Green Functions for "One-Body" Equations Describing the Disordered State

Shigeji Fujita

The rigorous evolution equations for the one-body Green's functions (${g}^{>},{g}^{<}$) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, $N$ (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (${g}^{>},{g}^{<}$).

e.H. Hellal

Roger Balian, Lewis H. Nosanow, N. R. Werthamer

Shigeji Fujita

The rigorous evolution equations for the one-body Green's functions ( g > , g < ) describing an imperfect gas in a finite volume are not closed (an analogue of B–B–G–K–Y hierarchy for the many-body distribution functions). The analysis of these equations are made in terms of proper connected diagrams. It is shown that in the bulk limit: N (particle number)→∞, \(\varOmega\)(volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.

В. Л. Бонч-Бруевич, Sergei Vladimirovich Tyablikov, Raymond J. Seeger

Victor Dansage, V. Ballenegger, Angel Alastuey

Abstract We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.