Why pair production can not take place in vacuum?

The most basic conservation laws are conservation of energy, conservation of momentum and conservation of angular momentum. These laws are a result of the invariance of the fundamental equations of motion to moving the origin or rotating the axes of our coordinate system. In other words, the fundamental equations should look the same, regardless of how we choose our coordinate system. The mathematical theorem which relates the existence of these conservation laws to the invariance of the equations is called Noether's Theorem. 

The reason why a gamma-ray cannot create electron-positron pairs in a vacuum is that there is no way to do this without violating the conservation laws. Another fundamental invariance of the equations is Lorentz invariance, which means that the fundamental equations should look the same in all coordinate systems which can be related by a Lorentz transformation. A Lorentz transformation relates the equations in one coordinate system with the equations in another coordinate system which is moving with a constant velocity with respect to the first one. Under a Lorentz transformation the gamma-ray undergoes a Doppler shift. That means its energy and momentum are changed. We can choose a Lorentz transformation under which the energy of the gamma-ray becomes too small to create an electron-positron pair. 


The physical process must be the same, regardless of which coordinate system we choose to describe it in. Therefore, it must be impossible for the isolated gamma-ray to create a pair in any coordinate system. Obviously, the argument is the same for the creation of proton-antiproton pairs, etc. If we started with two gamma-rays moving in opposite directions, then the gamma-rays could collide with each other, and create a particle- antiparticle pair. In this case, when we try to make a Lorentz transformation which Doppler-shifts the energy of one gamma-ray to low energy, the energy of the other gamma-ray will be Doppler-shifted to a higher energy. The sum of the energies of the two gamma-rays is the smallest in the coordinate system in which their total momentum is zero. Thus, if they can create an electron-positron pair in this coordinate system, they can do it in any Lorentz-transformed coordinate system. 

There must always be another particle present which can be manipulated so the total momentum of all particles is conserved before & after pair production.

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