Suppression of Bremsstrahlung at Non-Zero Temperature

The first-order bremsstrahlung emission spectrum is $\alpha d\omega \mathrm{/}\omega$ at zero temperature. If the radiation is emitted into a region that contains a thermal distribution of photons, then the rate is increased by a factor $1+N(\omega )$ where $N(\omega )$ is the Bose-Einstein function. The stimulated emission changes the spectrum to $\alpha Td\omega \mathrm{/}{\omega }^2$ for $\omega\ll T$. If this were correct, an infinite amount of energy would be radiated in the low frequency modes. This unphysical result indicates a breakdown of perturbation theory. The paper computes the bremsstrahlung rate to all orders of perturbation theory, neglecting the recoil of the charged particle. When the perturbation series is summed, it has a different low-energy behavior. For $\omega \ll \alpha T$, the spectrum is independent of $\omega$ and has a value proportional to $d\omega \mathrm{/}\alpha T$ .