hilbert space - What happens to the global $U(1)$ symmetry in alternative formulations of Quantum Mechanics?

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?