A method for choosing and deriving a suitable single-particle potential for many-body calculations at finite temperature is suggested. With this potential one can readily determine the self-consistent correlated chemical potential and consequently the resulting equations of state rigorously satisfy a number of basic thermodynamic relations. Application of our method to a Brueckner-Hartree-Fock nuclear matter calculation yields results in remarkably good agreement with the Hugenholtz--Van Hove theorem.
Conserving approximations lead to thermodynamically consistent expressions for Fermi-liquid properties such as compression modulus, symmetry energy and Landau parameters. We will emphasize the need for the precision experiment to determine the density of states at the Fermi surface by discussing nuclear matter approximations in terms of thermodynamic consistency (the Hugenholtz-Van Hove theorem and the virial theorem). The density of states is the primary self-consistent observable to be determined from nuclear many-body approximations. The precise measurement of this observable will establish a rigorous criterion on the validity of nuclear matter approximations.
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.
In describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations has been developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator
G, defined in the presence of a nonlocal external scalar field
U. Approximations for
G(U) are obtained by replacing the
G2(U) which appears in these equations by various functionals of
G(U). If the approximation for
G2(U) satisfies certain simple symmetry conditions, then the
G(U) thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L$, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal
f-sum rule.Several examples of conserving approximations are described. The Hartree approximation,
G2(U)=G(U)G(U), generates the random-phase approximation for
L. The Hartree-Fock approximation for
G(U) leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for
G(U) is obtained by expanding the self-energy to first order in the many-particle scattering matrix
T(U). This
T is obtained by summing ladder diagrams in which the sides of the ladder are composed of
G(U)′s. The resulting
L equation, which involves coefficients proportional to
∣T∣2, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential,
Vs(U). This potential is obtained by summing bubbles composed of two
G(U)′s. The resulting
L equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to
∣Vs∣2.
Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.
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.