In the usual definition, gauge theory is something that has redundancy built in. Various solutions are just different manifestations of different gauge choices. Well of course physical observables must be gauge invariant.
I wonder if QM itself is a gauge theory since it uses rays in Hilbert space. Each ray contains states that only differ by a (constant?) phase and all these states describe the same physical system. Thus the phase is a gauge choice.
If you look at the continuity equation for probability, you'll see that any (constant) phase cancels out, thus this is the EOM for physical degrees of freedom while the schrodinger equation is the equation for the gauge dependent entity (by explicitly specifying a gauge, or a phase in this case).
As in E&M it is usually easier to calculate using a particular gauge, and I think that's the reason why it's easier to solve Schrodinger equation rather than the continuity equation.
Promoting the phases to depend on spacetime will create multitudes of problems that might be the topic of future posts.
Well, that's all the 2 cents I have for now
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