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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

Dennis

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Quantum field theory tells you how the final state of the two electrons is probabilistically related to their initial state. It doesn’t give you a tidy classical-like picture of what happens in between. In fact, it gives you a very messy and unintuitive but amazing picture that is infinitely more interesting than the classical one.

According to the path integral formulation (which your QFT course may not have covered), every possible evolution of the electron and photon fields consistent with the initial and final states occurs! The complex probability amplitudes for the infinite possibilities all add up to produce a net amplitude from which the probability of various final states can be calculated.

So envisioning a virtual photon as a wad of silly putty shot from one to the other, emitted at a particular time and place, and absorbed at a particular time and place, is a misleading and far-too-classical picture to have in your mind.

In Feynman diagrams, you are actually integrating over all possible times and places for electron-photon interactions. So you really should imagine scenarios like one of the electrons travels to Mars and back while the other plays loop-de-loop around the lab, and an infinite number of other crazy possibilities.

That said, the amplitudes for the crazy stuff tend to destructively interfere, and the net amplitude ends up having the classical trajectory, plus “nearby” ones (say, a few Compton wavelengths away), as its dominant contributions.

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  • $\begingroup$ Hi G., thank you for the response. So maybe regarding intermediate states, if it were possible to measure them, we could obtain various results, as in basic quantum mechanics, with different probabilities? (If so is there a way to calculate the probability? I haven't gotten that far in my reading.) BTW, It's still difficult for me to picture the path integral formulation as anything more than a way to arrive at a result, as opposed to a description of something which is actually occurring, in any sense of that word. Maybe that is a deeper level discussion... $\endgroup$ – Dennis Aug 8 at 16:10
  • $\begingroup$ If you measured the intermediate results, they would no longer be intermediate results. The probabilistic results of any measurement can be calculated but what is happening between measurements cannot, except as a superposition of states in general. $\endgroup$ – G. Smith Aug 8 at 17:11
  • $\begingroup$ Maybe I miscommunicated. I think I understand the basic postulate of measurement in QM and superposition of states. My lack of understanding here is more about time evolution within a QFT process. Scattering seems to have a duration, wherein fields interact over a period of time. I can't see explicitly where time enters into the interaction terms, so not sure what is being said about the evolution of the system. For example: is there some point during the process where the beam has partially deflected, but not fully as it would as t->inf? Does that make sense? Thanks for your patience on this $\endgroup$ – Dennis Aug 8 at 18:10
  • $\begingroup$ It does not make sense to me, for the reasons I explained in my answer. “Partial deflection” only has meaning to me in classical physics. Let’s see if it makes sense to someone else. $\endgroup$ – G. Smith Aug 8 at 19:41
  • $\begingroup$ Okay, I see, thank you. In basic QM we explicitly wrote down things like the coefficients of the + and - z-direction spin states in a beam of particles passing through an x-directed B field, as a function of time. Maybe I was looking for something like that. Anyway, still confused by the idea that if a third beam of particles arrives during a scattering event, given beams of finite extent, surely the timing of its arrival would affect the final state. For free particles interacting nature this seems like the norm. ? $\endgroup$ – Dennis Aug 8 at 20:40

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