Is it true that both linear momentum and energy cannot be conserved if the production of an electron-positron pair via photon annihilation were to occur in free space?
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There are two ways that I could interpret this question. If you're interested in the interaction
$$ \gamma \to e^+e^- $$
taking place in free space, that's forbidden. The reason is that the electron-positron pair always has a rest frame where the momentum is zero, but the photon has no such rest frame.
If, on the other hand, you're asking about
$$ \gamma \gamma \to e^+ e^- $$
that is allowed. The matrix element is exactly the same as for the more familiar process $e^+e^-\to\gamma\gamma$, at least in the limit where time-reversal is a good symmetry of quantum electrodynamics. However the phase space for the $\gamma\gamma$ process is very different and it has never been observed with free photons. Much more common is an interaction between a hard (MeV+ energy) photon and the virtual photons that make up the electric field around atomic nuclei.
rob
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I guess that depends on what you mean by free space. Yes that is true that a gamma ray in a perfect vacuum can not decay by pair production since it can not conserve both energy and momentum. It will undergo pair production however if near a nucleus. A nucleus will enable energy and momentum to be conserved.
Natsfan
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