Free particle spectrum?
For simplicity, let's consider the total absorption of a photon by an electron in the lab frame, where the electron is initially at rest. For a photon of energy Eγ = pc and momentum p = Eγ/c , the final total energy of the recoiling electron would be
Ee= m0c2 + Eγ.
But the momentum of the recoiling electron would be p = Eγ/c,
leading to a total energy of
[Ee]2= [m0c2]2 + [pc]2= [m0c2]2 + [Eγ]2
These two equations are irreconcilably different, because energy and momentum cannot both be conserved..