Statistical mechanics of the electron-phonon system

Abstract

The electron-phonon system has received considerable attention in connection with the phenomenon of superconductivity, where the condensation of electron pairs is believed to arise from the attraction produced by the phonons () The system is usually described by the Frolich Hamiltonian (), where the phonons are represented by a scalar field coupled to the electron density. Although this interaction has been widely used for approximate calculations, it has never been proved that it provides a statistical mechanical description in the usual sense, in particular that it allows the infinite volume limit to be taken for the thermodynamic quantities. Here, we shall consider this problem, concentrating on the statistical mechanics of the electrons. We therefore us the method of Feynman (), in which the statistical operator of the electrons is given by a path integral representation and the phonon field is eliminated at the expense of introducing a non instantaneous interaction between the electrons. Except for this last feature, the situation is then similar to the one encountered in the case of quantum systems with instantaneous potentials (). The Hamiltonian we start from is