Green Function Method for Electron Gas. I

Abstract

The Green function method is applied to the problems of electron gas. It is emphasized that there is a perfect parallelism between the Green function methods in· zero-temperature problems and in finite-temperature problems of systems in thermal equilibrium. Further it is pointed out that the Green function method is also useful for the calculation of transport quantities. Much advances have been made recently in the study of many-body problems in quantum mechanics and quantum statistics. Particularly noteworthy is the com­ putation of the exact correlation energy of an electron gas at high density by Gell­ Mann and Bruecknerl) (GB). The high density result of GB has subsequently been derived by other methods by Sawada et al}) Hubbard,3) and others.4) The field theoretical techniques are quite useful in summing certain subseries of terms in the so-called linked-cluster expansion. Recently Galickij and Migdal,5) and Klein and Prahge 6 ) have applied the Green function method to many-body problems in quantum mechanics. This method is quite suitable for the study of many-body problems. From the one-particle Green function one may compute the energy and damping of quasiparticles and from the two-particle Green function one may obtain the potential energy due to two-body forces and the energy spectrum of excitations which are not describable as a sum of energies of quasi-particles. Montroll and \Vard 7 ) have developed a generalization of the cluster integral theory of Mayer to deal with the quantum statistics of many-particle systems. They have shown that in the case of an electron gas the classical limit of the contri­ bution of ring integrals to the grand partition function yields the Debye-Huckel theory, while the low temperature limit leads to the GB result. Matsubara 8 ) has applied a formalism of quantum field theory to the calculation of the grand partition function in quantum statistics. Matsubara's method has been developed recently by Bloch and Dominicis,9) Fradkin/O) and others.H) The ap­ plicability of the field theoretical techniques to quantum statistics is based on the existence of the theorem corresponding to that of Wick