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I am confused about the field transformation under conformal transformation. Consider the scale transformation of field $\phi$ (not necessarily scalar)

In CFT of Francesco et al, formula (2.121), the transformation is $$ \vec{x}\rightarrow \vec{x}'=\lambda x,\,\,\,\phi(\vec{x}) \rightarrow \phi'(\lambda \vec{x}) =\lambda^{-\Delta}\phi(\vec{x})$$

In the AdS/CFT review of AGMOO https://arxiv.org/abs/hep-th/9905111, page 33, the transformation is $$ x^{\mu}\rightarrow\lambda x^{\mu},\,\,\,\phi(x) \rightarrow \phi'(x) =\lambda^{\Delta}\phi(\lambda x)$$

Are these two kinds of transformation same and why?

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    $\begingroup$ Seems to me like a problem of active vs. passive symmetry transformations. Can you look these notions up? $\endgroup$ – DanielC Mar 18 '18 at 2:03

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