Comments on Intensity-Dependent Frequency Shift in Compton Scattering and Its Possible Detection

We show that the Hamiltonian of an electron coupled to an external radiation field is not Hermitian within the space of Volkov solutions with scattering boundary conditions. This is demonstrated by explicitly showing that the inner product of two states which are asymptotically orthogonal to each other fails to vanish at a finite time. It is further demonstrated that if the momentum operator is Hermitian within the (asymptotically) free-electron states, it fails to be Hermitian within the fully developed Volkov states. Consequently the parameters $p$ and $p^{\prime }$ entering into the Volkov solution with scattering boundary conditions cannot be interpreted as the momenta of the incident and emerging electron, respectively. The only consistent use of the Volkov solutions is for nonscattering situations, viz., the emission or absorption of a photon by an electron which is never decoupled from the external field. In the latter case the parameters $p$ and $p^{\prime }$ which enter into the energy-momentum conservation formula refer to average initial and final electron momenta. A consistent interpretation requires us also to compute the dispersion in the electron momenta. On this basis we also obtain a dispersion in the scattered frequency, a physically detectable quantity, which is much larger than the intensity-dependent frequency shift. On the basis of general quantum principles we also demonstrate: (a) that the Feynman-Dyson $S$-matrix technique is the correct procedure to use inasmuch as it gives agreement with energy-momentum conservation and unitarity, (b) that Kibble's explanation of energy-momentum conservation is incompatible with quantized perturbation theory, and (c) that in the case of external transverse fields the perturbation expansion yields a polynomial in the field strength. An experiment involving frequency mixing is suggested as a means of obtaining a maximum effect in the "beat" frequency due to the nonlinearity parameter ${\nu }^2$.