Quantum Statistics of Interacting Particles; General Theory and Some Remarks on Properties of an Electron Gas

Abstract

A systematic generalization of the Mayer cluster integral theory has been developed to deal with the quantum statistics of interacting particles. The grand partition function appears in a natural way and the cluster integrals are integrals over propagators which are derived from the Green's function solution of the Bloch equation (which follows from the Schroedinger equation by replacing it/ℏ by β = 1/kT). Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram in (β, r) space. The generalization of classical ring cluster integrals has been analyzed. It is shown that in the case of the electron gas the classical limit of the contribution of these integrals to the grand partition function yields the Debye-Huckel theory while the low temperature limit leads to the Gell-Mann—Brueckner equation for the correlation energy of the ground state. A prescription is given for the construction of the cluster integral associated with any given diagram.