From what I understand, a global symmetry is a symmetry that holds at every point in space-time. On the other hand, a gauge symmetry is a redundancy in the description of a physical state.
Now, when localizing a global symmetry, we usually impose $\partial_\mu \rightarrow D_\mu = \partial_\mu + A_\mu$. For example, for scalar field theory we might have
$$\mathcal{L} = |\partial_\mu \phi|^2 \rightarrow |D_\mu \phi|^2$$
Now, this Lagrangian still enjoys the global symmetry $\phi \rightarrow e^{i\theta}\phi$ for constant $\theta$ after localizing the global symmetry as the global symmetry is a subset of the local symmetry. However, in performing this localization my understanding is that you end up gauging the symmetry.
- How can I see that the global symmetry was originally not a gauge symmetry, but after localization is a gauge symmetry? Naively I expected $\phi \rightarrow e^{i\theta} \phi$ to correspond to physically equivalent states similar to how quantum states are only defined up to an overall phase factor.
- Are there general principles in which one can easily see that some symmetry is in fact a gauge symmetry?
