- #1
jdstokes
- 523
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[SOLVED] Mandl and Shaw 5.1
To show
[itex]-\frac{1}{2}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}(\partial_\mu A^\mu)^2 \quad \mathrm{and} \quad -\frac{1}{2}\partial_\nu A_\mu \partial^\nu A^\mu[/itex]
represent the same Lagrangian it suffices to show that
[itex]\partial_\nu A_\mu\partial^\mu A^\nu - \partial_\nu A^\nu \partial_\mu A^\mu[/itex] is at most a 4-divergence.
The trouble is, I have no idea why this would be the case. Is this a matter of utilizing the product rule in some clever way?
Edit: Yes it is: factor out [itex]\partial_\nu[/itex].
To show
[itex]-\frac{1}{2}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}(\partial_\mu A^\mu)^2 \quad \mathrm{and} \quad -\frac{1}{2}\partial_\nu A_\mu \partial^\nu A^\mu[/itex]
represent the same Lagrangian it suffices to show that
[itex]\partial_\nu A_\mu\partial^\mu A^\nu - \partial_\nu A^\nu \partial_\mu A^\mu[/itex] is at most a 4-divergence.
The trouble is, I have no idea why this would be the case. Is this a matter of utilizing the product rule in some clever way?
Edit: Yes it is: factor out [itex]\partial_\nu[/itex].
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