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    $\begingroup$ Strictly speaking, Noether's theorem applies only to global symmetries, and what you wrote down there is not a global symmetry since $\chi$ depends on spacetime. I don't understand your last question - the Yang-Mills Lagrangian is manifestly Lorentz invariant, and the quantity associated with the Lorentz symmetry is, as always, the stress-energy tensor. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 14, 2015 at 17:11
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    $\begingroup$ Thanks for your comment. There is something I don't understand : so if here $A_\mu\rightarrow A_\mu +\partial_\mu\chi$ is not a global symmetry, then why the transformation $x^i\rightarrow x^i+\delta x^i\,,\,\phi_\mu\rightarrow \phi_\mu +\delta\phi_\mu $ should be called global symmetry since it's coordinate dependant too? Has the difference between local and global symmetry something to do with the fact that conservation laws can be derived on-shell or off-shell? $\endgroup$
    – dolun
    Commented Jan 14, 2015 at 19:04
  • $\begingroup$ For the last question, I was just feeling that Lorentz invariance would "mix" with gauge invariance to ensure charge conservation, but I was surely mistaking. $\endgroup$
    – dolun
    Commented Jan 14, 2015 at 19:05