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Consider the Lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu \nu}$ with $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} $.

After deriving the Euler-Lagrange equations for this Lagrangian density, I am asked to verify that the action remains invariant under constant shifts $x^{\mu} \to x^{\mu} + \epsilon^{\mu}$ and derive the corresponding Noether current. But I have trouble working out the result of this spacial translation.

Looking at my lecture notes I find that a four vector transforms as $A'^{\mu}(x') = \frac{\partial x'^{\mu}}{\partial x^{\lambda}}A^{\lambda}(x)$. But what I wrote down at first sight was that for small translations $A'^{\mu} = A^{\mu}(x^{\sigma} + \epsilon^{\sigma}) = A^{\mu}(x^{\sigma}) + \frac{\partial A^{\mu}(x^{\sigma})}{\partial x^{\sigma}}\epsilon^{\sigma}$, a procedure that I have used before as well. I am failing to see where I made a mistake, or to see how these descriptions are compatible. Conceptually I am quite confused about how vector/tensor fields transform under this translation now, and I do not know how to proceed. Any help would be much appreciated.

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As you noted $A'^{\mu}(x') = \frac{\partial x'^{\mu}}{\partial x^{\lambda}}A^{\lambda}(x)$ is how a four-vector transform under Lorentz transformations, but a translation $x^\mu+\epsilon^\mu$ is not a Lorentz transformation.

Some people call translations Galilean Transfromations. The group of Lorentz transformations + Translations is called Poincarè Group.

So, in conclusion the transformation you should use is $A'^{\mu} = A^{\mu}(x^{\sigma} + \epsilon^{\sigma}) = A^{\mu}(x^{\sigma}) + \frac{\partial A^{\mu}(x^{\sigma})}{\partial x^{\sigma}}\epsilon^{\sigma}$

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