Without electromagnetic coupling, the QM charged particle wave function is not invariant under a local gauge transformation — one with a phase that depends on space (space-time): \begin{equation} \psi \mapsto e^{-i\alpha(x)}\phi \end{equation} In order to turn on the electromagnetic coupling, we make the replacement $$\hat{\vec{p}} \mapsto \hat{\vec{p}} - \frac{ie}{c}\vec{A}$$ This means that, in the Schrodinger equation is gauge invariant because the term arising from $\vec{A}$ is cancelled through the action of $\hat{\vec{p}} = i\hbar\partial$ on the phase factor $e^{-i\alpha(x)}$ in the wave function. The math trick is clear. But physics cannot be understood from it, namely, why the requirement of local gauge invariance requires 1) the charge conservation 2) and the emergence of some kind of carrier of interaction.
I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses and sheer volume of education into the physical tools and mindset that the other students who took the course had, like Noether's theorem, Lagrangian and Hamiltonian mechanics, statistical methods, and so on.  The courses themselves went well enough. My mathematical experience more or less made up for a lacking physical understanding. However, I still haven't found an elementary explanation of gauge invariance (if there is such a thing). I am aware of some examples, like how the magnetic potential is unique only up to a (time-)constant gradient. I also came across it in linearised general relativity, where there are several different perturbations to the spacetime metric that give the same observable dynamics.  However, to really understand what's going on, I like to have simpler examples. Unfortunately, I haven't been able to find any. I guess, since "gauge invariance" is such a frightening phrase, no one use that word when writing to a high school student.  So, my (very simple) question is: In many high school physics calculations, you measure or calculate time, distance, potential energy, temperature, and other quantities. These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above? Or are these two different concepts?
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