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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.

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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.

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  • $\begingroup$ in the second paragraph: what do you mean by the statement that an observer is only capable of transforming the field locally? Why is this a direct consequence of (of course resonable) assumptions that locality principle & Lorentz invariance should be satisfied by the theory? $\endgroup$
    – user267839
    Commented Jul 2, 2023 at 23:07
  • $\begingroup$ Does it mean that if we require locality & Lorentz invariance, then for a (matter) field with dynamics described by a Lagrangian having a global symmetry the concept of observables would be not well defined? Or do I misunderstand this point? $\endgroup$
    – user267839
    Commented Jul 2, 2023 at 23:24
  • $\begingroup$ @user267839 It's perfectly possible to have a local and Lorentz invariant theory with a global symmetry and well-defined observables. As I said, the historical motivation of Yang and Mills to consider local symmetries is no longer considered to be correct, even though the resulting Yang-Mills theory is correct. There's a lot of examples of wrong motivations leading to correct results in the history of QFT, like the Dirac's sea of negative energy fermions. $\endgroup$
    – Andrew
    Commented Jul 3, 2023 at 13:48
  • $\begingroup$ yes, but I would like from didactical point of view firstly take a glace at the original motivation, before proceeding to the fact that it finally turned out to be wrong. Do you know the original source where the motivation is discussed? My concern is really just about the correct phrasing of the original motivation (which as you said turned out to be wrong). Could you check if I phrase the orginal motivation correctly? $\endgroup$
    – user267839
    Commented Jul 3, 2023 at 14:18
  • $\begingroup$ So far I understand it, Yang and Mills originally seeked for a theory beeing local and Lorentz invariant. And they conjectured that such theory cannot have global symmetry if it should be compatible with phyical observations, that's what you mean in your answer? My concern is just that I not understand your phrase that " an observer is only capable of transforming the fields locally" as statement. Could you formalize it? Does the statement as such says in physical terms that " any observable couldn't be invariant with respect to glocal symmetries, only the local ones"? $\endgroup$
    – user267839
    Commented Jul 3, 2023 at 14:38

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