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Progress of Theoretical Physics

Path Integral on the Coset Space of the SO(2N) Group and the Time-Dependent Hartree-Bogoliubov Equation

S. Nishiyama

Published Jul 1, 1981• 14 citations
Publisher

Abstract

A quantized time-dependent Hartree-Bogoliubov (TDHB) theory of many fermion systems with pair correlations is presented. It is obtained not by using a priori quantized method but by using the path integrals on the coset space SO(2N)/U(N). Then we can get another version of the TDHB equation expressed through the coset variables in a natural way as a classical limit.

DOI
10.1143/ptp.66.348

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A possible quantization is proposed with respect to time-dependent Hartree-Bogoliubov theory based on the SO(2N+1) algebra which was presented by Fukutome, Nishiyama and the present author. The adopted quantization procedure is an extension of Marshalek and Holzwarth's idea used in the case of the conventional time-dependent Hartree-Bogoliubov theory based on the SO(2N) algebra. As a natural consequent of the quantization, Holstein-Primakoff type boson expansion theory can be derived in a unified form of even and odd particle system.

Progress of Theoretical Physics•Feb 1, 1980•2 citations
A quantized time-dependent Hartree-Bogoliubov (TDHB) theory of many fermion systems with pair correlations is presented. It is obtained not by using a priori quantized method but by using the path integrals on the coset space SO(2N)/U(N). Then we can get another version of the TDHB equation expressed through the coset variables in a natural way as a classical limit.
SO(2N)U(N)\frac{{\rm SO}(2N)}{U(N)}U(N)SO(2N)​ Riccati–Hartree–Bogoliubov equation based on the SO (2N) Lie algebra of the fermion operators
In this paper we present the induced representation of SO (2N) canonical transformation group and introduce [Formula: see text] coset variables. We give a derivation of the time-dependent Hartree–Bogoliubov (TDHB) equation on the Kähler coset space [Formula: see text] from the Euler–Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati–Hartree–Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO (2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.
An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.
We study the variational properties of the action functional appeared in the time-dependent Hartree-Bogoliubov (TDHB) theory, Through the calculus of the second variation on the coset manifold SO(2N)/U(N), the Jacobi equation is obtained. Assuming the periodic Jacobi field, we derive in a natural way the equation for the quasi-particle random phase approximation (RPA) describing the collective excitation around certain static HB fields. The present method of obtaining the SO(2N) RPA may be useful to get the SO(2N+1) RPA.
A possible quantization is proposed with respect to time-dependent Hartree-Bogoliubov theory based on the SO(2N+1) algebra which was presented by Fukutome, Nishiyama and the present author. The adopted quantization procedure is an extension of Marshalek and Holzwarth's idea used in the case of the conventional time-dependent Hartree-Bogoliubov theory based on the SO(2N) algebra. As a natural consequent of the quantization, Holstein-Primakoff type boson expansion theory can be derived in a unified form of even and odd particle system.