Pair production can't occur in vaccum because momentum cannot be conserved because the photon does not have a rest frame of reference, that means that the produced pair has a center of mass frame of reference where the total momentum is zero, so momentum is not conserved.

Now it is said that momentum is conserved when a third body like a nucleus takes the momentum for itself, but still, this is where my problem is, the nucleus still has a frame of reference where the momentum is zero, so how is it conserved?

• Write your formulas to illustrate your thoughts. Commented May 23, 2019 at 15:38
• There are two inertial frames in which the nucleus has zero momentum for some time interval: there is the frame in which the nucleus starts at rest and then starts to move after the interaction, and there is the frame in which the nucleus starts with nonzero momentum and then comes to rest after the interaction. There is no (inertial) frame in which the nucleus has zero momentum the whole time. Commented May 23, 2019 at 15:42

You have to have clear in mind that momentum and energy conservation hold in a single inertial frame, so one must chose where the sums will be.

In the inertial frame where the nucleus is at rest before the interaction, momentum conservation will give it a recoil after the interaction.

In the feynman diagram an exchange with a virtual photon transfers energy and momentum to the nucleus, in whatever inertial frame one wants to calculate conservation of energy and momentum.