# Equivalence betwen diferent definitions of charge

1. Jun 7, 2007

### Sauron

I am trying to fit together a few definitions of charge which are being commonly used.

On one side we have the Noether charge associated with any invariance. Sure most of you know how it goes, you have a Lagrangian invariant under a (Lie) symmetry group and for every group generator (Lie algebra element) you have a conserved current. The integral of that conserved current is a charge.

On other side you have, for the same theory, that for every element of the Cartan subalgebra (maximum number of commuting elements) of the Lie algebra you have an associated charge. The noncommuting elements of the algebra act as ladder operators that raise or lower the charge associated with a particular state (actually a root state).

How are related these two notions of charge? (if related at all). I mean, the cartan subalgebra obviously has less elements that the whole algebra so that there are less "cartan" charges than Noether charges, what point am I missing?

A third related question is the notion of electric and magnetic charges. If I begin with an U(1) theory, like classical electromagnetism I have from Noether theorem only one possible charge, (and in these case also one possible "cartan" charge ). But even though it is stated that particles can have electric and magnetic charge (or both in dyonic objects). I guess that the reason is that magnetic charges come from the hodge dual, that is we have really two potential vectors, the standard one of electromagnetism and it´s hodge dual, that is, we have two gauge fields each with it´s own U(1) Noether charge, and so we can have particles charged respect to one or other (of both). That´s what I think, but I would like that someone would confirmate me it.