Every explanation of the Aharonov–Bohm effect that I have seen seems to justify the phase that shows up due to different paths that the particles (electrons) take to reach some point in space.
How does this make any sense in (standard = Kopenhagen) Quantum Mechanics where there are no trajectories?
In the double slit experiment (standard = Kopenhagen) Quantum Mechanics does not allow any trajectories so why is it with the Aharonov–Bohm effect legit to reason based on trajectories?
The Bohm-Aharonov effect (well, really the double-slit experiment) does not concern one single path, but rather a sum over all paths, each being weighted by a phase (of unit modulus). That is what gives the interference pattern. The presence of a magnetic vector potential with non-zero circulation about a particular region - that region being the solenoid's interior in the case of the AB-effect - changes the respective phases of all those paths, and this shows up in the interference pattern.
