# Quantum statistics of interacting particles

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## Summary

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 Schrödinger equation by replacing*it/h* by β*= 1/kT).* Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram. The indistinguishibility of particles causes each particle (or vertex) in a Mayer cluster integral to be replaced by a « toron » which can be described in the following manner through a Feynman diagram. Consider an*(r, β)* space in the form of a torus of tubal circumference β. A toron of order*n* is represented by a closed path which loops the torus*n* times so that a cut at constant β’ (with 0 〈 β’*〈 β =* 1/kT) on the torus identifies the position of*n* particles at β’ but which in the absence of a cut gives no indication of where one particle ends and another begins. The grand partition function is a sum over all graphs in which torons are connected by interaction lines that represent quanta of energy and momentum exchanged through collisions. Self interaction lines on torons must be included. The cluster integrals associated with rings of torons have 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 Brückner equation for the correlation energy of the ground state. A prescription was given for the construction of the cluster integral associated with any given diagram.

A complete account of this work is published in the Jan. 1957 issue of*Journ. of Fluid Physics.*