In several different articles about the AdS/CFT correspondence, it is stated that one can show that the fifth coordinate $z$ on the AdS side, in coordinates such that the AdS metric becomes: $$ds^2 = \frac{L^2}{z^2}(dt^2-d(x^1)^2 - d(x^2)^2 - d(x^3)^2- dz^2)$$ can be interpreted as an inverse energy scale for the 4D theory on the boundary, unfortunately the root article they link to, i.e. this review, states vacuously that this is clear from looking at the conformal transformations of Poincaré coordinates. Can somebody clarify this a bit further? Unfortunately, I'm not extremely familiar with conformal field theory and I'm just trying to get a global heuristic understanding of the correspondance for the moment so that I can understand the ideas behind a phenomenological AdS/QCD model I will working with.
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2$\begingroup$ On a CFT, the energy is given by the dilatation operator that scales the coordinates - $x^\mu \to \lambda x^\mu$. The isometry of AdS that acts as a dilation on the boundary is one in which $z \to \lambda z$. This is the sense in which $z$ is the inverse energy scale. $\endgroup$– PraharCommented Sep 24, 2015 at 13:33
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$\begingroup$ @Prahar Is this statement coordinate independent, i.e., what the global coordinates? $\endgroup$– ungeradeCommented Dec 4, 2019 at 12:37
2 Answers
There are various ways of arguing that $z$ should be an inverse energy scale, none of them to my knowledge very precise. In fact, it's not clear to me that it is possible to make a precise relationship between $z$ and energy scale.
That said, perhaps the simplest way of arguing for the relationship is to note that for massive particles, AdS is like a well: the larger $z$, the deeper the particle is in the well and the less potential energy it has.
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$\begingroup$ A paper which has come out since this answer was posted might help: arxiv.org/abs/1610.08516 $\endgroup$ Commented Jul 13, 2022 at 20:12
Expanding on the comment of Prahar:
CFT is invariant under scaling $x^\mu \to \alpha x^\mu$. Length scale is inverse in energy thus energy scales as $E \to E/\alpha$.
Now let's see the metric in Poincare coordinate $$ dS^2=\frac{\ell^2}{z^2}(\eta_{\mu\nu} dx^\mu dx^\nu +dz^2 ).$$
Scale transformation in CFT side translates to diffeomorphism invariance in gravity side with $x^\mu \to \alpha x^\mu$ and $z \to \alpha z$. Thus $z \sim E^{-1}$. This has physical implication. UV divergences in CFT thus implies IR divergences in gravitational theories if integrated over the holographic $z$ direction.
