A Generalized Mean Field Approach to the Polaron

The motion of an electron in a polar crystal has often been treated in a number of previous works motivated by its increasing interest in transport phenomena in semiconductors, which play an important role in the miniaturization of electronic devices. In addition to its possible connection with the real behavior of an electron and a crystal, this model represents one of the simplest examples of the interaction of a particle and a field. The electron interacts with ions, which are not rigidly fixed. Thus, the electron polarizes the lattice in its neighborhood, and if the electron moves about, the polarization state moves with it. This electron, together with its distorted environment, is a polaron. The precise motion of the polaron in a quantum-mechanical framework is exceedingly complicated. The path integral variational method has been successfully applied to the analysis of this idealized problem, see for example the work of Bogoliubov, Feynman or Höhler. More recently the polaron model has also been treated from different points of view by Accardi et al., Donsker and Varadhan, Löwen, . . . for example). In the paper of Bechouche et al. a rigorous presentation of the time evolution of the mean field approach to the polaron is given. A new term is added in the equation for the potential by assuming that the action of the electric field is not instantaneous. This term gives a potential involving memory effects.