Many-particle Green's functions

Lorenz S. Cederbaum

The theory of Green's functions is briefly reviewed. It is discussed which properties of a quantum system and which physical processes can be described by Green's functions. A general approximation scheme to evaluate Green's functions and propagators is presented. Particular attention is paid to the one-particle Green's function and to the particle–particle propagator which is a relevant component of the two-particle Green's function. Several illustrative applications are presented and related to experiment.

J. Schirmer

T.T.S. Kuo, Yiharn Tzeng

We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.

Ervin Kaminski Lenzi, R. S. Mendes, A. K. Rajagopal

The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-particle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages.