The contest is a flat $4d$ Minkowsky space. A conformal transformation is a diffeomorphism $\tilde x(x)$ such that the metric transforms as \begin{equation*} \tilde g_{\tilde \mu \tilde \nu} = w^2(x) g_{\mu \nu} \end{equation*} This implies that $ \left| det \left( \frac{\partial \tilde x}{\partial x} \right) \right| = |w| $.
Tensors transform as required by the change of coordinate.
The lagrangian is the usual $\mathscr{L}= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A^{\mu}j_{\mu}$.

Now, IF the tranformed action is $ \tilde{S}= \int_{\tilde M} \sqrt{|\tilde g| }\tilde{ \mathscr{L}}\, d\tilde x^0 \cdots d\tilde x^4 \, $ $\,$ , I can show, by hand or by principle, that $\tilde S=S$. For example because $\mathscr{L}=\tilde{\mathscr{L}}$, and $\sqrt{|\tilde g| }$ compensates the change of variables

Is it correct?

If that is the case, by doing this I noticed that I never used the fact that the transformation is conformal, and in fact, since the action is an integral of a scalar function, it should be invariant under every general change of coordinates.
So, for second part of the question, considering conformal transformations as opposed to generic diffeomorphisms for this theory,

is the significance of conformal transformations linked to the fact that they are a group with a finite number of generators, so that we can obtain a conserved current via Noether's first theorem?

sorry if this sounds trivial, but at least for the first part I've seen two methods of proving it and neither of those was convincing, so I feared that I might be missing something.


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