I'm wondering on what principles Noether's theorem foots. More precisely:
The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful considerations, however, seem to untangle them as special field variations. So what's going on here truly?
Answer
In a comment you write
space time symmetries don't fit into the framework of the action since the action is a functional on the fields only not also on space time (space time here appears merely as a dummy variable
This isn't quite right. A given spacetime transformation often induces a transformation on fields themselves, and in this way, spacetime transformations fit into the framework of the action.
This is most easily and explicitly illustrated by way of a simple example.
Example. Consider a theory of a single real scalar field on R3,1 (Minkowski space). Let F denote the space of fields considered in the theory (which usually consists of e.g. smoothness assumptions and assumptions about the behavior of the fields at infinity). The action functional will be a function S:F→R.
Now, on the one hand, the Lorentz group SO(3,1) acts in a natural way on R3,1, namely through the group action ρ:SO(3,1)→Sym(R3,1) defined as follows: ρ(Λ)(x)=Λx,
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