What is the physical constraint that gauge invariance is a required condition for electromagnetic fields? What would happen if the electromagnetic fields were not gauge invariant?
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By disposing of the $U(1)$ gauge symmetry, we also dispose of the global version of the $U(1)$ symmetry which gives rise to both a conserved current and conserved charge due to Noether's theorem.
Specifically, every continuous global symmetry, by Noether's theorem, gives rise to a conserved current $j^{\mu}$ which satisfies the continuity equation,
$$\partial_\mu j^{\mu} = \frac{\partial j^{0}}{\partial t} + \nabla \cdot \vec{j} = 0.$$
The conserved Noether charge is defined as,
$$Q = \int_V d^3 x \, \, j^{0},$$
and by demanding $\partial_t Q = 0$, we see this implies,
$$\frac{\partial Q}{\partial t} = -\int_V d^3x \, \, \nabla \cdot \vec{j} = -\int_{\partial V} \vec{j} \cdot ndS = 0,$$
or in other words the flux of $\vec{j}$ is zero, and hence $Q$ is conserved locally. By losing $U(1)$ gauge symmetry, we lose a redundancy in the description of the system, but also a conservation law.
JamalS
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