How to show that the annihilation of an electron-positron pair under emission of one real photon (mass zero) is forbidden by energy-momentum conservation, the emission of two photons is allowed?

  • $\begingroup$ I know the linked duplicate asks for $\gamma\to e^+ e^-$ instead of $e^+ e^-\to\gamma$, but that doesn't make a difference for the answer. $\endgroup$ – ACuriousMind Jan 25 '16 at 22:27

The electron-positron pair has a center-of-mass reference frame where the momentum is 0.

Obviously, there exists no one-photon system with positive energy which has 0 momentum, as the energy-momentum relation for a photon is $E = p c$.

  • $\begingroup$ Is there a way to get it mathematically? $\endgroup$ – user83866 Jan 25 '16 at 22:39
  • $\begingroup$ @user83866 $p=\frac{E}{c}$ for a single photon cannot equal 0, but $p_{epcm}=0$. That's the math. $\endgroup$ – Bill N Jan 25 '16 at 22:43
  • $\begingroup$ @BillN Is it possible show why this statement is true or it is an experimental fact which is not defined theoretically? $\endgroup$ – user83866 Jan 25 '16 at 22:58
  • $\begingroup$ What don't you understand about something non-zero not being equal to zero? This IS the theoretical result. $\endgroup$ – Bill N Jan 25 '16 at 23:20

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