I'm going back over some graduate courses I took 20 years ago. I no longer work in physics and never worked in particle theory (which will be obvious).
I'm having trouble trying to understand the dynamics of a process like Moller scattering, where beams of electrons deflect each other.
I remember calculating the cross section for Moller scattering using the mechanics of QFT. We set up an initial state of incoming electrons as wave packets in some narrow momentum band; an operator acting on this initial state as a sum of time translation and the interaction between the electrons; and the final state as wave packets of outgoing electrons with a new momentum (different angle). The interaction part of the operator arises from the exchange of photons by the electrons, which can occur in various ways. We sum up the contributions from each way and calculate the amplitude, which is angle-dependent and agrees with experiment. Right? Or at least right-ish?
(I read that according to electroweak theory, there is also a Z-boson exchange. Hopefully that is not really relevant to my question, so let's leave it for now.)
A simple mental picture of electrons exchanging photons (as in the Feynman diagram) appeals to my mostly classical understanding of the world. The photon sort of shoots over from electron to electron, carrying momentum from one to the other - I guess like spaceship shooting a wad of silly putty that attaches to a second ship, so that both are deflected.
Except, this mental picture is confusing me. Do we have any conception of an intermediate state in this process? When or how exactly do the exchanges happen? Say the electron beams are tightly constrained along the direction of motion, so well localized spatially and in time. I guess in this case their momentum is relatively diffuse; does this mean the elementary QFT treatment (above) is not relevant? Is there a point when such beams "start" to interact, or "have mostly" interacted?
In practice, a tightly localized packet of electrons (or in the limit, a single pair of particles) would still deflect; how and when would this happen exactly, on a physical level? Say that as the beams approach each other, a third, much higher energy charged beam suddenly comes flying in and disrupts the process. Wouldn't the final state depend on when exactly the third beam arrived, with respect to some kind of intermediate state for the initial two beams?
Thank you for any help