QED: basic process which violates energy-momentum conservation

Question

Why is the following process not possible?

My book says it's because it violate energy and momentum conservation. Can someone explain me explicitly why? Why couldn't an energetic electron not radiate an photon and continue to propagate with a smaller momentum?

Answer 1

Put the initial electron in rest, the four momentum is $(m,0,0,0)$. The emission photon is $(E,0,0,E)$. The electron after emitting photon is $(m−E,0,0,−E)$. By $p^2=m^2$, we have $(m−E)^2−E^2=m^2$ thus $−2mE=0$, which is not possible.