How are real particles created? 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)?
 A: The difference between real and virtual particles in QFT has to do with whether the particles are represented by an internal line or whether they have at least one free end. 
For real particles, their momenta and energy need to be considered in the overall conservation of energy and momentum. This is represented, ad-hoc so far as I understand, by the dirac delta in the propagator integral. For the internal lines, their momenta integrals are done over all of phase space.
So if you ask "How are real particles created in QFT?" As with everything else quantum, the answer to "how" is "it was possible and then it happened." If you mean "what is the difference between the mathematical treatment of virtual and real particles?" I would say, "there are extra constraints on the energy-momentum vector for real particles."
A: We measure real particles in the laboratory, and model their existence and behavior using quantum field theory and the Feynman diagrams that give the values for measurable quantities: crossections, decays, angular distributions... .
Quantum field theories are  useful in many domains of physics, from nuclear models to condensed matter  to particle physics.
In particle physics it is posited that each particle ( and implied antiparticle) in the elementary particles table generates a field that fills up all space time

an electron field, a positron field an electron neutrino field, etc for the ennumerated particles.
On these fields , which mathematically are the plane wave solutions ( no potential) of the corresponding quantum mechanical equation ( Dirac, Klein Gordon, quantized Maxwell) act creation and annihilation operators. Where they act the corresponding particle is created, or destroyed. 
The Feynman diagrams show this at the vertices,

This diagram is for computing the interaction of e- e- scattering. At the vertices the electrons go in time in the y direction, by consecutive creation and annihilation , and at the vertices  a photon is created which is exchanged between them, and is anihilated , in a virtual mathematical space, as everything is under an integral.
This integral is important because it allows for the existence of real electrons input and output. Remember a plane wave covers from -infinity to infinity, and cannot be a useful representation of a particle. The uncertainty principle mathematically leads to wave packets representing the real paticles , in this case electrons in electrons out after scatter. The mathematical formalism of calculating measurable quantities using feynman diagrams is shown here , clarifying the need for wavepackets for modeling measureable quantities of real particles.
A: Which QFT textbooks have you read? I'll follow Secs. I.2-I.4 of Quantum Field Theory in a Nutshell by Anthony Zee. For a source current $J$ of the simplest kind of particle, an elementary spin-$0$ particle, the transition amplitude is $Z(J)=Z(0)\exp iW(J)$ with $i^2=-1$ and $$W=\int\dfrac{d^4 k}{(2\pi)^4}\dfrac{J^\ast (k) J(k)}{k^2-m^2+i\epsilon}.$$
(The details are slightly more complicated for other kinds of particle, but not in ways that matter here.)
Here $J(k)$ is the $k$-space representation of the current, $k^2:=k^\mu k_\mu$, and $\epsilon$ is a small real parameter we can set to $0$ in some calculations to recover a finite physical result. The denominator is a complex number of very little imaginary part, so the "real" case $k^2=m^2$ maximises the integrand's modulus. Virtual $k^\mu$ far from this condition make a heavily suppressed but non-zero contribution to the phase $W$, and classical physics neglects this option altogether. You may want to compare the real case to resonance in classical physics: factors of $c,\,\hbar$ let us think of $k^\mu$ as either a four-momentum or a four-frequency, analogous to a resonant frequency.
Just as quantum mechanics reveals classical states for a system comprising a given number of particles can be superposed, quantum field theory allows even the particle number to be uncertain. We cannot reduce the problem to saying, "there are this many particles of each species": instead we superpose over such options. Virtual particles characterise this, but to speak of them "becoming real" warrants a bit of clarification. The number of real particles can change (if the Lagrangian contains such terms as to allow it) due to $k^\mu$-transferring interactions of one or more quantum fields, including with existing real particles, which are really just one way that fields manifest themselves.
A: ''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.
A: *

*leptons (electron) and quarks, that build up matter, were created with pair creation.

*that means, that a matter-antimatter pair can be created out of vacuum (and can annihilate too into vacuum), this pair creation (and annihilation) is going on in every neutron and proton all the time, because neutrons and protons are made up of not just valence quarks, but a sea of quark-antiquark pairs, the net of that sea are the 3 valence quarks 

*now why do we see more matter then antimatter? that is baryon asymmetry

*now you are saying that it is virtual particles, actually virtual particles (their mass is off shell) are a mathematical way to describe the forces (EM, strong, weak, gravity) that act between particles 
