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Progress of Theoretical Physics

Utility of Dirac Quantization of Classical System Involving Both Collective and Independent-Particle Degrees of Freedom: A Schematic Model

A. Kuriyama, M. Yamamura

Published Mar 1, 1981• 4 citations
Publisher

Abstract

The Dirac quantization method of canonical system with constraints is applied to a classical image of many-fermion system expressed with collective and independent-particle variables. As an illustration, the Lipkin model is adopted. The results are consistent with those coming from the specific properties of the SU(2)-algebra.
DOI
10.1143/ptp.65.1094

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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.
In a previous paper, a quantum-mechanical formulation involving both mean fields and independent-particle fields in many-fermion systems was proposed using the path-integral technique. Then the semiclassical calculation of the energy spectra was performed, and the quantization rule was derived by applying a stationary phase approximation on the path integral. In this paper, a boson-fermion expansion is derived from our formulation using Dirac quantization. As an illustration, the Lipkin model is utilized.
The problem of connecting systems with different numbers of degrees of freedom is discussed. Constraints appropriate for ``Bose'' and ``Fermi'' quantization are used to construct algebras of Dirac brackets associated with special solutions of the nonlinear complex oscillator. The constraints are shown to provide a basis for characterizing the elementary excitations of the oscillators. An alternative notion of quantization through a correspondence with an enveloping subalgebra of the Dirac brackets is introduced, a notion which simplifies the operator-ordering problem implied by the original Dirac brackets. The infinite- and the two-dimensional representations of the subalgebra are utilized to illustrate the quantization technique.
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.