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Progress of Theoretical Physics

Derivation of Random Phase Approximation on the Basis of the Generalized Schwinger Representation of the Fermion-Pairs

Y. Mizobuchi, Seiya Nishiyama, M. Yamamura

Published Jan 1, 1977• 4 citations
Publisher

Abstract

A derivation of the random phase approximation is given, which is based on the generalized Schwinger representation of the fermion-pairs. The method is made of two stages : First, the truncation of the boson space is done, and secondly comes the separation of boson operators into static and fluctuating parts. The pairing vibration in the super phase and the so-called quadrupole oscillation are treated by the method, which leads to the same results as those of RPA. Then the relations of these bosons to ones appearing in RPA are clarified.

DOI
10.1143/ptp.57.96

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A derivation of the random phase approximation is given, which is based on the generalized Schwinger representation of the fermion-pairs. The method is made of two stages : First, the truncation of the boson space is done, and secondly comes the separation of boson operators into static and fluctuating parts. The pairing vibration in the super phase and the so-called quadrupole oscillation are treated by the method, which leads to the same results as those of RPA. Then the relations of these bosons to ones appearing in RPA are clarified.
A new boson expansion theory based on the random phase approximation is presented. The boson expansions are derived here directly in the random phase approximation representation with the help of a technique that combines the use of the Usui operator with that of a new bosonization procedure, called the term-by-term bosonization method. The present boson expansion theory is constructed by retaining a single collective quadrupole random phase approximation component, a truncation that allows for a perturbative treatment of the whole problem. Both Hermitian, as well as non-Hermitian boson expansions, valid for even nuclei, are obtained.
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.
Explicit construction of Dyson, Holstein-Primakoff, and Schwinger representations for random-phase approximation phonon operators is given within Lie algebraic framework. It is shown that these representations emerge as exact boson realizations of quadrupole collective algebra, but the enforcement of SU(6) symmetry involves important constraints embodied in definite nonlinear conditions imposed on random-phase approximation phonon amplitudes. The constructed Schwinger representation could be employed to provide alternative approach to the interacting boson model parameters, avoiding standard procedures of mapping the shell-model SD subspace into the sd boson space.
We have proposed a theory called the resonating Hartree-Bogoliubov (Res HB) theory to describe large quantum fluctuations in a superconducting Fermion system. One needs also to deal with small quantum fluctuations which correspond to those described by the conventional random phase approximation (RPA). We can derive a new approximation called the Res HB RPA from time dependent small fluctuations of the Res HB ground state. The theory can give a unified description of collective excitations arising from large and small quantum fluctuations and their coupling in a superconducting system. For instance it may be applicable to rotational and vibrational excitations in nuclei with resonance of different deformed shapes.
The second random-phase approximation (SRPA) is the simplest and most natural extension of the RPA. It enlarges the space of the elementary modes introduced to describe the collective states by adding 2 particle -2 hole excitations to the 1 particle -1 hole ones of the RPA. In deriving the SRPA equations, use is made, as in the RPA, of the so-called quasi-boson approximation (QBA) where expectation values in the ground state of the system are approximated by their values in the uncorrelated reference state. This, however, has been shown to imply a degree of approximation worse than that in the RPA. It is, therefore, necessary to improve the QBA by considering a reference state which contains some correlations. Having in mind to perform such calculations for realistic systems, we consider a simple extension of the SRPA in which the reference state contains 2 particle - 2 hole correlations. The quality of such an extension is tested by applying it to a solvable three-level model and found to be good.