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Apr 18, 2013 at 9:26 comment added Tomáš Brauner Thanks for the comments, Luboš! Here is an answer to some of your points. (2) One can derive Noether currents by variation w.r.t. the gauge field, although that's not my main reason to gauge the symmetry, see next. (3) You can do this, but it's much less restrictive than gauge invariance and tells you nothing about how the external fields such as $A$ appear in a low-energy EFT, see e.g. the paper hep-ph/9311264 by Leutwyler. One can also add other external fields; the reason why I couple "gauge" fields to conserved currents is that I'm interested in low-energy EFTs for Goldstone bosons.
Apr 18, 2013 at 9:02 comment added Luboš Motl 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.
Apr 18, 2013 at 9:00 comment added Luboš Motl 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.
Apr 17, 2013 at 7:40 comment added Tomáš Brauner 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? :)
Apr 16, 2013 at 17:28 comment added Luboš Motl 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?
Apr 16, 2013 at 9:20 comment added Tomáš Brauner 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.
Apr 16, 2013 at 8:39 history answered Luboš Motl CC BY-SA 3.0