Is conformal transformation a coordinate transformation or not? I have been looking through several questions and answers on conformal transformation in the stack exchange community. While they have helped me gain a better understanding of the basic issues, I am also somewhat confused. Here are couple of them:

*

*While reading the post, Conformal transformation/ Weyl scaling are they two different things? Confused!, I became convinced of the fact that Conformal transformation is an actual transformation of the co-ordinates, leaving the metric intact while Weyl transformation is an actual transformation of metric, leaving the co-ordinates intact. However in the answer, What is the significance of the conformal invariance of electrodynamics in a covariant formulation?, they convey a completely opposite message. Thus I end up being confused.


*I understand that, tracelessness of the stress-energy tensor is said to be implied by the conformal invariance, which is sustained if the theory is massless. However, when proving this statement, what is actually used, is the Weyl transformation. This makes me think that, whether the invariance under Weyl transformation automatically implies conformal invariance!
 A: Preliminaries

*

*Any symmetry transformation acts ONLY on the fields, never ever on the coordinates. Often, transformations of coordinates is used as a TOOL to describe how the fields themselves transform. For instance, under translations, we have
$$
U(a) \phi(x)U(a)^{-1} = \phi'(x).
$$
Notice that the transformation has not acted on the coordinates. However, the coordinate transformation is often invoked to describe $\phi'$ in terms of $\phi$ as
$$
\phi(x) = \phi'(x+a) .
$$
This does not mean that coordinates were acted upon by $U(a)$. The way we should truly interpret this formula is that
$$
\phi'(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!} a^{\mu_1} \cdots a^{\mu_k} \partial_{\mu_1} \cdots \partial_{\mu_n}  \phi(x) .
$$
so that everything on both sides is at the same point $x$.


*Every field theory has some dynamical fields $\phi(x)$ and some background fields ${\bar \phi}(x)$. Symmetry transformations act ONLY on dynamical fields. Background fields are to be treated as ${\mathbb C}$-numbers on the Hilbert space so for example
$$
U \phi(x) U^{-1}  = \phi'(x) , \qquad U {\bar \phi}(x) U^{-1}  = {\bar \phi}(x) .
$$
One of the many background fields are the metric $g_{\mu\nu}(x)$. In many cases (especially at a beginner level), the metric is the only background field. For simplicity, I will assume that this is the case in the rest of this post.
Diffeomorphisms
Under diffeomorphisms, the fields transform as
$$
\phi(x) \to \phi'(x) , \qquad g_{\mu\nu}(x) \to g'_{\mu\nu}(x)
$$
where
$$
\phi(x) = D \left( \frac{ \partial f^\mu(x) }{ \partial x^\nu} \right) \cdot \phi'(f(x)) , \qquad g_{\mu\nu}(x) = \frac{ \partial f^\alpha(x) }{ \partial x^\mu} \frac{ \partial f^\beta(x) }{ \partial x^\nu}  g'_{\alpha\beta}(f(x))
$$
Here, $D$ is the representation of $GL(d,{\mathbb R})$ under which the field $\phi$ transforms.
The action is invariant under such a transformation. However, this is NOT a symmetry of the theory since the background fields are also transforming. However, if we consider only a subset of diffeomorphisms $f^\mu(x)$ such that
$$
g'_{\mu\nu}(x) = g_{\mu\nu}(x)
$$
then $f^\mu(x)$ generate symmetries of the theory. These are called isometry transformations
Weyl Transformations
Diffeomorphisms preserve the action in ANY local field theory. Some quantum field theories also admit another transformation which preserves the action known as Weyl transformations under which
$$
\phi(x) \to \phi_\Omega(x) , \qquad g_{\mu\nu} \to \Omega(x)^2 g_{\mu\nu}(x) 
$$
It is often the case that $\phi_\Omega(x) = \Omega(x)^{-\Delta} \phi(x)$, but this may not always be true (e.g. stress tensor in two dimensions).
Weyl transformations are NEVER symmetries of the theory since the background field always transforms.
Conformal Transformations
While Weyl transformations are NOT symmetries of the theory, its existence as an invariant transformation of the action allows us to EXTEND the isometry symmetry we discussed earlier. We do this by considering the following sequence of transformations
$$
g_{\mu\nu}(x) \stackrel{\text{diff}}{\to} \Omega_f(x)^2 g_{\mu\nu}(x)  \stackrel{\text{Weyl}}{\to} g_{\mu\nu}(x)
$$
In this way, this composite transformation IS a symmetry of the theory since the background field are invariant. This composite transformation is known as "conformal transformation".
Summary
In a summary, a conformal transformation involves TWO transformations - a diffeomorphism which transforms the metric by a conformal factor and a Weyl transformation that removes that factor.
In other words, to answer OPs question - a conformal transformation consists of both a (special) diffeomorphism AND a Weyl transformation. However, in many texts, the diffeomorphism part of a conformal transformation is -- unforunately and mistakenly -- also referred to as a conformal transformation.
A: A conformal transformation is a change of coordinate $x\to x'$ that changes the metric in a very particular way. It is invariant up to a phase:
\begin{equation}
g_{\mu \nu}(x)\to g'_{\mu\nu}(x')=\Lambda(x)g_{\mu \nu}(x)
\end{equation}
Hence, after a conformal transformation you have new coordinates and a new metric, with that particular form.
A conformal transformation is different from a generic change of coordinates, because a generic change of coordinates will do change the metric, but not (necessarily) in that particular way.
A conformal transformation is also different from a Weyl transformation, since the latter is a change in the metric itself, but the coordinates are untouched.
However, if we have a theory which is invariant under both coordinates transformation and Weyl rescaling, then this theory is conformal invariant (an example is string theory). In fact, first I could change the coordinates. Then I could perform a Weyl transformation and change the metric as I like. The overall final result would be the one that I would have been obtained if I had done a conformal transformation right from the start.
A: Conformal transformation/mapping is a term from the complex analysis. In its original sense it is indeed a coordinate transformation:
The conformal property may be described in terms of the Jacobian derivative matrix of 
a coordinate transformation. The transformation is conformal whenever the Jacobian at 
each point is a positive scalar times a rotation matrix (orthogonal with determinant 
one). Some authors define conformality to include orientation-reversing mappings whose 
Jacobians can be written as any scalar times any orthogonal matrix.[1]

These transformations are widely used in physics, e.g., for solving electrostatic problems, elastic problems, in field theory, etc.
