What is the actual definition of conformal invariance?

Question

I've seen a large variety of slightly different definitions of conformal invariance. For simplicity I'll only consider scale invariance, which is already confusing enough. Some of the definitions are:

1. The action stays the same under a step of RG flow. (During this step, one has to perform a rescaling of the form $\phi'(x) = \Omega(x)^{-\gamma} \phi(x)$ where $\gamma$ is the engineering dimension corrected by the anomalous dimension.)
2. The partition function stays the same if the metric $g_{\mu\nu}$ is replaced with $\Omega(x)^2 g_{\mu\nu}$.
3. The action stays the same if the field $\phi(x)$ is replaced with $$\phi'(x') = \Omega(x)^{-\Delta} \phi(x).$$
4. The partition function stays the same if the field $\phi(x)$ is replaced with $$\phi'(x') = \Omega(x)^{-\Delta} \phi(x).$$

It is not obvious to me that all four of these definitions are equivalent, or even if they are at all. Furthermore I'm not sure if $\gamma$ is supposed to be $\Delta$, or whether $\Delta$ is simply the engineering dimension, or if it's something else entirely. However, all sources I've seen simply choose one of these four definitions to be the official one, then use the other three interchangeably.

What is the 'proper' definition of a CFT, and which of the other ones are equivalent and why?

Answer 1

The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a field) is transformed by the standard conformal factor.

All other statements you mention are proxies for this, equivalent under additional assumptions only. For example, since you asked about conformal invariance but stated things about scale invariance only, you always need to add the requirement of Poincare co/invariance of the respective fields. Apart from that, you need to assume a specific field contents, and for 3. also that the CFT is described by an action (many are not). 2. and 4. must be assumed together.

Condition 1. is a consequence of the fact that since CFTs are scale invariant, they are fixed points of the renormalization group flow. Conversely, fixed points are scale invariant, but it is an open question whether that implies conformal invariance. But scale invariant QFTs are typically conformal.

In any case, for scaling the physical (i.e., renormalized) fields, $\Delta$ must be the true dimension, including anomalous terms (if there are any).