Radiation reaction in nonrelativistic quantum electrodynamics

Abstract

We derive the Heisenberg operator equation of motion for a nonrelativistic point electron coupled to the quantized electromagnetic field, including radiation reaction. The derivation proceeds in close analogy with the classical theory of extended charges (with the Compton wavelength formally playing the role of a size parameter), and we give a systematic treatment of the classical problem, showing explicitly from the equation of motion that the classical theory shows no runaway solutions or preacceleration when the electron size exceeds the classical electron radius. In the quantum-mechanical case, we show that the electrostatic self-energy of a point electron is zero and that, for values of the fine-structure constant $\ensuremath{\alpha}\ensuremath{\lesssim}1$, the equation of motion admits neither runaway solutions nor noncausal motion. Furthermore, the correspondence limit of the solutions to the quantum-mechanical equation of motion agrees with that of the Lorentz-Dirac theory in the classical regime, but without the imposition of additional conditions and with no possibility of observable noncausality. Thus, a consistent picture of a classical point electron emerges in the correspondence limit of the quantum-mechanical theory.