TITLE: How are real particles created?
QUESTION [1 upvotes]: The textbooks about quantum field theory I have seen so far say
that all talk in popular science literature
about particles being created spontaneously out of
vacuum is wrong. Instead, according to QFT those virtual particles are
unobservable and are just
a mathematical picture of the perturbation expansion of the propagator.
What I have been wondering is, how did the real particles, which are observable, get created? How does QFT describe pair production, in particular starting with vacuum and ending with a real, on-shell particle-antiparticle pair?
Can anybody explain this to me and point me to some textbooks
or articles elaborating on this question (no popular science, please)?

REPLY [1 votes]: ''The textbooks about quantum field theory I have seen so far say that all talk in popular science literature about particles being created spontaneously out of vacuum is wrong.''
And they are right doing so. See also my essay 
https://www.physicsforums.com/insights/physics-virtual-particles/
''How does QFT describe pair production, in particular starting with vacuum and ending with a real, on-shell particle-antiparticle pair?''
It doesn't. There are no such processes. Pair production is always from other particles, never from the vacuum or from a single stable particle.
''I cannot find a calculation for an amplitude <0|e+e-> or something like that.''
Because this amplitude always vanishes. 
All nonzero amplitudes must respect the conservation of 4-momentum, which is impossible for <0|e+e->. You can see this from the delta-function which appears in the S-matrix elements. It follows from this formula that the requested amplitude vanishes, since delta(q)=0 when q is nonzero.
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