Photon Radiation from Van Allen Belt The Van Allen belts  contain charged particles, and the geomagnetic field drives them back and forth, thus accelerating them.
My questions:
Will that cause the charged particles to lose energy by means of radiating a photon? Or will they be locked in some quantum state so that they don't lose energy?
If they do radiate some photons, then
Is it possible to take an image of these belts (even if not in the visible band)? And are there such images anywhere to see?
 A: In principle trapped particles in the Van Allen belts will radiate away energy in the form of electromagnetic waves due to their acceleration by magnetic fields. However, the accelerations are relatively small and are not likely to be directly detectable or have a major impact on their motion. This could be quantified by considering their cyclotron emission due to gyromotion (the highest frequency/strongest radiating component), $P = - \frac{2 \sigma_t B^2 KE}{m c \mu_0}$ (where $B$ is magnetic field strength, $KE$ is kinetic energy, $m$ is mass, and $\sigma_t$ is the Thomson cross section). Even assuming the strongest value for Earth’s magnetic field, $B = 65000$ NT, the time constant associated with this process for electrons is about 20 years. That’s much longer than the time scales for other loss processes at work in the Van Allen belts. Moreover, the frequency of this cyclotron radiation would be in the MHz range ($f = \frac{qB}{2\pi m}$), meaning that it would be difficult to observe particle-like behaviour.
It's worth noting that interaction of particles with electromagnetic waves from other sources can be important for the dynamics of the Van Allen Belts. This can scatter previously trapped particles onto paths which take them into the atmosphere where they are lost through collisions.
