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We are taught that gauge transformations in classical and quantum mechanicsdo not change the physics of the problem. Nevertheless here we discuss threebroad scenarios where under gauge transformations: (i) conservation laws arenot preserved in the usual manner; (ii) non-gauge-invariant quantities can beassociated with physical observables; and (iii) there are changes in thephysical boundary conditions of the wave function that render itnon-single-valued. We give worked examples that illustrate these points, incontrast to general opinions from classic texts. We also give a historicalperspective on the development of Abelian gauge theory in relation to ourparticular points. Our aim is to provide a discussion of these issues at thegraduate level.

Gauge transformations and conserved quantities in classical and quantum mechanics