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I find that the gauge symmetries of the lagrangian are a topic that gets obfuscated quite a bit. I'm trying to understand the big picture of this in QED. My understanding is that:

  1. Gauge derives its name ultimately from that thing that I call the
    classical gauge transformation of classical electromagnetism: $\mathbf{A} \rightarrow \mathbf{A}' + \nabla f$.
  2. In QED, the spacetime equivalent of this is $A_\mu \rightarrow A_\mu + \frac{1}{g}(\partial_\mu \Lambda)$.
  3. Basic spacetime symmetry requires us to have global symmetry, for which we have $\phi \rightarrow e^{i \theta} \phi$.
  4. Other spacetime symmetries require us to have local symmetry, for which we have $\phi \rightarrow e^{i \theta (x)} \phi$.
  5. You have to have a gauge covariant derivative, thus we have $\partial_\mu \rightarrow \partial_\mu - igA_\mu$.

Now my question is this: are any of the above gauge symmetries & derivatives (that is, 2 through 5) redundant? For example, does rule 5 above actually the same as rule 2?

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The 2) is Lorentz invariant way to write the 1).

The 5) is a way in which one has to modify derivative in order to make the theory gauge invariant. For the simple case of complex scalar field $\phi$ with $U(1)$ gauge symmetry, the lagrangian looks as follows: $$ \mathcal{L} = \partial_\mu \phi^{*} \partial^{\mu} \phi + \ldots $$ Applying an infinitesimal position-dependent transformation $\phi \rightarrow e^{i \theta}\phi$, one gets in the lowest order: $$ \delta \mathcal{L} = i \partial_\mu \theta (\phi^{*} \partial^{\mu} \phi - \phi \partial^{\mu} \phi^{*}) $$ So to cancel this change in the Lagrangian one needs to promote $\partial_\mu$ to $\partial_\mu + i A_\mu$, with the aforementioned transformation properties : $A_\mu \rightarrow A_\mu + \partial_\mu \theta$. So that the simultaneous transformation of both fields would leave the Lagranigian invariant.

About 3) and 4) - one should distinguish between global and gauge symmetry, because they have different meaning. The global symmetry - is a set of transformations, that leaves our theory invariant, and the gauge symmetry, as is often said, is not actually a symmetry, but a redudancy in description of the theory.

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