

Published Dec 1, 1981• 6 citations
Publisher

Abstract

With the use of the ordinary and the Grassmann variables, a classical theory for many-fermion system with particle-hole correlations is presented in a canonical form. This system involves many valence particles/holes in addition to many particle-hole pairs. The particle-hole pairs are assigned as the collective degree of freedom and the others as the independent-particle one. In order to avoid the double counting in the number of the degrees of freedom due to the introduction of the collective variables, certain constraints are introduced. The quantization is performed with the aid of the Dirac theory for canonical system with constraints. It is shown that the result is completely equivalent to that obtained with the use of the boson expansion method by Marshalek.

DOI
10.1143/ptp.66.2147

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Journal of Physics C Solid State Physics•Apr 30, 1982•22 citations
With the use of the ordinary and the Grassmann variables, a classical theory for many-fermion system with particle-hole correlations is presented in a canonical form. This system involves many valence particles/holes in addition to many particle-hole pairs. The particle-hole pairs are assigned as the collective degree of freedom and the others as the independent-particle one. In order to avoid the double counting in the number of the degrees of freedom due to the introduction of the collective variables, certain constraints are introduced. The quantization is performed with the aid of the Dirac theory for canonical system with constraints. It is shown that the result is completely equivalent to that obtained with the use of the boson expansion method by Marshalek.
A classical theory for many-fermion system is presented in the canonical form by introducing the ordinary and the Grassmann variables, the number of which exceeds “apparently” the original one. The basic idea is a translation of a theory proposed by the present authors for the case of the particle-hole pairs to the case of the Hartree-Bogoliubov type pairs. Certain constraints governing the variables are introduced and the double counting in the number of the degrees of freedom concerning the extra variables can be avoided. The quantization is performed with the aid of the Dirac bracket to take account of the constraints. It is shown that final result is completely identical with the compact form of the boson expansion given by Marshalek.
A canonical theory with constraints is proposed, in the classical framework, for describing pairing rotation and intrinsic degrees of freedom. The basic idea is a direct translation of a theory developed by the present authors for the case of particle-hole pair vibration. The total particle-number and the phase angle for the rotation and the Grassmann numbers for the intrinsic degrees of freedom are used as the canonical variables, the number of which exceeds apparently the original one. Certain constraints play the role of avoiding the double counting of the degrees of freedom concerning the extra variables. Hamiltonian and other physical quantities can be expressed in terms of the canonical variables and their forms can be well understood in the language of two-dimensional rotor model.
A microscopic theory of collective and independent-particle motions in many-fermion system is developed in the framework of Dirac's canonical theory with constraints.The basic idea is an extension of the conventional time-dependent Hartree-Fock theory based on the use of the Grassmann variables and, in certain sense, it supplements incomplete parts of the paper presented by the present authors.Canonicity condition to certify the canonical property of the variables, equation to determine collective submanifold and constraints to govern the collective and the Grassmann variables are given.
The many-fermion problem is formulated in terms of boson variables that represent collective and particle-hole excitations of the system. An expansion scheme is presented which in a systematic and physically reasonable way allows the inclusion of the boson degrees of freedom in increasing detail. The RPA canonical transformation can be readily performed among the discretised boson operators. The authors apply the scheme to the homogeneous electron gas and use the exp S method to derive explicit expressions for the static properties for the system.