In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.
Derivative expansion for the one-loop effective Lagrangian in QED