In a previous post [ Noether theorem, gauge symmetry and conservation of charge ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied to a global $U(1)$ gauge symmetry, or via the covariant (or minimal, or Weyl, or ...) substitution with a $U(1)$ local gauge symmetry and antisymmetry of the electro-magnetic field applied to the equations of motion. I nevertheless still get confused about something: what is a charge ? More precisely, how to charge a field ?
Indeed, if one believes that only the global gauge construction is able to demonstrate the conservation of the charge, one has to admit that the charge is not defined that way ! The same demonstration applies for the conservation of the particle number. Indeed, I used in [ Noether theorem, gauge symmetry and conservation of charge ] a Lagrangien for uncharged particles to show the conservation of a particular current through Noether theorem.
So the question becomes: how to charge this particle ? In particular: how to charge this particle and still conserve the global gauge ?
NB: In the local gauge construction, it seems to be easier to give a charge to the system, but then only the second Noether theorem applies, and people get annoyed by that.
To conclude, a (possibly important) remark about my point of view: I would like to understand the problem of "charging the field" in a condensed matter perspective, when I believe we have no a priori idea of the charge of the quasi-particles appearing in our effective theories. Can we overcome this problem ? Of course, any point of view (especially the opinion of QFT physicists who might have already tackled this problem) is warmly welcome :-) And the question "how to charge a field ?" is just a warm-up for this more complicated one "how to define the charge of an effective, emergent field ?" which is entirely subsidiary question for the moment.
IMPORTANT EDIT I just became aware of this question [ How does non-Abelian gauge symmetry imply the quantization of the corresponding charges? ] which is strongly related, and well answered. I nevertheless think one can continue to discuss here about the condensed matter problem. Indeed, there are some people who believe that a superconductor (among other emergent phases of matter) have charge neutral excitations [see e.g. http://dx.doi.org/10.1103/PhysRevB.41.11693 which is too old to be on arXiv, but this one: http://arxiv.org/abs/cond-mat/0404327 rephrases it in a different way.]. So the question follows: how can these excitations at low temperatures become again charged at higher temperature ? How can we measure charge current from an uncharged excitation ? How can we "charge" a field after all ?