- #1
- 3,081
- 1,600
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
<br /> \mathcal{J} \equiv \Theta -\xi\cdot L<br />
(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
<br /> \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
<br /> \delta * F = * \delta F + \frac{1}{2}(g^{\alpha\beta}\delta g_{\alpha\beta}) * F<br />
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
<br /> \Theta = -\frac{1}{2}\delta A \wedge *F<br />
So I would say that my Noether current is given by
<br /> \mathcal{J} = -\frac{1}{2}\Bigr(\delta A - \xi\cdot F \Bigr)\wedge * F<br />
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?
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
<br /> \mathcal{J} \equiv \Theta -\xi\cdot L<br />
(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
<br /> \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
<br /> \delta * F = * \delta F + \frac{1}{2}(g^{\alpha\beta}\delta g_{\alpha\beta}) * F<br />
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
<br /> \Theta = -\frac{1}{2}\delta A \wedge *F<br />
So I would say that my Noether current is given by
<br /> \mathcal{J} = -\frac{1}{2}\Bigr(\delta A - \xi\cdot F \Bigr)\wedge * F<br />
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?