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Why is the following process not possible?

process

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?

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    $\begingroup$ Put initial electron in the rest $(m,0,0,0)$. 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. $\endgroup$ – user26143 Nov 17 '13 at 8:39
  • $\begingroup$ Related: physics.stackexchange.com/q/12488/2451 , physics.stackexchange.com/q/13513/2451 , physics.stackexchange.com/q/22916/2451 , and links therein. $\endgroup$ – Qmechanic Nov 17 '13 at 9:35
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    $\begingroup$ @user26143: can you make that an answer. Comments are transient things and can disappear, but an answer will remain for future generations of budding physicists :-) $\endgroup$ – John Rennie Nov 17 '13 at 10:11
  • $\begingroup$ @John Rennie Surely $\endgroup$ – user26143 Nov 17 '13 at 13:57
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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.

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