I have a question regarding Noether's theorem. In our introductory QFT class (which is based on the book by Michele Maggiore) we have derived the Noether currents in the same form as displayed in this post: Question about Noether theorem In this formula, there are contributions from two different kinds of transformations: a transformation of the field alone and a transformation of the coordinates.
My problem is: I don't understand the meaning of the transformation of coordinates. I have tried to understand the derivation from different QFT books (and I haven't found the same derivation twice, which doesn't make it easier) in the hope that I then would better understand the premises, but unfortunately I have not succeeded so far.
Also Peskin/Schröder for example only discuss transformations of fields and don't mention the transformation of coordinates at all. Poincare symmetry, which is in most books treated as a transformation of the coordinates, can be treated also as a transformation of fields, as shown in the answer to the following question for pure translations: Noether's Theorem: Foundations. Like the guy who asked that question, I think that the coordinates entering the action are only dummy variables. So what is then the meaning of the coordinate transformations in the prevalent formulation of Noether's theorem? Maybe someone can give a concrete example to illustrate the idea.