Electron interactions: Part I. Field theory of a degenerate electron gas

Abstract

The method of Gell-Mann and Brueckner for treating electron interactions in a degenerate electron gas is generalized using the Feynman-Dyson techniques of field theory. A Feynman propagator is constructed for the effective interaction between electrons which takes into account the polarizability of the medium of unexcited particles in the Fermi sea. The well-known plasmon excitation appears as a singularity in this propagator. The plasmon is seen to be a correlated, resonant oscillation of the electron density field which is damped by transferring its energy to less correlated, multiple excitations. General expressions for the plasmon dispersion relation and for the plasmon level width are derived in terms of the polarizability of the many-body medium. The self-energies of the lowest states of the electron gas are discussed by using the adiabatic theorem. This enables us to derive an exact expression for the ground state energy in terms of the polarizability. A formal calculation of the ground state energy to third order in the inter-electron spacing is carried out. Because of the degeneracy of the excited states of the noninteracting system, the adiabatic transforms of these states are not stationary states of the interacting system. However, as the momenta of the excited particles approach the Fermi momenta these states become asymptotically stationary. For states with only a few excited particles present an independent particle model is valid with the result that only the Feynman propagator for the physical oneparticle state is needed. This propagator, which is corrected for the virtual polarization of the medium by the particle, provides all the information concerning the energies and damping of the single particle states.