# Lagrangian of electrodynamics

## Main Question or Discussion Point

Hi, I have a computational question which concerns forms. I want to compute the variation of the electrodynamic Lagrangian, seen here as an n-form:

$$L = -\frac{1}{2}F \wedge *F$$

with F=dA. I want to derive the Noether-current from this Lagrangian. The symmetrytransformation we are concerned with are coordinatetransformations induced by Lie-derivatives acting on A. A general variation of L can be composed as

$$\delta L = E \cdot\delta A+ d\Theta$$

where $$\Theta$$ are the boundary terms and E are the equations of motion for the vector potential A. If we now have a vector field $$\xi$$ we can construct the Noether current

$$\mathcal{J} \equiv \Theta -\xi\cdot L$$

(where the dot indicates contraction with the first index of L) such that

$$d\mathcal{J} = - E\delta A$$

If the equations of motion hold, then there can be a Noether charge Q such that

$$\mathcal{J} = dQ$$

I want to verify this for the electrodynamic Lagrangian given above, and I have the suspicion that for this particular Lagrangian we can't construct this Q ( so that the current $$\mathcal{J}$$ isn't exact, but it should be closed). But I'm a little stuck with the calculation. A variation of L gives me

$$\delta L = -\frac{1}{2} (\delta F \wedge *F + F \wedge \delta *F)$$

which can be worked out, with F=dA, as

$$\delta L = -\frac{1}{2}[d(\delta A \wedge *F) + \delta A \wedge d*F + F \wedge \delta *F ]$$

I'm interested in the A-field. I thought that

$$\delta * F = * \delta F + \frac{1}{2}(g^{\alpha\beta}\delta g_{\alpha\beta}) * F$$

and the metric-part is going to give me the energy-momentum tensor of the electromagnetic field, which we can disregard. I recognize in this variation

$$\Theta = -\frac{1}{2}\delta A \wedge *F$$

So I would say that my Noether current is given by

$$\mathcal{J} = -\frac{1}{2}\Bigr(\delta A - \xi\cdot F \Bigr)\wedge * F$$

but if I take the exterior derivative of this, it doesn't give me the form I want; It's not exact if the equations of motion for A hold.

So my questions are :

1)what is the corresponding Noether current for the electrodynamic Lagrangian associated with diffeomorphism-invariance of the action?

2) Is this current exact?

Related Differential Geometry News on Phys.org
thanks.........................

hi

thanks..for all and i wait more...........

Try consulting Eguchi,Gilkey and Hanson:"Gravitation,gauge theories and differential geometry",Physics Reports,Vol.66,6,pp.213-393,December 1980.This is a nice handbook-style article that you may already be familiar with.