My recent answer to the question Scattering, Perturbation and asymptotic states in LSZ reduction formula got me thinking again about wave packets and the LSZ reduction formula.
In my answer, I claim that the LSZ reduction formula really applies to particles which are widely separated wave packets, come to interact over a finite volume, and then leave. Of course, in the limit that the wave packets are sharply peaked in momentum space, the volume over which they are interacting is infinite, but in real life, this is never really the case.
In reality, the LSZ reduction formula allows you to compute the cross sections for when the particles are spread out much beyond their Compton wavelength and they hit dead on. It doesn't address the possibility that the peaks of the wave packets glance by each other.
I don't know anything about collider physics, but presumably, people know about this sort of thing. Is there any way to predict what will happen when particles "glance" by each other, where by "glance" I mean that the position uncertainty of the particles is not much larger than their Compton wavelengths and they do not collide head on? What effect does this have on cross sections? Is there a good reason that people don't worry about this possibility?
