Electric Dipole Approximation and the Canonical Formalism in Electrodynamics

We use the electric dipole approximation to study the problem of finding commuting solutions of coupled equations of motion. We point out that for a charged particle in an external radiation field, the solutions of the coupled equations cannot be considered independent in the sense of commuting with one another if the homogeneous solutions are assumed to have the commutation properties of uncoupled variables. We explicitly treat the case of a charged free particle and a charged harmonic oscillator in an external radiation field. We indicate that for a retarded (advanced) self-field, the free particle fits into a canonical formalism while the oscillator does not. For a stationary self-field, both the free particle and the oscillator fit into a canonical formalism. We show that the Fourier transforms of the configuration space solutions (based on $e^{i\mathrm{k}·\mathrm{x}}$ and $e^{i\omega t}$) do not exist. In the latter connection, we point out that earlier treatments of the oscillator by Sokolov and Tumanov and Norton and Watson contain misleading results as a consequence of their using Fourier transforms.