  As a preliminary step, the radiation produced by a classical charged current
coupled to a quantized $A_{\mu}$ is solved. To each order in $\alpha$, all
infrared divergences cancel between the virtual $\gamma$'s and the real
$\gamma$'s absorbed from the plasma or emitted into the plasma. When all orders
of perturbation theory are summed, the finite answer predicts a suppression of
radiation with $\omega< \alpha T$. The analysis of QED then consists of two
steps. First, a general probability at $T\neq 0$ is organized so that all the
virtual $e^{\pm},\gamma$ are in the amplitudes and all the real
$e^{\pm},\gamma$ are in the phase space integrations. Next, the cancellations
of IR divergences between virtual and real are demonstrated.
