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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:
[tex]L = -\frac{1}{2}F \wedge *F[/tex]
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
[tex]\delta L = E \cdot\delta A+ d\Theta[/tex]
where [tex]\Theta[/tex] are the boundary terms and E are the equations of motion for the vector potential A. If we now have a vector field [tex]\xi[/tex] we can construct the Noether current
[tex]
\mathcal{J} \equiv \Theta -\xi\cdot L
[/tex]
(where the dot indicates contraction with the first index of L) such that
[tex]d\mathcal{J} = - E\delta A[/tex]
If the equations of motion hold, then there can be a Noether charge Q such that
[tex] \mathcal{J} = dQ [/tex]
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 [tex]\mathcal{J}[/tex] isn't exact, but it should be closed). But I'm a little stuck with the calculation. A variation of L gives me
[tex] \delta L = -\frac{1}{2} (\delta F \wedge *F + F \wedge \delta *F)[/tex]
which can be worked out, with F=dA, as
[tex]
\delta L = -\frac{1}{2}[d(\delta A \wedge *F) + \delta A \wedge d*F + F \wedge \delta *F ][/tex]
I'm interested in the A-field. I thought that
[tex]
\delta * F = * \delta F + \frac{1}{2}(g^{\alpha\beta}\delta g_{\alpha\beta}) * F
[/tex]
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
[tex]
\Theta = -\frac{1}{2}\delta A \wedge *F
[/tex]
So I would say that my Noether current is given by
[tex]
\mathcal{J} = -\frac{1}{2}\Bigr(\delta A - \xi\cdot F \Bigr)\wedge * F
[/tex]
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?
[tex]L = -\frac{1}{2}F \wedge *F[/tex]
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
[tex]\delta L = E \cdot\delta A+ d\Theta[/tex]
where [tex]\Theta[/tex] are the boundary terms and E are the equations of motion for the vector potential A. If we now have a vector field [tex]\xi[/tex] we can construct the Noether current
[tex]
\mathcal{J} \equiv \Theta -\xi\cdot L
[/tex]
(where the dot indicates contraction with the first index of L) such that
[tex]d\mathcal{J} = - E\delta A[/tex]
If the equations of motion hold, then there can be a Noether charge Q such that
[tex] \mathcal{J} = dQ [/tex]
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 [tex]\mathcal{J}[/tex] isn't exact, but it should be closed). But I'm a little stuck with the calculation. A variation of L gives me
[tex] \delta L = -\frac{1}{2} (\delta F \wedge *F + F \wedge \delta *F)[/tex]
which can be worked out, with F=dA, as
[tex]
\delta L = -\frac{1}{2}[d(\delta A \wedge *F) + \delta A \wedge d*F + F \wedge \delta *F ][/tex]
I'm interested in the A-field. I thought that
[tex]
\delta * F = * \delta F + \frac{1}{2}(g^{\alpha\beta}\delta g_{\alpha\beta}) * F
[/tex]
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
[tex]
\Theta = -\frac{1}{2}\delta A \wedge *F
[/tex]
So I would say that my Noether current is given by
[tex]
\mathcal{J} = -\frac{1}{2}\Bigr(\delta A - \xi\cdot F \Bigr)\wedge * F
[/tex]
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?