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  • $\begingroup$ Hi Luboš, thanks for the comprehensive answer! Concerning the first six paragraphs: I do know these things :) As to the rest: I'm afraid I had something else in mind. I want to add an auxiliary gauge field, i.e. no new physical degrees of freedom. The point is that I want to construct a generating functional for the conserved currents of the theory. This is just a mathematical trick; the physical symmetry of the theory remains global. E.g. for a massless Lorentz scalar, the shift symmetry can be gauged as $\mathscr L=\frac12(\partial_\mu\phi-A_\mu)^2$, but this doesn't work in my case. $\endgroup$
    – Tomáš Brauner
    Commented Apr 16, 2013 at 9:20
  • $\begingroup$ Thanks, Tomáš! Unfortunately, I can't help you with this because I don't understand your theory that is both gauge-symmetric as well as physically globally symmetric only. It's like the Princess Koloběžka (Scooter?) the First, right? ;-) youtube.com/watch?v=mBC9vr3nuiI What does it mean for a field to be called a "gauge field" if the normally associated with it gauge symmetry doesn't exist at all? $\endgroup$
    – Luboš Motl
    Commented Apr 16, 2013 at 17:28
  • $\begingroup$ Suppose you have an action $S[\phi]$ with a global symmetry. I'm looking for an action $S'[\phi,A_\mu]$ such that $S'[\phi,0]=S[\phi]$. If I manage to make $S'[\phi,A_\mu]$ gauge-invariant, then $e^{-W[A_\mu]}=\int[d\phi]e^{-S'[\phi,A_\mu]}$ will be a generating functional of Green's functions of conserved currents of the original theory. That's why I say that $A_\mu$ is not a dynamical field and the physical symmetry, obtained by setting it to zero, is still global. You see the scooter there? :) $\endgroup$
    – Tomáš Brauner
    Commented Apr 17, 2013 at 7:40
  • $\begingroup$ Dear Tomáši, first, it's easy to add fields so that the new theory will have a gauge symmetry but in a typical case, one doesn't get an interesting theory because the new gauge symmetry just removes some degrees of freedom and it's more useful to erase them immediately with the symmetry, anyway. Second, I don't understand why you would consider extra fields to find out whether the original theory has conserved currents. If $S$ admits conserved currents, it does, otherwise it doesn't. $\endgroup$
    – Luboš Motl
    Commented Apr 18, 2013 at 9:00
  • $\begingroup$ Third, you may always add $J\cdot A$ fields to an action so that you obtain a generating functional for correlation functions of $J$ in the original theory: the theory with the extra term doesn't have to have a gauge symmetry. The field $A$ is auxiliary. Fourth, treating it as auxiliary is less constraning because if it is dynamical, you have to impose the eqn of motion from varying $A$. Fifth, the equation of motion is $J=0$ unless you manage to write new "kinetic" terms for $A$ as well which is what makes the gauge symmetry physical and interesting but it's not guaranteed to exist. $\endgroup$
    – Luboš Motl
    Commented Apr 18, 2013 at 9:02