The application of ray representations of translation groups to the motion of an electron in a crystal lattice

Abstract

We consider the motion of an electron in one and three dimensions when the electron is coupled to an external electromagnetic field. We characterize the electromagnetic field under the requirement that the dynamical system be invariant under translations in time and space. In addition to considering the case in which the translations in space can be arbitrary, we consider the case in which the dynamical system is to be invariant under space translations which are a multiple of a given length. This latter case is the case of an electron moving in a crystal. It is found that when all translations are permitted, the electromagnetic field is constant in space and time. In the case where space translations are multiples of a given length, the electromagnetic fields (not potentials!) are periodic in space. The paradox of characterizing the motion of an electron in a crystal when the system is subject to a constant external electromagnetic field and thus where the potential is not periodic is thereby resolved. Ray representations of the translation group play an essential role in the treatment.