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    $\begingroup$ Would it be accurate to say that one could rephrase your question as follows: "Under what conditions can one take a Lagrangian leading to an action which possesses a global Lie-algebraic symmetry and find a new Lagrangian depending on a gauge field that leads to an action with local Lie-algebraic symmetry such that the new Lagrangian agrees with the old Lagrangian when the gauge field is set to zero and the symmetry parameter is constant on spacetime?" $\endgroup$
    – joshphysics
    Commented Apr 16, 2013 at 9:02
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    $\begingroup$ Yes, this is exactly what I have in mind! I just wanted to avoid being too formal. But sometimes it is better if one wants to be precise :) $\endgroup$
    – Tomáš Brauner
    Commented Apr 16, 2013 at 9:07
  • $\begingroup$ Ok well in that case, I'm not sure Lubos' answer is very satisfying. I have to think a bit more about this, but I think the answer, if phrased in the manner of my first comment above, is that given any action of a Lie algebra on the fields that is either (i) an internal symmetry or (ii) a spacetime symmetry or both, making the symmetry parameter local, introducing a gauge field, introducing the corresponding covariant derivative, and replacing partial derivatives with covariant derivatives will do the trick. $\endgroup$
    – joshphysics
    Commented Apr 16, 2013 at 9:19
  • $\begingroup$ Replacing ordinary derivatives with covariant ones is indeed what one usually does. But that doesn't work in this trivial example. The covariant derivative for the shift symmetry will be something like $D_\mu\psi=\partial_\mu\psi-A_\mu$, and the term with the time derivative will not be gauge invariant. (There are theories whose action is gauge invariant yet the gauge field does not appear solely in covariant derivatives. An example is the low-energy effective theory for a ferromagnet. Generally this happens when the Lagrangian changes under the gauge transformation by a surface term.) $\endgroup$
    – Tomáš Brauner
    Commented Apr 16, 2013 at 9:38
  • $\begingroup$ I think that one can gauge this symmetry, which is the responsible of the renormalizability of Proca theory physics.stackexchange.com/q/16931 Have you read the opposite? Ref or link? $\endgroup$
    – Diego Mazón
    Commented Apr 17, 2013 at 6:19