On the Heitler Theory of Radiation Damping

Heitler has developed a method of including radiation damping in scattering problems. It involves an integral equation for the scattered field amplitude far from the scatterer. We have derived this integral equation classically for a special group of problems. Our derivation allows a comparison between the Heitler method and the Wentzel-Dirac $\lambda$-process. Two simple problems are discussed, involving the interaction of an oscillator with scalar and vector meson fields, respectively. Both problems have exact solutions, which we compare with approximate solutions by the Heitler method. The Heitler method preserves the shape of the resonance but neglects the shift in resonance frequency caused by the additional mass of the oscillator coming from the coupling to the meson field. The strong radiation damping encountered in vector meson theory appears in the second example, although we assumed no spin interaction. The damping can be traced to the Lorentz condition $\Sigma \stackrel{4}{i=1\mathrm{}}\frac{\partial U_i}{\partial x_i}=0\mathrm{}$ on the 4-vector field quantity $U_i$.