quantum mechanics - Why do Global phase transformations lead to the conservation of charge?

I am currently writing a report on the basics of quantum Gauge Invariance and there is a concept I am struggling with.

An early part of my discussion in the report concerns charge conservation in QM and I want to briefly explain how it arises from invariance under a Global Phase transformation $e^{i\theta}$. I understand why we have invariance, as $$|\langle\psi\lvert\psi\rangle|^2=|\langle{\psi}^{'}\lvert{\psi}^{'}\rangle|^2$$ where $ \lvert{\psi}^{'}\rangle$=$e^{i\theta}\lvert\psi\rangle$ where $\lvert\psi\rangle$ is the wavefunction for some charged particle, but I don't understand why charge conservation would intuitively arise from this invariance.

I have seen this likened to how the arbitrary nature of the potential scale leads to charge conservation via the argument that if charge were not conserved, neither would energy hence charge must be conserved. But I can't really grasp how some phase change would in any way correspond to shifting our potential arbitrarily.

Answer

The ordinary "phase change" in QM does not lead to charge conservation. This is simply because all states in QM have this sort of arbitrary phase, whether they are charged or uncharged, whether we consider the electromagnetic field or not. It's simply a conseqence of "states" actually being rays in Hilbert space, and not single vectors.

Charge conservation arises from another symmetry: If $Q$ is the electrical charge operator, then states transform under the transformations induced by this operator by $\mathrm{e}^{\mathrm{i}Qt}$, which is a simple phase transformation only for eigenstates of $Q$, i.e. states with definite charge.

You cannot properly explain charge conservation in ordinary QM - there you simply have to accept that there is a charge operator $Q$ that commutes with the Hamiltonian, and is hence conserved in all meaningful senses. If you go to QFT, then the quantum versions of Noether's theorem, the Ward-Takahashi identities, apply to the global version of the $\mathrm{U}(1)$ symmetry and are the correct statement of charge conservation. Note that it is the global symmetry, not the gauge symmetry, that leads to conservation both in the classical and in the quantum case (cf. e.g. this answer by Qmechanic) - a pure gauge symmetry has no true physical content, and cannot lead to conservation laws.