In Wikipedia is stated that the quantum-mechanical paths are not allowed to selfintersect (create loops). (It’s stated in the description of the first figure of five paths.) On the other hand, loops are considered in the Aharanov-Bohm effect. I assume that loops are actually allowed, right? All the textbooks that I‘ve read „are afraid“ to also draw the unconventional paths an electron can make through the double slit. Let's clear this up once and for all: Are the blue and green paths allowed?
Yes, loops are perfectly fine. But we never draw a "typical" path from the path integral anyway.
Formally, the path integral is over the Wiener space of continuous paths (see also this answer of mine for the construction) between two points. There is no further restriction, they just need to be continuous. The differentiable paths have in fact measure zero, so the "typical" path you draw shouldn't be smooth, but have at least one kink.
At the physical level of rigor we don't really want to draw those because it would draw attention to the fact that the typical physics derivation of the path integral is merely heuristic and not a proof without improving anything else about our understanding of the path integral.
