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Physical Review

The Collective Description of Many-Particle Systems (A Generalized Theory of Hartree Fields)

T. Kinoshita, Y. Nambu

Published May 1, 1954• 22 citations
Publisher

Abstract

A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the $S$ matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.

DOI
10.1103/physrev.94.598

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ON THE CORRELATION EFFECT IN MANY-BODY SYSTEMS

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Using the generalized method of self-consistent field developed by one of the authors[4], we discuss various correlation phenomena of many-body systems with a unified viewpoint. The general method of treatment is proposed and compared with the method of perturbation. From the approximation with two-body correlation we obtain, as particular applications, the Bruckner theory of nuclear matter and the Bohm-Pines theory of the collective oscillation of electron,gases. Finally the superconductivity and zero-point oscillations in non-ideal gases are discussed by solving, the bound state problem with the present method and the Gell-mann Low procedure. It is concluded that the method is very effective in treating correlation phenomena.

Acta Physica Sinica•Jan 1, 1960•0 citations
A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the SSS matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.
The Hartree-Fock (HF) approximation is generalized to the study of n -particle states, taking correlation effects in interacting quantum-mechanical many-particle systems systematically into account. The case of n = 2 is developed explicitly. It is shown that the form of the HF equations remains essentially the same as in the usual formulation, but with appropriate interpretations of various parameters in terms of n -particle, rather than single-particle, states. We show results of calculations for model systems that illustrate the power of the method when compared to ordinary versions of the HF treatment.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
Using the generalized method of self-consistent field developed by one of the authors[4], we discuss various correlation phenomena of many-body systems with a unified viewpoint. The general method of treatment is proposed and compared with the method of perturbation. From the approximation with two-body correlation we obtain, as particular applications, the Bruckner theory of nuclear matter and the Bohm-Pines theory of the collective oscillation of electron,gases. Finally the superconductivity and zero-point oscillations in non-ideal gases are discussed by solving, the bound state problem with the present method and the Gell-mann Low procedure. It is concluded that the method is very effective in treating correlation phenomena.