# Electron/Positron pair production in free space?

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?

• May 29, 2017 at 10:41

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.

$$\gamma \gamma \to e^+ e^-$$
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.