Trying to understand “recursion at the lowest level of matter”

Question

I'm reading the book Gödel, Escher, Bach: an Eternal Golden Braid and on Chapter V, Hofstadter talks about different examples of recursive structures and processes. By page 142 of the 20th anniversary edition, he starts talking about recursion at the lowest level of matter, that is, according to him, related to the structure of elementary particles.

To explain his point, he draws a little example I'm unable to understand due to, most likely, my poor knowledge of physics in general. He starts by limiting himself to two kinds of particles: electrons and photons: "Imagine first a dull world where a bare electron wishes to propagate from point A to point B. A physicist would draw a picture like this:"

Based on this, my initial set of questions are:

• He says in such a world there is an easy mathematical expression to represent this line, does anybody know which expression is that?
• And how exactly is an electron simply traveling from point A to B?

Next, in a world where electrons and photons interact, the electron is now capable of emitting and then reabsorbing "virtual photons - photons which flicker in and out of existence before they can be seen.":

Electron and photon interaction:

"Now as our electron propagates, it may emit and reabsorb one photon after another, or it may even nest them, as shown below:"

Also here he says that "the mathematical expressions corresponding to these diagrams - called Feynman diagrams - are easy to write down." So my last set of questions are:

• How do these diagrams relate to Feynman ones?
• I'm not quite sure I actually understand them, could someone help me with a gentle introduction to the topic?
• More importantly to the book reading at hand, where is the recursion in all this? I fail to see it.

For those interested, I found all of this part of chapter V online.

Edit: I accepted the answer given by Gugg below. For completion, follow the interesting discussion we had on chat as well.

Answer 1

Just to start things off a very incomplete answer (and maybe faulty too):

1. These are Feynman diagrams.

2. The "recursion" idea is as follows. The electron can move from A to B in the simplest possible way (the mentioned line). But, for which you would draw a slightly more complicated diagram, it can also have 2 things happening to it (emit and absorb a photon). And maybe 4 things happening to it (emit, emit, absorb, absorb or another sequence of events). And maybe 6 things. Or 8. And in all those scenarios the photons may have things happening to them before they get reabsorbed. (There's the recursion, because new electrons can form there, together with positrons, which would be subject to all these things too.) In fact, and I believe this the be the main point, all of these possibilities (diagrams) happen together, but the main contributions to the electron getting to B are from those diagrams where relatively few things happen. By calculating the contributions of the simpler diagrams to "the big picture" you get a good approximation of "the big picture".

I'll await better and more complete answers (or minus points for this one), before removing this one.