So , everywhere I have looked for pair production it is stated that it cannnot happen in vacuum . Most proofs I have seen for that , state that the conservation of energy and momentum cannot be true at the same time without a body with which the photon will interact in the first place . In most of these proofs the momentum of the electron and the positron is considered the same value p . Why that ?
My explanation why pair production cannot happen in vacuum is based in the existence of a COM frame . We consider that a COM exists for every isolated system , so one will exist for the pair production in vacuum . But if that's true the spatial momentum of the photon will be zero , which is impossible . So if we consider that the photon first interacts with another object (for example with a nucleus that initially has zero velocity ) then we can find a COM . Is this explanation ok ?