In reading through old course material, I found the assignment (my translation):
Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.
My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.
How to solve this task?
Answer
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.
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