Why can't a single photon produce an electron-positron pair?

Another way of solving such problems is to go to another reference frame, where you obviously don't have enough energy.

For example you've got a $5 MeV$ photon, so you think that there is plenty of energy to make $e^-e^+$ pair. Now you make a boost along the direction of the photon momentum with $v=0.99\,c$ and you get a $0.35 MeV$ photon. That is not enough even for one electron.

Other way to see why this is impossible is to look at inverse process: why annihilating positron and electron can't give up only one photon? Imagine these two particles at rest near each other (or look at center-of-mass system). They will annihilate giving 1MeV of energy, but single photon can't pick this energy up by itself because it would also have E/c of momentum and starting setup, the two charged particles, had non. You need two photons that move in opposite directions.

Check if momentum can be conserved. That ought to do the trick.