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Progress of Theoretical Physics

Applicability of the Canonical Quantization Procedure for the Collective Hamiltonian Derived by the Selfconsistent-Collective-Coordinate Method

Masayuki Matsuo, K. Matsuyanagi

Published Aug 1, 1985• 18 citations
Publisher

Abstract

We investigate the validity of the canonical quantization of the collective Hamiltonian derived by the selfconsistent-collective-coordinate method. Special attention is paid to the boundary condition for the (η*, η) expansion, which fixes the canonical coordinate system to carry out the quantization procedure. We remove the ambiguity of the boundary condition by requiring that the boson representations obtained by the canonical quantization agree, to the order 1/Ω, with those derived by the modified-Marumori boson-expansion method in a certain limit. Numerical examples for an exactly solvable model show that our quantization procedure provides us with remarkably accurate results.

DOI
10.1143/ptp.74.288

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Journal of the Physical Society of Japan•Aug 15, 1986•1 citation
We investigate the validity of the canonical quantization of the collective Hamiltonian derived by the selfconsistent-collective-coordinate method. Special attention is paid to the boundary condition for the (η*, η) expansion, which fixes the canonical coordinate system to carry out the quantization procedure. We remove the ambiguity of the boundary condition by requiring that the boson representations obtained by the canonical quantization agree, to the order 1/Ω, with those derived by the modified-Marumori boson-expansion method in a certain limit. Numerical examples for an exactly solvable model show that our quantization procedure provides us with remarkably accurate results.
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
We develope a collective description of the many-particle system interacting through an arbitrary two-body potential. The description is given both in terms of the second quantized form and in terms of an auxiliary field analogous to the longitudinal electromagnetic field. As Bohm-Pines theory of the collective description has some unsatisfactory features concerning the subsidiary condition imposed on the auxiliary field, we propose here a strict method of treatment on the subsidiary condition. To obtain the useful result of the collective representation, we have to take the three steps of the canonical transformations. Though the first two transformations are the same as those of B-P, the third transformation, which is not applied by B-P and is characteristic of our theory, is necessary for obtaining a meaningful result. By using both the strict subsidiary condition and the three canonical transformations, the collective field coordinate and the particle coordinate are separated not only in the total Hamiltonian but also in the subsidiary condition. Moreover, we are able to clarify the correct meaning of the subsidiary condition and clarify also the connections between our theory and the theory of Sawada-Brueckner-Fukuda-Brout.
The selfconsistent-collective-coordinate method is applied to anharmonic quadrupole vibrations in a degenerate many j-shell model with the pairing-plus-quadrupole force. By comparing the result obtained by this method with that obtained by the boson-expansion method, it is shown that 1) the selfconsistent-collective-coordinate method yields, after a suitable canonical quantization, the result that agrees to order (j + 1/2)-1 with that obtained by the boson-expansion method, in the limit that the time-dependent Hartree-Bogoliubov space is truncated to the multiphonon space built out of the 2+ phonons, and 2) the couplings between the RPA 2+ phonons and the other modes (the pairing rotations and the noncollective modes) considerably modify the fourth-order boson Hamiltonian obtained by the conventional boson-expansion method.
Quantum statistical mechanics of soliton systems is developed in the path integral formalizm by making use of the non-canonical collective coordinate method. The free energy is calculated in the ideal-gas approximation up to the order of two-loop. Explicit calculations are given for the sine-Gordon model. The classical limit of the present quantum calculation agrees with the exact result of the low-temperature expansion of the transfer-integral method. Numerical calculation reveals the quantum effect on the specific heat.
The collective coordinate method is applied to develop the statistical mechanics of soliton systems. The present perturbation expansion is a direct extention of that done in the quantum field theory and is applicable to the quantum case. A new type of divergence which has never met so far in the many body theory appears in the present treatment but it can be removed by an appropriate treatment giving reasonable results. As the first paper of this series the classical case is investigated and explicit calculation is done in the classical sine-Gordon model. The results agree with the exact low temperature corrections obtained by the transfer-integral method and with those obtained by Miyashita and Maki who applied the collective coordinate approach in a different way.