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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?

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Qmechanic
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Xhorxho
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    The Wikipedia article only states that paths are not allowed to self-intersect under a diagram where one of the coordinates is time! – dan-ros Dec 02 '23 at 11:09

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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.

ACuriousMind
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  • I have reread your last paragraph several times. I am not sure I understand what you mean. Do you mean that paths with kinks would be misleading in some sense? Could you clarify your thoughts, please? – GiorgioP-DoomsdayClockIsAt-90 Dec 02 '23 at 14:44
  • @GiorgioP-DoomsdayClockIsAt-90 Have you read the answer I linked about how $\dot{x}$ doesn't actually really mean anything in this context? If we drew the paths with kinks, we'd provoke questions like "But what does $\dot{x}$ mean for this path? It's not differentiable at the kink!", and to answer those we'd have to do the proper mathematical construction of the Wiener measure instead of the usual physics derivation of the path integral - which the texts presumably want to avoid, otherwise they'd just do it in the first place. – ACuriousMind Dec 02 '23 at 14:57
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    In fact, under the Wiener measure, the "typical" path has an infinite number of kinks. With probability one, every segment of the path contains kinks—kinks everywhere! – Buzz Dec 02 '23 at 16:47