# What is the actual definition of conformal invariance?

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?

• Here's a thought about $3$ and $4$: In pure Yang-Mills, the action is scale-invariant but the partition function is not, not even if the $\Delta$s are allowed to be different, so $3$ doesn't imply $4$. In other words, we can't define a CFT using $3$, unless we're talking about a classical CFT. Is this thought relevant to your question? (And I'm assuming that $\Omega$ is the scale factor that relates $x'$ to $x$.) – Chiral Anomaly Dec 22 '18 at 16:37
• @DanYand Indeed, I'm sure (3) is the odd one out. (I included it because some particularly sloppy texts don't even distinguish between the classical and quantum cases.) But I'm still not sure about how to relate (1), (2), and (4). – knzhou Dec 22 '18 at 16:56
• @DanYand In fact, I am not sure if people on this site even generally agree on the definition of a conformal transformation. In a frustrating hour I've found "active diffeomorphism", "passive diffeomorphism", "diffeomorphism except you pushforward everything except the metric", "diffeomorphism except you pushforward only the metric", and "diffeomorphism except composed with a Weyl transformation". These can't possibly all be equivalent, but I can't tell which is meant because every source has vague notation that allows all of them. – knzhou Dec 22 '18 at 17:01

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).

• Could you give some description of how the others are equivalent, and which assumptions are necessary? – knzhou Dec 27 '18 at 0:12
• @knzhou: 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, and are then probably equivalent for a field theory with field $\phi$ (or fields $\phi$ and $g$) to be covariant, depending on how general objects you call partition functions. I don't know of any precise conditions. – Arnold Neumaier Dec 27 '18 at 9:31
• What is most mysterious for me by far is condition 1, do you know how that could be linked to any of the others? Thanks for the help so far! – knzhou Dec 27 '18 at 12:16
• This definition is a good one, but it leaves out the case of a theory where conformal invariance is broken spontaneously, while the other consequences (such as the existence of various conserved currents, the ope, the soft theorems for the dilaton etc) are still there – TwoBs Dec 27 '18 at 13:56
• "Condition 1. is a consequence of the fact that CFTs are fixed points of the renormalization group flow. " But that's exactly what my issue is -- how do we know that? I have never seen any source ever justify this in any way. – knzhou Dec 27 '18 at 16:22