Zusammenfassung
We are taught that gauge transformations in classical and quantum mechanics
do not change the physics of the problem. Nevertheless here we discuss three
broad scenarios where under gauge transformations: (i) conservation laws are
not preserved in the usual manner; (ii) non-gauge-invariant quantities can be
associated with physical observables; and (iii) there are changes in the
physical boundary conditions of the wave function that render it
non-single-valued. We give worked examples that illustrate these points, in
contrast to general opinions from classic texts. We also give a historical
perspective on the development of Abelian gauge theory in relation to our
particular points. Our aim is to provide a discussion of these issues at the
graduate level.
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