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Progress of Theoretical Physics

A Quantal Theory of Pairing Rotation, Pairing Vibrations and Independent-Particle Motions

A. Kuriyama, M. Yamamura

Published Apr 1, 1982• 0 citations
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Abstract

A microscopic theory is proposed to describe not only the pairing rotation, the pairing vibrations and the independent-particle motions, but also their mutual couplings in a consistent manner. With the aid of a BCS-type wave packet and a quasi-particle coherent state, the quantal system is translated into a corresponding classical one. First, we develop a canonical form in a classical image. This classical system obeys certain constraints, by which the double counting in the degrees of freedom is avoided. Then, the quantization is performed with the use of the Dirac theory of a canonical system with constraints. The original fermion operators themselves are expressed in terms of three types of degrees of freedom, which correspond to the pairing rotation, the pairing vibrations and the independent-quasi-particle motions, respectively.

DOI
10.1143/ptp.67.1122

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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 motion in many-fermion system is developed in the framework of the classical theory. The basic idea is an extension of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.
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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.
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.