"Slightly off-shell"? I'm not new to QFT, yet there are some matters which are quite puzzling to me. I often come across the statement that real particles (the ones we actually measure in experiments, not virtual ones) are "slightly off-shell". What does this actually mean? To my knowledge, something being off-shell means that it violates the relativistic energy-momentum relation. But how can this be possible for particles we actually consider to be "physical"? Please fill me in on the mathematical/experimental backing for such a statement to any degree of detail necessary.
 A: See for instance the comments on my answer to Are W & Z bosons virtual or not?. 
Basically the claim is that the observed particle represents a path internal to some Feynman diagram and accordingly there is a integral over it's momentum.
I'm not a theorist, but as far as I can tell the claim is supportable in a pedantic way, but not very useful.
A: This aspect of quantum field theories also finds expression in the "infraparticle" approach. The basic idea is to discuss the soft photon field and a "bare" particle together, and call it a "dressed" particle. The propagator of a dressed electron can be computed in the infrared to be of the form $\frac{k\cdot\gamma+m}{(k^2-m^2+\mathrm{i}\epsilon)^{1-\alpha/\pi}}$, instead of the bare particle propagator $\frac{k\cdot\gamma+m}{k^2-m^2+\mathrm{i}\epsilon}$. In the ultraviolet, however, at short distances, where perhaps one might want this point of view to hold more than at large distances, this approximation falls apart.
The "slightly off-shell" that you speak of is equivalent to the fact that the effective propagator is not (the Feynman propagator version of) a delta function in momentum space.
A lot of this was worked out in the 50s and 60s, but my understanding is that it has been displaced by the success of the mathematics of the renormalization group. You could look at section II of Thomas Appelquist and J. Carazzone, Phys. Rev. D 11, 2856–2861 (1975), "Infrared singularities and massive fields", which is a brief, quite readable review at that time (which is where I found the propagator above). Still, there are recent references on the Wikipedia page.
I'm a little out of my depth here, but this is as well as I can express it here. Of course, according to Feynman we're all a little out of our depth with QFT.
A: The reason people say this is because all particles you see are absorbed after a finite time, and the notion of on-shell is asymptotic. The finite time means that they are really internal lines in a diagram, and so ever-so-slightly off shell. The exactly on-shell S-matrix is an asymptotic quantity, relevant only in the holographic limit.
A: For a real particle to be off-shell, it is sufficient to be in an external field of some sort. It is not necessary to be "absorbed". "Absorbed" are virtual particles, for example, "virtual photons" who describe non propagating fields like a Coulomb one.
For a Compton scattering, the real electron is "off-shell" during interaction with the real photon, i.e., while being in its field.
