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In QFT we're interested in the symmetries of our theory (encoded in the invariance of the Lagrangian under symmetries) because they let us study conserved currents of the theory by Noether's theorem.

For example, the case of a complex scalar field which is invariant under $ \operatorname{U}(1) $ transformations. This symmetry leads to conservation of electric charge which implies that the field has said charge.

Noether's first theorem, however, says that for any continous global symmetry there is a conserved charge.

And while the $ \operatorname{U}(1) $ symmetry of the scalar field is global, the $ \operatorname{U}(1) $ symmetry of QED which gives rise to the charge of fermions is a gauge symmetry, which is required to be local.

How is it possible for the QED symmetry (and for any gauge symmetry actually) to give conserved currents if they're local?

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There's no conserved charge associated with the local symmetry. The reason for that is that gauge "symmetry" is not a physical symmetry. The current for the EM Lagrangian is $J^\mu=\partial_\nu F^{\nu \mu}$ which is 0 under the EOM. Gauge invariance provides us with different descriptions of the same system as opposed to different systems with symmetric dynamics.

The charge that we associate with a physical quantity is the global $U(1)$ charge.

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  • $\begingroup$ This isn't entirely true. The charge $\int_{S^2} F$ for electromagnetism or the so-called Komar mass in general relativity are conserved charges associated to gauge symmetries. These are however associated to lower degree conserved currents (e.g. $2$-forms in four dimensional spacetime instead of 3-forms), and they are only well-defined in "favourable" asymptotic situations. $\endgroup$
    – Bence Racskó
    Commented Apr 11 at 7:12
  • $\begingroup$ @BenceRacskó The reason the charge you define has a physical meaning is that the system possesses an asymptotic translational symmetry as opposed to a gauge symmetry which is not a real symmetry. $\endgroup$
    – Lucky Charms
    Commented Apr 11 at 13:59
  • $\begingroup$ @LuckyCharms thank you for your answer. I still don't understand something, if gauge symmetry is more of a redundance than a real symmetry, how is it possible that it gives rise to fields with significant physical meaning, like the EM field in QED ? $\endgroup$
    – Tomás
    Commented Apr 12 at 5:01
  • $\begingroup$ @Tomás In field theory we "embed" particles in fields, and in the case of the photon, it turns out that when we use a four-vector field we are giving it some extra unphysical degrees of freedom (a spin-0 mode and a longitudinal polarization). If we want to get any physics out of the theory we have to isolate the physical degrees of freedom and propagate them properly. Gauge invariance doesn't give you any new physics, however, it can be useful when trying to isolate the physical degrees of freedom. $\endgroup$
    – Lucky Charms
    Commented Apr 12 at 19:59

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