Radiative Corrections to the Theory of an Electron Gas

Radiative corrections to the Bohm-Pines dispersion equation and the Pines-Schrieffer-Drummond coupled pair of quasilinear equations are examined in the light of the Weisskopf-Wigner theory of line broadening. The principal results are the natural broadening and the Lamb shift of the particle-wave resonance condition. The dimensionless width of the nonresonant or adiabatic interaction is of the order of $\alpha \left(\frac{c}{\nu }\right)$ for electron-plasmon interaction and $\alpha \left(\frac{\nu }{c}\right)$ for electron-photon interaction, where $\alpha$ is the fine-structure constant. The Lamb shift of the resonance condition $\omega -\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{v}}\approx 0$ is of the order of $\frac{q^2{\omega }^2}{\pi \mu {\nu }^3}$ for electron-plasmon interaction and $\frac{q^2{\omega }^2}{\pi \mu c^2\nu }$ for electron-photon interaction. Here $\omega$ and $\overrightarrow{\mathrm{k}}$ are the frequency and wave vector, respectively, of the photon or the plasmon and $q$, $\mu$, and $\overrightarrow{\mathrm{v}}$ are the charge, mass, and velocity of the electron, respectively.