I have been very confused by this after some recent reading. So as far as I know, a conformal transformation (according to the definition in di Francesco et. al.'s book on CFT) is an active coordinate change $x \to x'(x)$ whose effect is to change the metric by a Weyl rescaling \begin{equation} g_{\mu\nu}(x)\to g'_{\mu\nu}(x') = \Omega(x)g_{\mu\nu}(x)\,. \end{equation} So that would mean that a theory that is invariant under the above diffeomorphism is a conformal field theory. However, an alternate definition is invariance under Weyl rescalings which have nothing to do with coordinates but are just local rescalings of the metric \begin{equation} g_{\mu\nu}(x)\to \Omega(x)g_{\mu\nu}(x)\,. \end{equation} On the other hand, my professor also told me that the true symmetry is diffeomorphism invariance and the Weyl rescaling so that we get back the old metric. My question is: doesn't changing the metric change the theory? I am very confused about the proper definition of conformal symmetry/invariance, and it would be great if someone could clarify it for me. I have looked at the answer here but I'm unsure if I understand how one can change the metric by rescaling and still have the same theory.
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$\begingroup$ Related (disclaimer - these are my answers): physics.stackexchange.com/a/635336/8821 and physics.stackexchange.com/a/613017/8821 $\endgroup$– PraharCommented May 6 at 2:56
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/449838/2451 , physics.stackexchange.com/q/449882/2451 , physics.stackexchange.com/q/612945/2451 , and links therein. $\endgroup$– Qmechanic ♦Commented May 6 at 5:50
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$\begingroup$ @Prahar thanks but why does di Francesco only talk about the diffeomorphism and not the Weyl rescaling when he talks about conformal invariance? $\endgroup$– QFTheoristCommented May 6 at 7:27
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1$\begingroup$ I'm not sure. Francesco (and many other similar textbooks) are a bit misleading in this regard. $\endgroup$– PraharCommented May 6 at 11:38
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