Let us suppose that the e+ e− pair is produced by an isolated photon of energy E = h ν and momentum h ν/c. Let the electron and positron be emitted with momentum p− and p+, their total energy being E− and E+. Energy conservation gives
hν0 = E+ + E− (1)
The momentum conservation implies that the three momenta vectors, p0, p+ and p− must form the sides of a closed triangle, as in Fig. 7.17. Now in any triangle, any side is equal or smaller than the sum of the other sides. Thus
Now the left hand side of the inequality is greater than the value of the radical on the right hand side. It must be still greater than the right hand side. We thus get into an absurdity, which has resulted from the assumption that both momentum and energy are simultaneously conserved in this process. Thus an electron–positron pair cannot be produced by an isolated photon.
