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Context: I ask this as a school teacher reaching past the boundaries of my expertise. A colleague was talking about the standard model with an advanced student, explaining how particles interact by exchanging gauge bosons, asking the student to imagine gauge bosons as little spheres. Of course they got to the issue that this mental model doesn't add up, since in classical mechanics such a process always leads to repelling forces, while the forces in electron-positron-scattering are attractive. He asked me on my opinion on how to rectify his explanation in such a way that a student could still create a mental image of the process.

This is what I came up with, and while I'm sure that it can't do the actual physics full justice, I would like you to point out precisely in what ways this is inaccurate, so that I can improve it in such a way that an advanced student can still have a mental image of the process.

The mental model: In classical physics, waves typically obey the superposition principle, that is, two waves don't interact, but can pass through each other without influencing each other. This is true for electromagnetic waves, but also all other waves the students might have encountered in class. It is not always true for water waves, however. For instance, two tall waves crashing into each other won't pass through each other unchanged since they lose energy due to turbulences. The correct description of their interaction requires an additional term besides just the sum of the two wave functions.

We can imagine something similar to be true for wave functions in QFT: As a particle like an electron is described as a wave function, the interaction of, say, two electrons is similar to that of two water waves in the sense that they do actually interact, yielding more complicated wave effects. A main difference is that unlike classical fields, which don't interact with other fields, the electron field interacts with the electromagnetic field as well, and the nonlinear effects come from one electron wave packet interacting with the em. field, which then interacts with the other electron wave packet. And since photons are electromagnetic wave packets, we can think of the em. part of the interaction as being comprised of photons.

So what do you think? How (in)accurate is this mental model of interactions mediated by gauge fields.

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A main difference is that unlike classical fields, which don't interact with other fields,

This is not true. You can absolutely define a "classical" version of the electron field, in the same way you can define a classical version of the electromagnetic field. And you can make them interact. In fact, this is usually how you go about defining a quantum theory. You start with a classical theory and then you make it quantum.

Here's what makes quantum interactions different from classical interactions (and thereby hard to visualise): In the quantum world, everything that can happen does happen. What do I mean by that? Consider the scattering of two particles. We can draw it in a little diagram like this:

enter image description here

Here's how to read such a diagram: Time is going from left to right. The straight lines represent two particles (for example an electron and a positron). They bump together to create a photon (the wiggly line in the middle). Then the photon again creates an electron-positron pair. Now, there are other things that can happen in between. For example, you can have something like this:

enter image description here

Here, the photon once again splits into an electron-positron pair in the middle which then annihilate into a photon again. Now, you can go crazy and imagine all possible things that can happen in between the two particles coming in and the two particles going out. All of these diagrams correspond to mathematical functions and in order to calculate the strength of your interaction, you have to add them all up. Every single diagram you can imagine.

So in some sense, all of the individual scattering situations you can imagine classically are happening at once, and adding them all up somehow gives you a result that agrees with reality.

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This is a very quantum mechanics thought. This doesn't fully grasp QFT, where the core effect is particle creation and annihilation and antiparticles. It doesn't explain for example how electrons and positrons annihilate to create a photon. In QFT we don't use wave functions (that's again, QM, although one can argue QFT is just QM with infinite degrees of freedom). Ignoring this, actually it isn't a bad model at all, especially for the propagation. The actual interaction is easier to describe with particles. You could say that they move as waves and exchange energy as particles. Then the QED interaction is described by its vertex, where two electrons and a photon interact. Note that you can think of positrons as electrons going back in time. So you can draw Feynman diagrams and say that the vertices (interactions) are the way you see them, but the way a particle moves from a vertex to another is like a wave, not as a particle going in straight lines. Describing interactions in terms of waves is quite hard.

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  • $\begingroup$ "Describing interactions in terms of waves is quite hard". The way I understood it, the virtual particles exchanged during an interaction aren't real anyway, but just a mathematical artifact arising from a specific way to approximate the results using a series expansion. So is a description with waves just hard, or is it conceptually problematic? Keep in mind that even though we do have some very advanced students, the chances that any of them can meaningfully handle Hilbert spaces and the Lagrangian formalism is slim to nonexistent anyway, so what's mathematically hard is less important here. $\endgroup$ Commented Jun 21, 2023 at 4:16
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I know that you are wanting to talk about

So what do you think? How (in)accurate is this mental model of interactions mediated by gauge fields.

And so we should be talking about accuracy of your mental model, but I would like to remind you that the original aim is

explaining how particles interact by exchanging gauge bosons[...]such a process always leads to repelling forces, while the forces in electron-positron-scattering are attractive. He asked me on my opinion on how to rectify his explanation in such a way that a student could still create a mental image of the process.

I would have to point out that it is perfectly ok that your mental image is a completely different character than the one that he gave, but it is not so much ok that your mental image does not even attempt to give an answer that his mental image does. You are totally allowed to abandon the mental image that he is conjuring, but you ought to be able to do more than his, maybe with omission of a few results, and not just achieve a lot less with nothing much to show for it.

Maybe you can come up with a better mental image than mine, I'm just suggesting: Instead of a wave being bunched up as a crest and pushed between two particles, maybe the exchanged boson is more like a trough, so that it could make a pull rather than always a push.

Of course, all of these are doomed to just be handwaving, and there will not be a real understanding without actually working through the maths and deeply understanding the maths, but such mental models are still helpful for the layperson. The horrible details are not their job to have to disentangle; that falls upon us, who are afflicted by the curiosity.

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I'm going to try and combine the previous answers into one that more directly answers your question.

First, I'm going to diverge a bit and say that honestly, your mental picture has the important parts right. Lenard is correct to say that classical nonlinear fields exist, but aside from that, yes, interactions between particles are precisely equivalent to and governed by the existence of nonlinearities in the underlying wave equation. A linear wave equation satisfies the principle of superposition and as such has no interactions.

Now, as you note, since the relevant nonlinearities are in coupling between the electron field and the electromagnetic field and since the particle of the electromagnetic field is the photon, you can think of the interaction as "trading a photon". But we can go farther. A "real" photon is a stable electromagnetic wave, but an unstable wave -- a transient disturbance in the EM field -- can still be thought of as a photon, just with the "wrong" properties. This is a virtual particle, and the farther the photon is from its "correct" properties -- zero mass and momentum equal to $\hbar k$ -- the less stable it is and the sooner it falls apart.

But for as long as it exists, a virtual particle can have any mass or momentum. In particular, it can have negative momentum, and thus when it hits another particle it causes a "backward" acceleration. This is basically the same as saying that if you're in a pool and you push a wave at someone, they can end up being pushed either direction depending on the shape of the wave.

(Some sources prefer invoking the uncertainty principle and say that the mass-energy/momenta can be "wrong" so long as it doesn't last very long ($\Delta t \leq \frac{\hbar}{2 \Delta E}$) or travel very far ($\Delta x \leq \frac{\hbar}{2 \Delta p}$) relative to the size of the "error". This is correct and mathematically equivalent to the above, but IMO less didactic; the uncertainty principle is just the same property of transient waves repackaged in the first place, after all.)

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