Classical theory of point dipoles

It is well known that a limit to the validity of quantum mechanics is set by its neglect of the effects of radiation damping, and an indication as to where this limit occurs is given in electron theory by the classical equation of Lorentz, which takes radiation damping into account. Now the interaction of the meson field with a neutron or proton contains an explicit dipole term, and this leads to scattering cross-sections for mesons which increase quadratically with the energy. A classical treatment of radiation damping for a dipole would thus give us the limits of the quantum theory due to neglect of this factor. A non-relativistic attempt has already been made by Heisenberg1 for a dipole of finite extension, but his results even in the limit of a point dipole do not agree with ours. We believe that this is because the usual method of calculation which he follows is inconsistent with the theory of relativity, for whereas the retardation of the field is taken into account, different parts of the dipole are assumed to move instantaneously in phase