I have read these posts in trying to answer my question: $k$-interval for First Brillouin Zone, Band gaps: are they at the centre or at the edge of the Brillouin zone?. I have also tried to find answers in Ashcroft and Mermin, Ibach and Luth, and Kittle to no avail.
In solid state physics, electron bands are usually introduced by deriving the nearly-free-electron dispersion, using Bloch's theorem. From there the professor folds the dispersion into the First Brillouin Zone (FBZ), and says 'voila: bands and gaps'! I understand from a math perspective that for Bloch wave-functions the k vector only adds a phase, and that it has no impact on physical observables. I am having a hard time understanding this physically. Once could say it is easy to unfold the bands back into the original picture in this case, but what about more complicated bands where the originating BZ is not so obvious?
My question is: why does folding the energy-momentum dispersion into the FBZ not destroy information about the absolute momentum of states? If it does destroy information, but that fact does not matter, why does it not matter?
A state with well defined momentum outside of the FBZ should be measure-able as such, correct? Then why does the FBZ band representation not limit the band diagram?