Can an accelerated "free" electron absorb a photon? I have read that an accelerated, free* electron can absorb a photon. Can anyone explain why this is true, and if any, provide mathematical proof?
*I guess it's technically not "free" anymore since its being accelerated
 A: Photons and electrons are quantum mechanical entities. Acceleration means interaction of the electron with a field, getting a dp/dt . The simplest Feynman diagram showing such an interaction is Compton scattering :

Total absorption would mean an incoming photon+ electron , and outgoing only an electron. This cannot happen because the electron has a fixed mass and  does not have excited states to absorb all the energy of the photon. What can happen is that most of the energy of the photon becomes kinetic energy of the electron, in any inertial frame, and correspondingly the photon can have very small energy , tending to zero but never zero.
If the outgoing (or incoming) photon becomes virtual, connecting with an electric or magnetic field, then the kinematics has to include the originator of the field in energy momentum considerations, and the electron can absorb all the energy of the incoming photon the energy/momentum balance in its rest mass system taken up by the generator of the field that gave the virtual photon.
A: It doesn't matter whether the electron is ``accelerated'' or not: this problem should be the same when we look at the initial electron rest frame -- it should obey Lorentz invariance. 
And in that frame, one can easily deduce that this is impossible, since the energy conservation and momentum conservation could not be satisfied at the same time. 
A: A free electron cannot emit a photon, but an accelerating electron can and does. Because quantum processes are reversible, conceptually, an electron accelerating in a magnetic (or electric) field can absorb photons as well. The conservation laws are obeyed due to the interaction with the field (e.g. created by electromagnets in particle accelerators).
Electron Acceleration by Photon Absorption in a Magnetic Field
