The scale transformation

Question

In this text, there are scale transformations which are given by Eq. (12.112)

$$x' = e^{-\rho}x, \quad \varphi{'}(x') = e^{d\rho}\varphi (x) \qquad (1)$$

Authors then calculate the dilatation current by using these transformations.

In this text, however, the transformations are different (Eq. (7.163)): up to different designations,

$$x' = e^{-\rho}x, \quad \varphi{'}(x') = e^{d\rho}\varphi (e^{-\rho}x) \qquad (2)$$

$(1)$ and $(2)$ have different variations $\delta \varphi$, and hence different expressions for dilatation currents. Which is correct?

Comments

the first, as the second effectively doesn't transform the argument of $\phi$. 

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@ArnoldNeumaier : could You please explain Your comment a little bit more?

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In simpler notation, (2) is asserting $\phi'(z)=e^{d\rho}\phi(z)$, which is not correct.

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To me it seems that in the second part of (1) only the field $\varphi$ undergoes a scaling transformation, whereas in (2) the field and spacetime get rescaled at the same time.

This would then mean that in some notation, the scaling current for (2) consists of a spacetime and an internal part, see eq (22) of this paper

$$\Theta^{\mu} = x_{\nu}\Theta^{\mu\nu} + \Sigma^{\mu}$$

whereas for (1) only the internal part $\Sigma^{\mu}$ is present. $\Theta^{\mu\nu}$ is the canonical energy-momentum tensor.

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''To me it seems that in the second part of (1) only the field φ undergoes a scaling transformation, whereas in (2) the field and spacetime get rescaled at the same time.''

yes, and this means that (2) is faulty, since if space-time is not transformed, the field shouldn't change either. It would make the scale transformation an internal symmetry, which is meaningless.