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.

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 $A'^{\mu} = A^{\mu}(x^{\sigma} + \epsilon^{\sigma}) = A^{\mu}(x^{\sigma}) + \frac{\partial A^{\mu}(x^{\sigma})}{\partial x^{\sigma}}\epsilon^{\sigma}$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.