EDIT : I think that the whole question can be summarized as « how do we know that every conformal transformation can be written under the form $e^{tX}$ for some operator $$X ? »
I'm reading the beginning of this which search for the conformal group. But my question is more general.
The "usual" way of presenting the matter about searching for the group of transformations having some properties is the following :
- We consider an infinitesimal transformation $x'_{\mu}=x_{\mu}+\epsilon_{\mu}$
- We express the constraint "neglecting" the quadratic terms in $\epsilon$
- We get an (differential) equation for $\epsilon$
- Solving provides the "infinitesimal" generators
- Exponentiating provides the group (finite transformations)
My question is : how to put the whole in a mathematically rigorous way ?
In the example of the conformal group of a riemannian manifold $M$, we are searching for the group of diffeomorphisms $\phi:M\to M$ that leaves unchanged the metric up to a scalar function.
More fundamental question : what is a "generator" in this context ?
I know two answers : 1. this is "close to the identity" 2. the tangent space to the group at the identity. Answer 1 is not mathematically rigorous (?). Answer 2 is only valid if we know that the searched group is a Lie group.
I guess that everybody make two implicit hypothesis :
- We are searching for a Lie group
- the transformation have to be analytical.
But with these two assumptions, at the end of the work, how are we sure to have found all the conformal transformations ?