# How to charge a field?

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 ?

• I'm worried that this question is based on some misconceptions about physics. For one thing, the Lagrangian you wrote down in that other question gives dynamics to a field whose excitations are charged particles. – user1504 Mar 28 '13 at 14:33
• @user1504 Ok, sorry, I think my question was not explicit enough. Perhaps more explicit, I know I can make a gauge theory with (global or local) symmetry $\Psi \rightarrow \Psi e^{\mathbf{i}q \phi}$. I want to know what constrains me to choose $q=e, e/2, 2e$ or even $q=1$ in which case there is no charge at all. $e$ is of course the electron charge. My question is even more subtle: what imposes the $q$ prior to any experiment ? – FraSchelle Mar 28 '13 at 16:04
• @Oaoa Nothing. In a pure U(1) theory charge is not quantized. This is (one) motivation for a grand unified theory (GUT) which embeds the U(1) of electromagnetism into a nonabelian group. I'm confused why you say $q=1$ corresponds to no charge at all? It corresponds to charge of 1. The transformation law (equivalently the charge) is part of the definition of the field $\Psi$. – Michael Brown Mar 28 '13 at 16:09
• @Oaoa You may find that you have an easier time communicating if you don't invent non-standard notations. If you take the variable $q$ and set it equal to the number $1$, you will get a model which describes particles whose charge is $1$. – user1504 Mar 28 '13 at 19:57
• The units of $\hat{n}$ and $\phi$ have to cancel out. You can put in a more explicit set of units if you want. $e^{i\hat{n} Q \phi}$, with $Q$ some reference charge setting your units. No amount of quibbling about units will change the fact that the argument of the exponential must vanish for charge zero particles. You have to set $\hat{n}$ to $0$ for charge $0$. – user1504 Mar 28 '13 at 20:09