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Fortschritte der Physik

Greensche Funktionen in der Theorie der Kernmaterie

M. K. Weigel, Gerd Wegmann

Published Jan 1, 1971• 35 citations
Publisher

Abstract

Abstract The aim of this article is the representation of the theory of nuclear matter with the technique of many‐body Green functions. We have treated the theory of Green functions thoroughly in the first chapters taking support of the functional method, which offers a convenient approach to the construction of the theory. After treatment of the properties of the exact two‐point function and of the effective single‐particle potential, preconditions and properties of the approximate solutions are discussed. Results achieved todate are treated in chapter 6. Brueckner theory and allied methods are not further discussed, since these are sufficiently known. The theory of linear response, being important for the semi‐phenomenological treatment of real atomic nuclei, is presented in appendix A.

DOI
10.1002/prop.19710190703

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Abstract The aim of this article is the representation of the theory of nuclear matter with the technique of many‐body Green functions. We have treated the theory of Green functions thoroughly in the first chapters taking support of the functional method, which offers a convenient approach to the construction of the theory. After treatment of the properties of the exact two‐point function and of the effective single‐particle potential, preconditions and properties of the approximate solutions are discussed. Results achieved todate are treated in chapter 6. Brueckner theory and allied methods are not further discussed, since these are sufficiently known. The theory of linear response, being important for the semi‐phenomenological treatment of real atomic nuclei, is presented in appendix A.
In this paper we present the theory of nuclear matter in the framework of relativistic nuclear field theory utilizing the technique of relativistic many-body Green's functions. The analytic properties of the Green's functions and related quantities are given and used to obtain the final set of equations for the nuclear matter problem. In detail, we have investigated theoretically and numerically the so-called ${\ensuremath{\Lambda}}^{00}$ and ${\ensuremath{\Lambda}}^{10}$ approximations. For comparison we also calculated the relativistic Hartree-Fock and the Brueckner-Hartree-Fock approximation, respectively. For masses and coupling strengths we used (phenomenological) parameter sets given by the Bonn and Groningen group. The results of the \ensuremath{\Lambda} approximations are similar to the relativistic Brueckner treatment.
The Green’s function method in the Kadanoff-Baym version provides a basic theory for nuclear dynamics which is applicable also to nonzero temperature and to nonequilibrium systems. At the same time, it maintains the basic many-body techniques of the Brueckner theory that makes reasonable a comparison of the numerical results of the two methods for equilibrium systems. The correct approximation to the spectral function which takes into account the widths of energy levels is offered and discussed, and the comparison of the values of binding energy in the two methods is produced.
The formal theory of Green functions is reviewed for many particle problems in nuclear physics or in solid state physics. In particular, one- and two-particle Green functions are discussed in detail. The Feynman amplitudes are also explained together with the effective potential and its applications. This article contains the following sections: §1. Preliminary remarks §2. Definition of Green functions §3. Structure of equations satisfied by Green functions 3.1 One-particle Green function 3.2 Two-particle Green function §4. Perturbation theory and its foundation §5. Spectral representation and its application §6. Connection with reaction matrix §7. Amplitudes and their matrix 7.1 Definitions of amplitudes and effective potential 7.2 Nuclear optical model 7.3 One electron in insulators and semi-conductors Appendix I. Functional differentiation Appendix II. Schwinger's dynamical principle Appendix III. Polaron problem
Abstract This chapter discusses the significance of two body interactions to many-body properties, beginning with a schematic model showing the relevance of short-range correlations beyond those of the Fermi gas. These correlations are studied within Brueckner theory and the help of the Bethe-Goldstone equation, mainly for nuclear matter proper but extensions to finite temperature are presented. Effective forces of Skyrme type are derived through an expansion of the G-matrix to small relative momenta. This scheme hinges essentially on the Brueckner-Hartree-Fock approximation, for which results for binding energy and equilibrium density are presented. An alternative variational method based on Jastrow functions is sketched, which demonstrates the necessity for considering effective three body forces. The concept of the nuclear equation of state is presented and parameterized by Skyrme forces with density dependent parameters. Finally, a semi-classical transport theory is presented which accounts for statistical collisions of the nucleons.
Widths of low-lying states in nuclei are of the order of 30 MeV. These large widths are a consequence of the strong interactions leading to a strongly correlated many body system at the typical densities of nuclear matter. Nevertheless ''traditional'' Brueckner calculations treat these states as quasiparticles i.e. with spectral functions of zero widths. The width is related to the imaginary part of the selfenergy and is included selfconsistently in an extension of the Brueckner theory using T -matrix and Green's function (GF) techniques. A more general formulation applicable also to non-equilibrium systems is contained in the Kadanoff-Baym (KB) equations while still maintaining the basic many-body techniques of Brueckner theory. In the present work the two-time KB-equations are timestepped along the imaginary timeaxis to calculate the binding energy of nuclear matter as a function of density, including the spectral widths self-consistently. These zero temperature calculations are compared with quasiparticle calculations. The inclusion of the selfconsistent widths are found to add several MeV to the binding. The spectral widths are due to the long-ranged correlations. Short ranged correlations decrease rather than increase the binding. The method is easily extended to nonzero temperatures where the importance of the widths are expected to increase.
The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.