# Intuition for symmetry currents

1. Feb 20, 2013

### electroweak

How should I think about symmetry currents?... in particular, when there are no fields to "carry the charge", eg in a pure Maxwell theory or, maybe, in a CFT of free scalars? Perhaps it would help if someone elucidated the connection between the "charge" in Noether's theorem and the "charge" in representation theory (ie how fields transform).

2. Feb 21, 2013

### tom.stoer

In Maxwell's theory w/o matter fields you cannot derive the matter current using symmetry considerations i.e. Norther's theorem, so you have to put it in by hand. But already then it must be conserved du to anti-symmetry of F.

Of course this is not the way we want to introduce the current. By inspection of the inhomogeneous Maxwell equations you derive (backwards) an enhanced Lagrangian now containing a coupling of the A-field with the current j. From this Lagrangian you CAN derive the current as Noether current.

Note that in many (all?) gauge theories conservation of the current follows from these two procedures: Noether's theorem and consistency condition due to anti-symmetry of F. The latter one says that you cannot couple a non-conserved current w/o making the theory inconsistent.

I think this always works regardless whether j is a single entity j or whether it is a more complicated expression based on more fundamental matter fields. Rather nice example is pure gluonic QCD w/o quarks; due to the non-abelian gauge group and the non-linearity of F you find a purely gluonic conserved current. Again both procedures work: the conservation follows directly from the anti-symmetry of F w/o using Noether's theorem; and of course you can apply Noether's theorem directly to derive the current.

3. Feb 21, 2013

### vanhees71

Sure, you can derive Noether's theorem for global symmetries by "gauging" the symmetry with external (i.e., background) gauge fields as external sources in the effective action. The functional derivative of this action with respect to these source fields gives you the expectation value of the conserved current (setting the auxiliary fields to 0 after taking the derivative). Using local gauge invariance of the theory with external gauge fields you can derive in an elegant way also the Ward-Takahashi identities for the effective action and thus for the proper vertex functions.

4. Mar 21, 2013

### Tarti

I hope I understand your question correctly that you are trying to find an intuition for Noether's theorem. As you state, I would always fall back to something that you really know, e.g. charge conservation in electrodynamics and how you can derive it from gauge invariance, not directly from the continuity equation.
What you basically have to do is to look what has to be "fixed" if a gauge transformation is applied to the action - and voila, you find charge conservation from this very first principle.