Gauge transformations with varying phase give us conservation of the charge density. Hence charged particles cannot move?

Question

I stumbled upon the following paragraph in Quark confinement and Topology of gauge theories by Polyakov

"Gauge invariance with constant phase $\Psi \to e^{i \alpha}$ lead to conservation of the total charge. Gauge transformations with a varying phase $$\Psi \to e^{i \alpha(x)}$$ will give us conservation of the charge density. But this in term means that the charged particle cannot move. The only thing which saves the electron from this fatal immobility is the degeneracy of the vacuum in QED, that is, its non invariant under gauge transformations."

Are these statements correct? For example, I've never heard before that charge density is conserved due to local gauge invariance. Or that the QED vacuum isn't invariant under gauge transformations.

(The paper has almost 1500 citations, so I suppose his statements are correct. But I have never seen them anywhere else or any concrete calculations which backs them up.)

Answer 1

As written, the claim is wrong.

Noether's theorem applied to gauge symmetries is more properly Noether's second theorem, and results in off-shell identities as opposed to the on-shell conservation laws of Noether's first theorem for global symmetries. That these identities are off-shell is another manifestation of gauge symmetries being a symptom of redundancy in our description of the physical system - off-shell identities are nothing else than dependencies between our chosen variables that have nothing to do with the dynamics of the system, and in principle one could use these identities to reduce the total number of variables, i.e. eliminate the redundancy.

As Qmechanic elucidates in detail in this excellent answer, the "second Noether current" vanishes off-shell and its charge is identically zero under reasonable assumptions, and for electrodynamics it is the trivial statement that $\partial_\mu \partial_\nu F^{\mu\nu} = 0$.

As for the claim that the QED vacuum is non-invariant under gauge transformations, that is of course also wrong. All physical states are invariant under gauge transformations by definition of a physical state, and vacuum should probably be a physical state. Even when the "gauge symmetry is spontaneously broken" (which is a phrase you definitely still hear even today), what is really being broken is its global part (it's local part cannot be broken, this is Elitzur's theorem). See also this excellent answer by Dominic Else.