quantum mechanics - Noether's Theorem: Foundations

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 $\mathbb R^{3,1}$ (Minkowski space). Let $\mathcal F$ 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 $S:\mathcal F\to \mathbb R$.

Now, on the one hand, the Lorentz group $\mathrm{SO}(3,1)$ acts in a natural way on $\mathbb R^{3,1}$, namely through the group action $\rho:\mathrm{SO}(3,1)\to \mathrm{Sym}(\mathbb R^{3,1})$ defined as follows: $$\begin{align} \rho(\Lambda)(x) = \Lambda x, \end{align}$$ where $\mathrm{Sym}(S)$ denote the set of bijections on a set $S$. On the other hand, this group action induces an action $\rho_\mathcal F$ of $\mathrm{SO}(3,1)$ on $\mathcal F$, the space of field configurations, as follows: $$\begin{align} \rho_\mathcal F(\Lambda)(\phi)(x) = \phi(\Lambda^{-1} x), \end{align}$$ which is sometimes written as $\phi'(x) = \phi(\Lambda^{-1} x)$ for brevity. It is this action of $\mathrm{SO}(3,1)$ on the fields that one would use to fit spacetime symmetries into the action framework. In particular, in this case we could say for example that $S$ is Lorentz-invariant provided $$\begin{align} S[\rho_\mathcal F(\Lambda)(\phi)] = S[\phi] \end{align}$$ for all $\Lambda\in\mathrm{SO}(3,1)$ and for all $\phi\in\mathcal F$. All of this can also be easily extended to theories of fields of more complicated types, like vector and tensor fields. In such cases, the action $\rho_\mathcal F$ will in general be more complicated because it will contain a target space transformation.