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Progress of Theoretical Physics

A Method of a Quasi-Linear Canonical Transformation for the Many-Body Problem. I

Shokichi Kanno

Published Apr 1, 1969• 0 citations
Publisher

Abstract

A general method of a quasi-linear canonical transformation is introduced as an approach to the many-body problems. In this method the n-body problem can be solved if the (n-1)-body problem is assumed to be solved. In the application of this method to the system of N bosons, there appears no logarithmic divergent term in the physical quantities which correspond to the self-energy operators Σ11 and Σ02 appearing in the method of Green's function.

DOI
10.1143/ptp.41.966

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A general method of a quasi-linear canonical transformation is introduced as an approach to the many-body problems. In this method the n-body problem can be solved if the (n-1)-body problem is assumed to be solved. In the application of this method to the system of N bosons, there appears no logarithmic divergent term in the physical quantities which correspond to the self-energy operators Σ11 and Σ02 appearing in the method of Green's function.
Many-body Green functions are a very efficient formulation of the many-body problem. We review the application of this method to nuclear physics problems. The formulas which can be derived are of general applicability, e.g., in self-consistent as well as in nonself-consistent calculations. With the help of the Landau renormalization, one obtains relations without any approximations. This allows to apply conservation laws which lead to important general relations. We investigate the one-body and two-body Green functions as well as the three-body Green function and discuss their connection to nuclear observables. The generalization to systems with pair correlations are also presented. Numerical examples are compared with experimental data.
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The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.
The theory of many-Boson systems was considered. A canonical transformation of the creation and annihilation operators was introduced in order to eliminate the singleparticle zero-momentam state and allow the application of field-theoretic penturbation techniques. These techniques were used to give a simpler derivation of a result due to Belyaev and to prove a theorem about the excitation spectrum of the system. General techniques for calculating the Green's functions are discussed. The excitation spectrum was calculated to second order for small wave number for the case of a repulsive square-well potential.
It is shown for systems of strongly interacting particles that in the limit of very many particles a transformation exists leading to an alternative problem which can be solved by a self-consistent field method. The transformed wave function describes "particles" moving in a collectively determined uniform potential; the transformation relating the transformed and original wave functions determines the amount of incoherence and correlation in the original function. The method appears to be more useful for fermion systems and is illustrated by applications to some aspects of nuclear structure.