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EDIT: so, the real question is, why did we need LPS invariance to be a feature of QFT in a first place??

Was that a mere reflection of the fact that properties of matter particles are not defined (i.e., superposition) at all points, so that we can't treat them in only 1 of possible ways they interact, but instead in sort of all (or any, LPS-wise) possible ways
$-$ is that the physical meaning of gauge bosons, what they are and what they do??


I've just learned from QFT, that EM field is a gauge field in a way that it is "compensating" (as they say) any local phase shifts (LPS) in wavefunctions of electrically charged particles - meaning, as I understand, that locally messed up complex sine wave restores all its smoothness.

So I've searched for example of LPS and got familiar with the Aharonov-Bohm effect on Wikipedia, where it says that:

This phase shift has been observed experimentally.

But... if EM field "compensates" any LPS, then how can any LPS could be "found experimentally" - ?

I mean, it is not a "compensation" of a shift - if that shift can still be found, right?

Or, does that mean that wavefunction of a particle really stays messed up (and, therefore, LPS could be observed), but if one looks at another wavefunction, the one that particle may somehow share with... EM field? Or its chunks (photons, I guess)
$-$ then one sees sine wave "restored all its smoothness".

..I mean, such combination of an electron field and EM field is quite unexpected to me; but may be that is why bosons in SM are called "gauge" after all.

Idk: I'm soo curios with this, I just can't keep going down that rabbit hole without clarification))

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