Promoting of global symmetry to local one: what's the motivation?

Question

Consider a typical free Lagrange function $\mathcal{L}$ from quantum theory with global $U(1)$-symmetry ( e.g. $ \mathcal{L}= (\partial_\mu \psi)(\partial^\mu \psi^*) + k^2|\psi|^2 $ ) with respect to the action

$\psi \rightarrow e^{-i\theta}\psi$

on matter field $\psi$ for constant $\theta$. Then we have a well known cooking recipe how to "promote" this global symmetry to a local one:

$\psi \rightarrow e^{-i\theta(x)}\psi $

ie $\theta(x)$ is space dependent. The usual differential $\partial$ is going to be replaced by the covariant one $D:=\partial_\mu - iqA_\mu$ with respect the gauge bosons $ A_\mu$ ( morally the element from Lie algebra of $U(1)$) to make the resulting Lagrangian invariant wrt now the local symmetry.

Naive question: What is this procedure good for, ie what is the advantage to turn global symmetry to a local one? Is there a deeper reason why local symmetry is more interesting for further consideration of the system?

Note that the question is not about how to perform this passing from global to local but really why it is important, so essentially about the raison d'etre.

Answer 1

I would argue the terminology "promoting a global symmetry to a local one" reflects the historical motivation, but not our modern understanding.

Historically (as far as I understand), Yang and Mills argued that because of locality and Lorentz invariance, no symmetry should truly be global, since an observer is only capable of transforming the fields locally. Following the chain of logic of a local symmetry leads you to introduce gauge fields associated with the local symmetry, that serve as connections in the gauge covariant derivative.

While Yang and Mills used this motivation that led them to make a fruitful discovery, in modern terms that is not how we think of things.

At least the way I think of it, the core physical principle is that we want to describe interacting spin-1 bosons. In order to make the interactions local, we make use of a gauge theory. The "making a global symmetry local" amounts to a formal trick that we know gives the right interactions for a gauge theory.

You can go a little deeper using the iterative Noether procedure. We start with a free theory of spin-1 bosons, then try to add consistent interactions between the bosons and with charged matter fields. I won't go through the argument in detail here, but it turns out that to have consistent couplings the gauge fields must couple to a conserved current. To zeroth order in the gauge fields, the conserved current is guaranteed by the presence of a global symmetry and Noether's theorem. An iterative procedure can then be set up to calculate the consistent interactions, which amounts to determining how the conserved current should be modified to include dependence on the gauge field. This leads to Yang Mills theories.