# Space translation of coordinates, classical field theory

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.

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