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Progress of Theoretical Physics

Equation of Collective Submanifold for Mixed States

Masatoshi Yamamura, João da Providência, A. Kuriyama, Carlos Fiolhais

Published Jun 1, 1989• 10 citations
Publisher

Abstract

With the aim of giving a microscopic description of various nuclear phenomena observed in highly excited states, the equation of the collective submanifold for mixed states is presented. The basic idea is a possible TDHF-like variation in the enlarged space which is adopted in the thermo field dynamics formalism. A set of equations, which determines the collective submanifold, is obtained. The form is analogous to that given in the conventional TDHF theory. At the small amplitude limit, an equation, the form of which is similar to the conventional RPA equation, can be derived in a natural way.

DOI
10.1143/ptp.81.1198

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•Jan 1, 1979•1 citation
With the aim of giving a microscopic description of various nuclear phenomena observed in highly excited states, the equation of the collective submanifold for mixed states is presented. The basic idea is a possible TDHF-like variation in the enlarged space which is adopted in the thermo field dynamics formalism. A set of equations, which determines the collective submanifold, is obtained. The form is analogous to that given in the conventional TDHF theory. At the small amplitude limit, an equation, the form of which is similar to the conventional RPA equation, can be derived in a natural way.
A classical microscopic theory of mixed states has recently been formulated in order to deal with the theoretical description of nuclear phenomena observed in highly excited states. In this theory, which is here briefly reviewed, the degrees of freedom of real fermions are doubled by fictitious fermions, in the context of the formalism of thermofield dynamics. The requirement of static stability of the thermal state of equilibrium plays, in the theory, a relevant role. The possibility of constraining the dynamics of the fictitious degrees of freedom, in such a way as to eliminate spurious thermal excitations, is discussed. The concept of canonical collective variables for mixed states is analyzed. Boson expansions appropriate to describe collective excitations of hot quantal systems are introduced and their properties investigated. It is shown that thermal boson expansions based on the formalism of thermofield dynamics are free from formal inconsistencies which may be present in boson expansions derived by an ad hoc quantization of conventional thermal mean field theories. The formalism is applied to the Lipkin model, and the correlation energy of a hot system is discussed.
A theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.A canonical transformation is investigated in the full time•dependent Hartree•Fock theory, i.e., in the classical image of boson expansion theory.With the aid of the transfor• mation, the whole system is separated into collective and intrinsic degrees of freedom.Under the condition of the unique separation, two types of equations of collective submanifold, which are canonically invariant, are obtained.One is of the same form as that of the conventional equation of collective submanifold.A principle of the specification of coordinate system is discussed.
Aiming at exploring microscopic dynamics responsible for the dissipative large-amplitude collective motion, the dynamic response and correlation functions are introduced within the general theory of nuclear coupled-master equations. The theory is based on the microscopic theory of nuclear collective dynamics which has been developed within the time-dependent Hartree-Fock (TDHF) theory for disclosing the complex structure of the TDHF manifold. A systematic numerical method for calculating the dynamic response and correlation functions is proposed. By performing numerical calculation for a simple model Hamiltonian, it is pointed out that the dynamic response function gives important information in understanding the large-amplitude dissipative collective motion which is described by an ensemble of trajectories within the TDHF manifold.
A systematic microscopic theory of large-amplitude collective motion, which has been recently developed within the framework of the time-dependent Hartree-Fock (TDHF) theory under INS-TSUKUBA joint research project, is reformulated in terms of classical mechanics in the TDHF symplectic manifold. This reformulation clarifies an essential concept in the theory, i.e., an approximate integral surface in the TDHF manifold, which is identified as a dynamical collective submanifold for the large-amplitude collective motion. It is shown that the approximate integral surface must satisfy three dynamical conditions; (a) a stationary condition (i.e., a maximal-decoupling condition), (b) a stability condition and (c) a separability condition between collective and “local” non-collective modes of motion. These three dynamical conditions make clear the concept of “local” intrinsic modes of motion and guarantee that the TDHF-trajectory representing the large-amplitude collective motion under consideration is successfully reproduced on the approximate integral surface.
By applying the microscopic theory of dynamical collective submanifold, which has been formulated by the present authors, to a system with a simple model Hamiltonian, characteristic features of large-amplitude motion described by the time-dependent Hartree-Fock (TDHF) theory are visualized. Our theory dynamically defines an “optimum” submanifold out of the TDHF symplectic manifold (phase space) in such a way that the Hamiltonian on the submanifold is stationary with respect to variations perpendicular to it. The theory also classifies the submanifold into three different regions in terms of stablility and separability conditions. It is displayed that these three regions characterize collective, dissipative and stochastic motions in the TDHF theory.
The aim of this series of papers is to propose a microscopic theory to go beyond the situations where collective motions are described by the random phase approximation, i.e., by small amplitude harmonic oscillations about equilibrium. The theory is thus ap­ propriate for the microscopic description of the large amplitude collective motion of soft nuclei. The essential idea is to develop a method to determine the collective subspace (or submanifold) in the many-particle Hilbert space in an optimal way, on the basis of a fundamental princi­ ple called the invariance principle of the Schrodinger equation. By using the principle with­ in the framework of the Hartree-Fock theory, it is shown that the theory can clarify the structure of the so-called phonon-bands by self-consistently deriving the collective Hamil­ tonian where the number of the physical phonon is conserved. The purpose of this paper is not to go into detailed quantitative discussion, but rather to develop the basic idea.