Gauge transformations and conserved quantities in classical and quantum mechanics

TL;DR

This paper explores unconventional effects of gauge transformations in classical and quantum mechanics, revealing scenarios where conservation laws, physical observables, and boundary conditions are affected, challenging traditional views.

Contribution

It presents a detailed analysis of gauge transformations leading to non-standard conservation laws and observable quantities, supported by worked examples and historical context.

Findings

Gauge transformations can alter conservation laws.

Non-gauge-invariant quantities may be physically observable.

Wave functions can change boundary conditions under gauge transformations.

Abstract

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.