# Conformally invariant theory. Relationship between conformal transformations and conformal rescaling (Weyl scaling)

So, I'm learning about Twistors, and in every book I've read they say the same:

"If a flat theory is Poincaré-invariant and it is invariant under conformal rescaling (Weyl scaling), it is then conformally invariant (in the sence of conformal transformations)"

First let me say that I can't find a proof about this, but for me this looks like another way of stating the Liouville Theorem (or maybe may be it's a consequence).

Second, Is this sentence the reason for study only conformal rescaling (in the flat space)?

• It should be Liouville theorem. What else is left to exhaust the comformal transformations otherwise? Commented Aug 3, 2016 at 8:16

The proof of this statement has only been established for 4-dimensions recently. See M. A. Luty, J. Polchinski, R. Rattazzi "The a-theorem and the Asymptotics of 4D Quantum Field Theory" JHEP01 (2013) 152

For two dimensions, there has been a robust proof for more than 26 years. See J. Polchinski, "Scale And Conformal Invariance in Quantum Field Theory," Nucl. Phys. B 303, (1988) 226.

• That can't be right. Penrose uses this in his book: "Spinor and twistor methods..." (1986). I mean, this guy don't use things that have not been proof.
– raul
Commented Mar 7, 2015 at 11:08
• There is a difference between something thought to be true because no body was able to come up with a counter example, and actually providing a robust mathematical proof. Commented Mar 7, 2015 at 11:10
• This is in the article of Luty, Polchinski: "Conformal transformations are the subgroup of Weyl X dieormorphisms that leave the at space metric invariant" How is this different of what I wrote? And they are not trying to proof this in the article you gave me, they are using it.
– raul
Commented Mar 9, 2015 at 2:02
• This is a direct consequence of the definition of what conformal transformations are... It has nothing to do with whether there can exist QFTs that are invariant under scaling transformation but not under conformal transformations. The latter is addressed in that paper, as mentioned in the abstract "We also give a non-perturbative argument that excludes theories with scale but not conformal invariance" Commented Mar 9, 2015 at 2:19
• I think there is a major confusion here. The paper talks about Scale invariance $\Rightarrow$ Conformal invariance, which is highly non trivial and not established in arbitrary dimension. On the other hand OP was asking about Weyl invariance $\Rightarrow$ Conformal invariance. Which is trivial because, as it was stated two comments above, Conformal $\subset$ Weyl. Commented Mar 1, 2019 at 23:48