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I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and I'm kind of stuck in calculating derivatives of the Langrangian density with respect to the derivatives of the field. For instance, the following Lagrangian (Maxwell in a vacuum):

$$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 = -\frac{1}{2}(\partial_\mu A_\nu)(\eta^{\mu \alpha} \eta^{\nu \beta} \partial_\alpha A_\beta) + \frac{1}{2}(\eta^{\mu \nu}\partial_\mu A_\nu)^2 $$

I'm supposed to calculate [itex]\partial \mathcal{L}/\partial(\partial_\mu A^\nu)[/itex]. Now, I have two ways to think about this, one of them is to think of each term as a homogeneous function of degree 2 of the variable [itex]\partial_\mu A_\nu[/itex], and using Euler's theorem on homogeneous functions, multiply by 2 and take the derivatives of each term with respect to [itex]\partial_\mu A_\nu[/itex] as if it wasn't being summed over, giving:

$$ \frac{\partial \mathcal{L}}{\partial(\partial_\mu A^\nu)} = -(\eta^{\mu \alpha} \eta^{\nu \beta} \partial_\alpha A_\beta) + (\eta^{\alpha \beta}\partial_\alpha A_\beta)(\eta^{\mu \nu}) = -\partial^\mu A^\nu + \eta^{\mu \nu} \partial_\alpha A^\alpha $$

This is a bit shady in my head, but apparently it works. The other way I think about it is significantly longer in terms of algebra, but basically involves actually differentiating with respect to a different set of indices, let's say [itex]\partial \mathcal{L}/\partial(\partial_\alpha A_\beta)[/itex], use the product rule and the fact that [itex]\partial(\partial_\mu A_\nu)/\partial(\partial_\alpha A_\beta) = \delta^\alpha_\mu \delta^\beta_\nu[/itex], which gives the same result, but seems unnecessarily long.

So, how do you think about it? What's the quickest but still somewhat rigorously pleasing way to think about it that gives the right answer?

Cheers

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# Calculating derivatives of a Lagrangian density

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