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There are infinitely many. But the cool math that Feynman worked out is that infinite diagrams (paths) kinda look random, unless they’re not random (aligned).
When you get enough infinite “randombess” you tend to cancel almost everything out (not actually random but phases that destructively interfere), so what you’re left with (when you “sum all paths”) are solutions that favour a particular direction or phase.
I absolutely love the concept of chaotic systems that mathematically stabilize towards order.
As I understand it, any path in a Feynman diagram can be replaced by its own separate diagram, and as long as there is an average of more than one diagram each of those paths could be replaced by, you're nesting greater than one to infinite depth. Or equivalently, raising greater than one possibilities to an infinite power. That's infinite.
Now, you might ask "How much of that could possibly happen in finite time?", but time is irrelevant here. As far as we know, all possible paths are taken.
"Copy-Paste Copy-Paste Copy-Paste Copy-Paste...!"
Just like the π-Bunny, it just keeps going and going and going...
Silly stuff aside, thank you for helping me see it in a new way, I had not considered nesting greater than one to infinite depth, deep in its' way as The Dirac Sea.