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I'm pretty new to quantum, without any formal education therein, so forgive my general layman ignorance when I ask how it is that point-like exchange particles can be modeled in Feynman diagrams as occupying multiple values of space at a single value of time (i.e. a situation in which the virtual photon created by an electromagnetic interaction between two electrons is graphed to be "flat"/0-slope along an x axis of space and a y axis of time).

Alternately, while a proton (a non-point composite particle) is comprised of three quarks and three gluons, all of which are themselves elementary point particles, a proton takes up a set amount of space. I understand that it is the strong force, acting via the gluons, that keeps the quarks in place and preserves this spatial shape,

Am I making the mistake of putting too much credence in apparent trajectory, which Feynman diagrams do not purport to represent? Or is it intentionally meant to depict the travel of certain exchange particles as being (in some non-literal way) instantaneous?

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The short answer is that Feynman diagrams definitely do not represent the trajectories of particles in spacetime; they are simply a way of writing down formulae that would otherwise be pretty hard to remember. The long answer is below.

Quantum Field Theory, which is the mathematical formalism behind particle physics, dictates that when we scatter two electrons off of each other they have certain probabilities of coming out at different angles. To get these probabilities we have to do a calculation which, unfortunately, is impossible to do analytically. What we usually do when confronted with such a situation is to do a series expansion: break the complicated expression up into an infinite sum of simpler terms and hope that by evaluating only some of them we get a good approximation to the probability for the process at hand. (You can take a look at the Wikipedia articles for anomalous magnetic moment to see just how well this idea worked.)

What Feynman and others did was to realize that each of the terms in the series could be represented with a little picture that we now call a Feynman diagram. The Feynman rules for a particular theory tell us how to convert the diagram into a mathematical expression that is more or less straightforward to evaluate. This is a very handy tool, because it's much easier to just draw lots of diagrams than to remember the complicated formulas off the top of your head.

In a way, when we draw Feynman diagrams we imagine that whatever process we're studying can take place through the exchange of one or more "virtual particles". But we shouldn't attach too much reality to this idea. First off, and this is the meat of your question, they don't represent the trajectories of particles because the particles do not have well defined trajectories. They are quantum objects, and in some sense they take every possible path, and these paths all contribute equally to the probability amplitude. (This is at the heart of the Feynman path integral.) So the answer to your question is that the particles aren't everywhere at the same time, because while the vertical axis sort of represents time, the horizontal axis does not represent space. This is related to the fact that the particles being interchanged are virtual: they can have any mass, move at any speed, and in general break lots of laws of physics. The reason they can do that, of course, is that they don't really exist.

There is another reason Feynman diagrams shouldn't be taken literally, which is this: even if each diagram was indeed a picture of what's really happening, how come there are infintely many of them? When two electrons scatter off of each other the main contribution is from one-photon exchange, but there's also diagrams with lots of photons exchanged at different places, diagrams with particle-antiparticle pairs being created and destroyed, and so on. To get the full probability we need to sum all of them, so we can't really claim that something definite is going on there.

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