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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.

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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.

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  • $\begingroup$ I think I was a little confused because one of the books I am currently using makes no reference to a charge operator and in it's discussion of global and local phase transformations simply references a global transformation $e^{-ie\alpha}$ where e is the charge of an electron and $\alpha$ is some scalar. It is an introductory particle Physics textbook so I suppose it makes sense for it to skim over this and not mention an operator. When I asked a lecturer about it they claimed charge conservation results from the arbitrary nature of the potential scale, referencing an argument by Wigner. $\endgroup$ – R.McGuigan Dec 14 '17 at 16:59
  • $\begingroup$ I'm wondering, would such an argument be accurate? I think the meatier parts of Gauge theories are beyond me for the time being so I wonder whether these arguments about charge conservation being a consequence of a potential shift would be adequate, along with discussion of a charge operator and the associated transformation. One more thing, my lecturer likens the phase shift to an arbitrary potential shift, is this also adequate? $\endgroup$ – R.McGuigan Dec 14 '17 at 17:03
  • $\begingroup$ @R.McGuigan Classically, "the arbitrary nature of the potential scale" is just the exact same as the gauge symmetry, and this has nothing to do with quantum mechanics or phases. Applying Noether's theorem to the global part of the symmetry yields charge conversation. $\endgroup$ – ACuriousMind Dec 14 '17 at 17:03
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    $\begingroup$ @ ACuriousMind Hmm, I suppose I am a little confused then, the quote from my lecturer was "Changing the electrostatic potential by a constant amount is a global transformation as it changes the potential everywhere and invariance under the global transformation is related to a conservation law, i.e. charge. ". Is my lecturer inaccurate? And could a transformation $e^{iQt}$ not be treated as some sort of phase transformation?References to QTF and Noether's theorem in QM are probably beyond my capacity to discuss currently, though I can appreciate that they would provide much better arguments. $\endgroup$ – R.McGuigan Dec 14 '17 at 17:21
  • $\begingroup$ @R.McGuigan Changing the potential by a constant is precisely the global part of the symmetry. The gauge symmetry is that you can add arbitrary gradients to the four-potential. $\endgroup$ – ACuriousMind Dec 14 '17 at 18:02

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