Re: Why pair-production cannot take place in vacuum?

Your question is not completely specified, because you have
not told me what the initial state is or what kind of pairs you want
to create. I will assume that you want to start with a gamma-ray (which
is a high energy photon) and create an electron-positron pair.

Allowed processes must obey the conservation laws.  This is actually a
definition of conservation laws.  However, all processes that we have
observed do obey these laws, so we assume that they will continue to
work.  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.

I hope this answers your question.