The global $U(1)$ symmetry in Quantum Mechanics corresponds to the freedom to shift the phase of the wave function $$ \Psi \to e^{i\varphi} \Psi \, $$ and can be used to understand the conservation of electric charge.
Where, if at all, does this symmetry show up in alternative formulations of Quantum Mechanics, such as
- the path integral formulation,
- the phase space formulation,
- the pilot wave formulation?
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