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Proceedings of the Physical Society

Green function decoupling and diagrams

P. Gluck

Published Sep 1, 1967• 3 citations
Publisher

Abstract

The case of phonon interactions is considered to demonstrate that the equation-of-motion method is equivalent to the diagram technique for finding the self-energy occurring in the evaluation of Green functions. Precise approximations are stated to establish the equivalence.
DOI
10.1088/0370-1328/92/1/326

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Using a diagram technique, equations of motion for N coherently driven two-level atoms are obtained from Green's functions. The longitudinal (Coulomb) dipole–dipole interaction is neglected. Three approximations (Born approximation, extended Born approximation, and screened interaction approximation) are considered. The existing theories are shown to be equivalent to the extended Born approximation. For a small sample, the equations of motion exhibit S conservation in this case. The screened interaction model predicts a different behaviour; in particular, S-breaking terms appear in the equations of motion for a small sample.
It is shown that any given decoupling of the equations of motion for the propagators of an interacting Fermi system is equivalent to a particular partial sum of Feynman diagrams. This result is established for many-time propagators in a straightforward way by iterating in the diagrammatic form of the equations of motion. The iteration method does not work for two-time propagators, but we are able to present an alternative argument which makes it plausible that the result holds true in this case as well. The decouplings leading to the Hartree-Fock, ladder, and ring diagram (RPA) partial sums are discussed in detail. It is conjectured that the reverse correspondence is not true, i.e. to any given partial sum there does not necessarily correspond a decoupling.