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Journal of Physics G Nuclear Physics

Path-integral approach to time-dependent self-consistent field theories

Hugo Reinhardt

Published Jun 1, 1979• 15 citations
Publisher

Abstract

With the aim of obtaining a field theory of large amplitude collective motion the author formulates the time-dependent Hartree approximation within a path-integral approach.
DOI
10.1088/0305-4616/5/6/001

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The geometry of the self-consistent collective-coordinate (SCC) method formulated within the frame-work of the time-dependent Hartree-Fock (TDHF) theory is investigated by associating the variational parameters with a symplectic manifold (a TDHF manifold). With the use of a canonical-variable parametrization, it is shown that the TDHF equation is equivalent to the canonical equations of motion in classical mechanics in the TDHF manifold. This enables us to investigate geometrical structure of the SCC method in the language of the classical mechanics. The SCC method turns out to give a prescription how to dynamically extract a “maximally-decoupled” collective submanifold (hypersurface) out of the TDHF manifold, in such a way that a certain kind of trajectories corresponding to the large-amplitude collective motion under consideration can be reproduced on the hypersurface as precisely as possible. The stability of the hypersurface at each point on it is investigated, in order to see whether the hypersurface obtained by the SCC method is really an approximate integral surface in the TDHF manifold or not.
The study of large-amplitude collective motion has been stimulated in Japan, by Professor T. Marumori through his paper published in 1977. Under this situation, a series of annual research projects were organized by the Research Institute for Fundamental Physics, Kyoto University, entitled “Large Amplitude Collective Motion and Microscopic Theory”, “Time-Dependent Self-Consistent Field Method and Quantization”, “Microscopic Theories of Large Amplitude Collective Motions”, and so forth from 1980 to 1985. The present volume of Progress of Theoretical Physics Supplement is edited to give a comprehensive review on the theoretical study reported mainly by the authors engaged in those research projects.
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.